Lie algebra extension#By semidirect sum

Due to the Lie correspondence, the theory, and consequently the history of Lie algebra extensions, is tightly linked to the theory and history of group extensions. A systematic study of group extensions was performed by the Austrian mathematician Otto Schreier in 1923 in his PhD thesis and later published.

Otto Schreier (1901 - 1929) was a pioneer in the theory of extension of groups. Along with his rich research papers, his lecture notes were posthumously published (edited by Emanuel Sperner) under the name Einführung in die analytische Geometrie und Algebra (Vol I 1931, Vol II 1935), later in 1951 translated to English in [https://www.amazon.com/Introduction-Modern-Algebra-Matrix-Theory/dp/0486482200 Introduction to Modern Algebra and Matrix Theory]. See {{harvnb|MacTutor|2015}} for further reference.{{harvnb|Schreier|1926}}{{harvnb|Schreier|1925}} The problem posed for his thesis by Otto Hölder was "given two groups {{mvar|G}} and {{mvar|H}}, find all groups {{mvar|E}} having a normal subgroup {{mvar|N}} isomorphic to {{mvar|G}} such that the factor group {{math|E/N}} is isomorphic to {{mvar|H}}".

Lie algebra extensions are most interesting and useful for infinite-dimensional Lie algebras. In 1967, Victor Kac and Robert Moody independently generalized the notion of classical Lie algebras, resulting in a new theory of infinite-dimensional Lie algebras, now called Kac–Moody algebras.{{harvnb|Kac|1967e}}{{harvnb|Moody|1967}} They generalize the finite-dimensional simple Lie algebras and can often concretely be constructed as extensions.{{harvnb|Bäuerle|de Kerf|ten Kroode|1997|loc=Chapter 19}}

Notational abuse to be found below includes {{math|e^X}} for the exponential map {{math|exp}} given an argument, writing {{mvar|g}} for the element {{math|(g, e_H)}} in a direct product {{math|G × H}} ({{math|e_H}} is the identity in {{mvar|H}}), and analogously for Lie algebra direct sums (where also {{math|g + h}} and {{math|(g, h)}} are used interchangeably). Likewise for semidirect products and semidirect sums. Canonical injections (both for groups and Lie algebras) are used for implicit identifications. Furthermore, if {{math|G}}, {{math|H}}, ..., are groups, then the default names for elements of {{math|G}}, {{math|H}}, ..., are {{math|g}}, {{math|h}}, ..., and their Lie algebras are {{math|g}}, {{math|h}}, ... . The default names for elements of {{math|g}}, {{math|h}}, ..., are {{math|G}}, {{math|H}}, ... (just like for the groups!), partly to save scarce alphabetical resources but mostly to have a uniform notation.

Lie algebras that are ingredients in an extension will, without comment, be taken to be over the same field.

The summation convention applies, including sometimes when the indices involved are both upstairs or both downstairs.

Caveat: Not all proofs and proof outlines below have universal validity. The main reason is that the Lie algebras are often infinite-dimensional, and then there may or may not be a Lie group corresponding to the Lie algebra. Moreover, even if such a group exists, it may not have the "usual" properties, e.g. the exponential map might not exist, and if it does, it might not have all the "usual" properties. In such cases, it is questionable whether the group should be endowed with the "Lie" qualifier. The literature is not uniform. For the explicit examples, the relevant structures are supposedly in place.

Lie algebra extensions are formalized in terms of short exact sequences.{{harvnb|Bäuerle|de Kerf|ten Kroode|1997}} A short exact sequence is an exact sequence of length three,

{{NumBlk2|

:|$$

\mathfrak h \; \overset i \hookrightarrow \; \mathfrak e \; \overset s \twoheadrightarrow \; \mathfrak g,

|1}}

such that {{mvar|i}} is a monomorphism, {{mvar|s}} is an epimorphism, and {{math|ker s {{=}} im i}}. From these properties of exact sequences, it follows that (the image of) $\backslash mathfrak\; h$ is an ideal in $\backslash mathfrak\; e$. Moreover,

:$$

\mathfrak g \cong \mathfrak e/\operatorname{Im} i = \mathfrak e/ \operatorname{Ker} s,

but it is not necessarily the case that $\backslash mathfrak\; g$ is isomorphic to a subalgebra of $\backslash mathfrak\; e$. This construction mirrors the analogous constructions in the closely related concept of group extensions.

If the situation in ({{EquationNote|1}}) prevails, non-trivially and for Lie algebras over the same field, then one says that $\backslash mathfrak\; e$ is an extension of $\backslash mathfrak\; g$ by $\backslash mathfrak\; h$.

The defining property may be reformulated. The Lie algebra $\backslash mathfrak\; e$ is an extension of $\backslash mathfrak\; g$ by $\backslash mathfrak\; h$ if

{{NumBlk2|

:|$$

0 \; \overset \iota \hookrightarrow \mathfrak h \; \overset i \hookrightarrow \; \mathfrak e \; \overset s \twoheadrightarrow \; \mathfrak g \; \overset \sigma \twoheadrightarrow \; 0

|2}}

is exact. Here the zeros on the ends represent the zero Lie algebra (containing only the zero vector {{math|0}}) and the maps are the obvious ones; $\backslash iota$ maps {{math|0}} to {{math|0}} and $\backslash sigma$ maps all elements of $\backslash mathfrak\; g$ to {{math|0}}. With this definition, it follows automatically that {{math|i}} is a monomorphism and {{math|s}} is an epimorphism.

An extension of $\backslash mathfrak\; g$ by $\backslash mathfrak\; h$ is not necessarily unique. Let $\backslash mathfrak\; e,\; \backslash mathfrak\; e\text{'}$ denote two extensions and let the primes below have the obvious interpretation. Then, if there exists a Lie algebra isomorphism $f\; \backslash colon\; \backslash mathfrak\; e\; \backslash rightarrow\; \backslash mathfrak\; e\text{'}$ such that

:$f\; \backslash circ\; i\; =\; i\text{'},\; \backslash quad\; s\text{'}\; \backslash circ\; f\; =\; s,$

File:Lie algebra extension figure 1.svg

then the extensions $\backslash mathfrak\; e$ and $\backslash mathfrak\; e\text{'}$ are said to be equivalent extensions. Equivalence of extensions is an equivalence relation.

A Lie algebra extension

:$$

\mathfrak h \; \overset i \hookrightarrow \; \mathfrak t \; \overset s \twoheadrightarrow \; \mathfrak g,

is trivial if there is a subspace {{math|i}} such that {{math|t {{=}} i ⊕ ker s}} and {{math|i}} is an ideal in {{math|t}}.

A Lie algebra extension

:$$

\mathfrak h \; \overset i \hookrightarrow \; \mathfrak s \; \overset s \twoheadrightarrow \; \mathfrak g,

is split if there is a subspace {{math|u}} such that {{math|s {{=}} u ⊕ ker s}} as a vector space and {{math|u}} is a subalgebra in {{math|s}}.

An ideal is a subalgebra, but a subalgebra is not necessarily an ideal. A trivial extension is thus a split extension.

Central extensions of a Lie algebra {{math|g}} by an abelian Lie algebra {{math|h}} can be obtained with the help of a so-called (nontrivial) 2-cocycle (background) on {{math|g}}. Non-trivial 2-cocycles occur in the context of projective representations (background) of Lie groups. This is alluded to further down.

A Lie algebra extension

:$$

\mathfrak h \; \overset i \hookrightarrow \; \mathfrak e \; \overset s \twoheadrightarrow \; \mathfrak g,

is a central extension if {{math|ker s}} is contained in the center {{math|Z(e)}} of {{math|e}}.

Properties

• Since the center commutes with everything, {{math|h ≅ im i {{=}} ker s}} in this case is abelian.
• Given a central extension {{math|e}} of {{math|g}}, one may construct a 2-cocycle on {{math|g}}. Suppose {{math|e}} is a central extension of {{math|g}} by {{math|h}}. Let {{math|l}} be a linear map from {{math|g}} to {{math|e}} with the property that {{math|s ∘ l {{=}} Id_g}}, i.e. {{math|l}} is a section of {{math|s}}. Use this section to define {{math|ε: g × g → e}} by

:$$

\epsilon(G_1, G_2) = l([G_1, G_2]) - [l(G_1), l(G_2)], \quad G_1, G_2 \in \mathfrak g.

File:Lie algebra extension figure 2.svg

The map {{math|ε}} satisfies

:$$

\epsilon(G_1, [G_2, G_3]) + \epsilon(G_2, [G_3, G_1]) + \epsilon(G_3, [G_1, G_2]) = 0 \in \mathfrak e.

To see this, use the definition of {{math|ε}} on the left hand side, then use the linearity of {{math|l}}. Use Jacobi identity on {{math|g}} to get rid of half of the six terms. Use the definition of {{math|ε}} again on terms {{math|l([G_i,G_j])}} sitting inside three Lie brackets, bilinearity of Lie brackets, and the Jacobi identity on {{math|e}}, and then finally use on the three remaining terms that {{math|Im ε ⊂ ker s}} and that {{math|ker s ⊂ Z(e)}} so that {{math|ε(G_i, G_j)}} brackets to zero with everything.

It then follows that {{math|φ {{=}} i⁻¹ ∘ ε}} satisfies the corresponding relation, and if {{math|h}} in addition is one-dimensional, then {{math|φ}} is a 2-cocycle on {{math|g}} (via a trivial correspondence of {{math|h}} with the underlying field).

A central extension

:$$

0 \; \overset \iota \hookrightarrow \mathfrak h \; \overset i \hookrightarrow \; \mathfrak e \; \overset s \twoheadrightarrow \; \mathfrak g \; \overset \sigma \twoheadrightarrow \; 0

is universal if for every other central extension

:$$

0 \; \overset \iota \hookrightarrow \mathfrak h' \; \overset {i'} \hookrightarrow \; \mathfrak e' \; \overset {s'} \twoheadrightarrow \; \mathfrak g \; \overset \sigma \twoheadrightarrow \; 0

there exist unique homomorphisms $\backslash Phi\; :\; \backslash mathfrak\; e\; \backslash to\; \backslash mathfrak\; e\text{'}$ and $\backslash Psi\; :\; \backslash mathfrak\; h\; \backslash to\; \backslash mathfrak\; h\text{'}$ such that the diagram

File:Lie algebra extension figure 3.svg

commutes, i.e. {{math|i' ∘ Ψ {{=}} Φ ∘ i}} and {{math|s' ∘ Φ {{=}} s}}. By universality, it is easy to conclude that such universal central extensions are unique up to isomorphism.

Let $\backslash mathfrak\; g$, $\backslash mathfrak\; h$ be Lie algebras over the same field $F$. Define

:$\backslash mathfrak\; e\; =\; \backslash mathfrak\; h\; \backslash times\; \backslash mathfrak\; g,$

and define addition pointwise on $\backslash mathfrak\; e$. Scalar multiplication is defined by

:$\backslash alpha(H,\; G)\; =\; (\backslash alpha\; H,\; \backslash alpha\; G),\; \backslash alpha\; \backslash in\; F,\; H\; \backslash in\; \backslash mathfrak\; h,\; G\; \backslash in\; \backslash mathfrak\; g.$

With these definitions, $\backslash mathfrak\; h\; \backslash times\; \backslash mathfrak\; g\; \backslash equiv\; \backslash mathfrak\; h\; \backslash oplus\; \backslash mathfrak\; g$ is a vector space over $F$. With the Lie bracket:

{{NumBlk2|:|$[(H\_1,\; G\_1),(H\_2,\; G\_2)]\; =\; ([H\_1,\; H\_2],[G\_1,\; G\_2]),$|3}}

$\backslash mathfrak\; e$ is a Lie algebra. Define further

:$i:\backslash mathfrak\; h\; \backslash hookrightarrow\; \backslash mathfrak\; e;\; H\; \backslash mapsto\; (H,\; 0),\; \backslash quad\; s:\backslash mathfrak\; e\; \backslash twoheadrightarrow\; \backslash mathfrak\; g;\; (H,\; G)\; \backslash mapsto\; G.$

It is clear that ({{EquationNote|1}}) holds as an exact sequence. This extension of $\backslash mathfrak\; g$ by $\backslash mathfrak\; h$ is called a trivial extension. It is, of course, nothing else than the Lie algebra direct sum. By symmetry of definitions, $\backslash mathfrak\; e$ is an extension of $\backslash mathfrak\; h$ by $\backslash mathfrak\; g$ as well, but $\backslash mathfrak\; h\; \backslash oplus\; \backslash mathfrak\; g\; \backslash neq\; \backslash mathfrak\; g\; \backslash oplus\; \backslash mathfrak\; h$. It is clear from ({{EquationNote|3}}) that the subalgebra $0\; \backslash oplus\; \backslash mathfrak\; g$ is an ideal (Lie algebra). This property of the direct sum of Lie algebras is promoted to the definition of a trivial extension.

Inspired by the construction of a semidirect product (background) of groups using a homomorphism {{math|G → Aut(H)}}, one can make the corresponding construct for Lie algebras.

If {{math|ψ:g → Der h}} is a Lie algebra homomorphism, then define a Lie bracket on $\backslash mathfrak\; e\; =\; \backslash mathfrak\; h\; \backslash oplus\; \backslash mathfrak\; g$ by

{{NumBlk2|:|$[(H\; ,\; G),\; (H\text{'}\; ,\; G\text{'})]\; =\; ([H\; ,\; H\text{'}]\; +\; \backslash psi\_G(H\text{'})\; -\; \backslash psi\_\{G\text{'}\}(H),\; [G\; ,\; G\text{'}]),\backslash quad\; H,H\text{'}\; \backslash in\; \backslash mathfrak\; h,\; G,\; G\text{'}\; \backslash in\; \backslash mathfrak\; g.$|7}}

With this Lie bracket, the Lie algebra so obtained is denoted {{math|e{{=}} h ⊕_S g}} and is called the semidirect sum of {{math|h}} and {{math|g}}.

By inspection of ({{EquationNote|7}}) one sees that {{math|0 ⊕ g}} is a subalgebra of {{math|e}} and {{math|h ⊕ 0}} is an ideal in {{math|e}}. Define {{math|i:h → e}} by {{math|H ↦ H ⊕ 0}} and {{math|s:e → g}} by {{math|H ⊕ G ↦ G, H ∈ h, G ∈ g}}. It is clear that {{math|ker s {{=}} im i}}. Thus {{math|e}} is a Lie algebra extension of {{math|g}} by {{math|h}}.

As with the trivial extension, this property generalizes to the definition of a split extension.

Example
Let {{mvar|G}} be the Lorentz group {{math|O(3, 1)}} and let {{math|T}} denote the translation group in 4 dimensions, isomorphic to {{math|($\backslash mathbb\{R\}^4$, +)}}, and consider the multiplication rule of the Poincaré group {{math|P}}

:$(a\_2,\; \backslash Lambda\_2)(a\_1,\; \backslash Lambda\_1)\; =\; (a\_2\; +\; \backslash Lambda\_2a\_1,\; \backslash Lambda\_2\backslash Lambda\_1),\; \backslash quad\; a\_1,\; a\_2\; \backslash in\; \backslash mathrm\; T\; \backslash subset\; \backslash mathrm\; P,\; \backslash Lambda\_1,\; \backslash Lambda\_2\; \backslash in\; \backslash mathrm\; O(3,1)\; \backslash subset\; \backslash mathrm\; P,$

(where {{math|T}} and {{math|O(3, 1)}} are identified with their images in {{math|P}}). From it follows immediately that, in the Poincaré group, {{math|(0, Λ)(a, I)(0, Λ⁻¹) {{=}} (Λ a, I) ∈ T ⊂ P}}. Thus every Lorentz transformation {{math|Λ}} corresponds to an automorphism {{math|Φ_Λ}} of {{math|T}} with inverse {{math|Φ_Λ⁻¹}} and {{math|Φ}} is clearly a homomorphism. Now define

:$\backslash overline\; \backslash mathrm\; P\; =\; \backslash mathrm\; T\; \backslash otimes\_S\; \backslash mathrm\; O(3,\; 1),$

endowed with multiplication given by ({{EquationNote|4}}). Unwinding the definitions one finds that the multiplication is the same as the multiplication one started with and it follows that {{math|{{overline|P}} {{=}} P}}. From ({{EquationNote|5'}}) follows that {{math|Ψ_Λ {{=}} Ad_Λ}} and then from ({{EquationNote|6'}}) it follows that {{math|ψ_λ {{=}} ad_λ. λ ∈ o(3, 1)}}.

Let {{math|δ}} be a derivation (background) of {{math|h}} and denote by {{math|g}} the one-dimensional Lie algebra spanned by {{math|δ}}. Define the Lie bracket on {{math|e {{=}} g ⊕ h}} byTo show that the Jacobi identity holds, one writes everything out, uses the fact that the underlying Lie algebras have a Lie product satisfying the Jacobi identity, and that {{math|δ[X, Y] {{=}} [δ(X), Y] + [X, δ(Y)]}}.{{harvnb|Bäuerle|de Kerf|ten Kroode|1997|loc=Example 18.1.9}}

:$[G\_1\; +\; H\_1,\; G\_2\; +\; H\_2]\; =\; [\backslash lambda\backslash delta\; +\; H\_1,\; \backslash mu\backslash delta\; +\; H\_2]\; =\; [H\_1,\; H\_2]\; +\; \backslash lambda\; \backslash delta(H\_2)\; -\; \backslash mu\; \backslash delta(H\_1).$

It is obvious from the definition of the bracket that {{math|h}} is and ideal in {{math|e}} in and that {{math|g}} is a subalgebra of {{math|e}}. Furthermore, {{math|g}} is complementary to {{math|h}} in {{math|e}}. Let {{math|i:h → e}} be given by {{math|H ↦ (0, H)}} and {{math|s:e → g}} by {{math|(G, H) ↦ G}}. It is clear that {{math|im i {{=}} ker s}}. Thus {{math|e}} is a split extension of {{math|g}} by {{math|h}}. Such an extension is called extension by a derivation.

If {{math|ψ: g → der h}} is defined by {{math|ψ(μδ)(H) {{=}} μδ(H)}}, then {{mvar|ψ}} is a Lie algebra homomorphism into {{math|der h}}. Hence this construction is a special case of a semidirect sum, for when starting from {{mvar|ψ}} and using the construction in the preceding section, the same Lie brackets result.

If {{math|ε}} is a 2-cocycle (background) on a Lie algebra {{math|g}} and {{math|h}} is any one-dimensional vector space, let {{math|e {{=}} h ⊕ g}} (vector space direct sum) and define a Lie bracket on {{math|e}} by

:$$

[\mu H + G_1, \nu H + G_2] = [G_1, G_2] + \varepsilon(G_1, G_2)H, \quad \mu, \nu \in F.

Here {{mvar|H}} is an arbitrary but fixed element of {{math|h}}. Antisymmetry follows from antisymmetry of the Lie bracket on {{math|g}} and antisymmetry of the 2-cocycle. The Jacobi identity follows from the corresponding properties of {{math|g}} and of {{math|ε}}. Thus {{math|e}} is a Lie algebra. Put {{math|G₁ {{=}} 0}} and it follows that {{math|μH ∈ Z(e)}}. Also, it follows with {{math|i: μH ↦ (μH, 0)}} and {{math|s: (μH, G) ↦ G}} that {{math|Im i {{=}} ker s {{=}} {(μH, 0):μ ∈ F} ⊂ Z(e)}}. Hence {{math|e}} is a central extension of {{math|g}} by {{math|h}}. It is called extension by a 2-cocycle.

Below follows some results regarding central extensions and 2-cocycles.{{harvnb|Bäuerle|de Kerf|ten Kroode|1997|loc=Chapter 18}}

Theorem

Let {{math|φ₁}} and {{math|φ₂}} be cohomologous 2-cocycles on a Lie algebra {{math|g}} and let {{math|e₁}} and {{math|e₂}} be respectively the central extensions constructed with these 2-cocycles. Then the central extensions {{math|e₁}} and {{math|e₂}} are equivalent extensions.

Proof

By definition, {{math|φ₂ {{=}} φ₁ + δf}}. Define

:$$

\psi: G + \mu c \in \mathfrak{e}_1 \mapsto G + \mu c + f(G)c \in \mathfrak{e}_2.

It follows from the definitions that {{mvar|ψ}} is a Lie algebra isomorphism and ({{EquationNote|2}}) holds.

Corollary

A cohomology class {{math|[Φ] ∈ H²(g, F)}} defines a central extension of {{math|g}} which is unique up to isomorphism.

The trivial 2-cocycle gives the trivial extension, and since a 2-coboundary is cohomologous with the trivial 2-cocycle, one has

Corollary

A central extension defined by a coboundary is equivalent with a trivial central extension.

Theorem

A finite-dimensional simple Lie algebra has only trivial central extensions.

Proof

Since every central extension comes from a 2-cocycle {{math|φ}}, it suffices to show that every 2-cocycle is a coboundary. Suppose {{math|φ}} is a 2-cocycle on {{math|g}}. The task is to use this 2-cocycle to manufacture a 1-cochain {{mvar|f}} such that {{math|φ {{=}} δf}}.

The first step is to, for each {{math|G₁ ∈ g}}, use {{mvar|φ}} to define a linear map {{math|ρ_G1:g → F}} by $\backslash rho\_\{G\_1\}(G\_2)\; \backslash equiv\; \backslash varphi(G\_1,\; G\_2)$. These linear maps are elements of {{math|g^∗}}. Let {{math| ν:g^∗ →g}} be the vector space isomorphism associated to the nondegenerate Killing form {{math|K}}, and define a linear map {{math|d:g → g}} by $d(G\_1)\; \backslash equiv\; \backslash nu(\backslash rho\_\{G\_1\})$. This turns out to be a derivation (for a proof, see below). Since, for semisimple Lie algebras, all derivations are inner, one has {{math|d {{=}} ad_Gd}} for some {{math|G_d ∈ g}}. Then

:$\backslash varphi(G\_1,\; G\_2)\; \backslash equiv\; \backslash rho\_\{G\_1\}(G\_2)\; =\; K(\backslash nu(\backslash rho\_\{G\_1\}),\; G\_2)\; \backslash equiv\; K(d(G\_1),\; G\_2)\; =\; K(\backslash mathrm\{ad\}\_\{G\_d\}(G\_1),\; G\_2)\; =\; K([G\_d,\; G\_1],\; G\_2)\; =\; K(G\_d,\; [G\_1,\; G\_2]).$

Let {{mvar|f}} be the 1-cochain defined by

:$f(G)\; =\; K(G\_d,G).$

Then

:$\backslash delta\; f(G\_1,\; G\_2)\; =\; f([G\_1,\; G\_2])\; =\; K(G\_d,[G\_1,\; G\_2])\; =\; \backslash varphi(G\_1,\; G\_2),$

showing that {{math|φ}} is a coboundary.

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Proof of {{mvar|d}} being a derivation}}

To verify that {{mvar|d}} actually is a derivation, first note that it is linear since {{mvar|ν}} is, then compute

:

&= K(d(G_1),[G_2, G_3]) + K(d(G_1), (G_3, G_1)) = K([d(G_1),G_2], G_3) + K([G_1, d(G_2)], G_3))\\

&= K([d(G_1), G_2] + [G_1, d(G_2)], G_3).\end{align}

By appeal to the non-degeneracy of {{mvar|K}}, the left arguments of {{mvar|K}} are equal on the far left and far right.

{{Hidden end}}

The observation that one can define a derivation {{math|d}}, given a symmetric non-degenerate associative form {{mvar|K}} and a 2-cocycle {{mvar|φ}}, by

:$K(\backslash nu(\backslash rho\_\{G\_1\}),\; G\_2)\; \backslash equiv\; K(d(G\_1),\; G\_2),$

or using the symmetry of {{math|K}} and the antisymmetry of {{mvar|φ}},

:$K(d(G\_1),\; G\_2)\; =\; -K(G\_1,\; d(G\_2)),$

leads to a corollary.

Corollary

Let {{mvar|L:g × 'g: → F}} be a non-degenerate symmetric associative bilinear form and let {{mvar|d}} be a derivation satisfying

:$L(d(G\_1),\; G\_2)\; =\; -L(G\_1,\; d(G\_2)),$

then {{mvar|φ}} defined by

:$\backslash varphi(G\_1,\; G\_2)\; =\; L(d(G\_1),\; G\_2)$

is a 2-cocycle.

Proof

The condition on {{mvar|d}} ensures the antisymmetry of {{mvar|φ}}. The Jacobi identity for 2-cocycles follows starting with

:$\backslash varphi([G\_1,\; G\_2],\; G\_3)\; =\; L(d[G\_1,\; G\_2],\; G\_3)\; =\; L([d(G\_1),\; G\_2],\; G\_3)\; +\; L([G\_1,\; d(G\_2)],\; G\_3),$

using symmetry of the form, the antisymmetry of the bracket, and once again the definition of {{mvar|φ}} in terms of {{mvar|L}}.

If {{math|g}} is the Lie algebra of a Lie group {{math|G}} and {{math|e}} is a central extension of {{math|g}}, one may ask whether there is a Lie group {{math|E}} with Lie algebra {{math|e}}. The answer is, by Lie's third theorem affirmative. But is there a central extension {{math|E}} of {{math|G}} with Lie algebra {{math|e}}? The answer to this question requires some machinery, and can be found in {{harvtxt|Tuynman|Wiegerinck|1987|loc=Theorem 5.4}}.

The "negative" result of the preceding theorem indicates that one must, at least for semisimple Lie algebras, go to infinite-dimensional Lie algebras to find useful applications of central extensions. There are indeed such. Here will be presented affine Kac–Moody algebras and Virasoro algebras. These are extensions of polynomial loop-algebras and the Witt algebra respectively.

Let {{math|g}} be a polynomial loop algebra (background),

:$\backslash mathfrak\; g\; =\; \backslash mathbb\{C\}[\backslash lambda,\; \backslash lambda^\{-1\}]\; \backslash otimes\; \backslash mathfrak\; g\_0,$

where {{math|g₀}} is a complex finite-dimensional simple Lie algebra. The goal is to find a central extension of this algebra. Two of the theorems apply. On the one hand, if there is a 2-cocycle on {{math|g}}, then a central extension may be defined. On the other hand, if this 2-cocycle is acting on the {{math|g₀}} part (only), then the resulting extension is trivial. Moreover, derivations acting on {{math|g₀}} (only) cannot be used for definition of a 2-cocycle either because these derivations are all inner and the same problem results. One therefore looks for derivations on {{math|C[λ, λ⁻¹]}}. One such set of derivations is

:$d\_k\; \backslash equiv\; \backslash lambda^\{k+1\}\backslash frac\{d\}\{d\backslash lambda\},\; \backslash quad\; k\; \backslash in\; \backslash mathbb\; Z.$

In order to manufacture a non-degenerate bilinear associative antisymmetric form {{math|L}} on {{math|g}}, attention is focused first on restrictions on the arguments, with {{math|m, n}} fixed. It is a theorem that every form satisfying the requirements is a multiple of the Killing form {{mvar|K}} on {{math|g₀}}.{{harvnb|Bäuerle|de Kerf|ten Kroode|1997}} Corollary 22.2.9. This requires

:$L(\backslash lambda^m\; \backslash otimes\; G\_1,\; \backslash lambda^n\; \backslash otimes\; G\_2)\; =\; \backslash gamma\_\{lm\}K(G\_1,\; G\_2).$

Symmetry of {{mvar|K}} implies

:$\backslash gamma\_\{mn\}=\backslash gamma\_\{nm\},$

and associativity yields

:$\backslash gamma\_\{m+k,n\}=\backslash gamma\_\{m,k+n\}.$

With {{math|m {{=}} 0}} one sees that {{math|γ_k,n {{=}} γ_0,k+n}}. This last condition implies the former. Using this fact, define {{math|f(n) {{=}} γ_0,n}}. The defining equation then becomes

:$L(\backslash lambda^m\; \backslash otimes\; G\_1,\; \backslash lambda^n\; \backslash otimes\; G\_2)\; =\; f(m+n)K(G\_1,\; G\_2).$

For every {{math|i ∈ $\backslash mathbb\{Z\}$}} the definition

:$f(n)\; =\; \backslash delta\_\{ni\}\; \backslash Leftrightarrow\; \backslash gamma\_\{mn\}=\backslash delta\_\{m+n,\; i\}$

does define a symmetric associative bilinear form

:$L\_i(\backslash lambda^m\; \backslash otimes\; G\_1,\; \backslash lambda^n\; \backslash otimes\; G\_2)\; =\; \backslash delta\_\{m+n,i\}K(G\_1,\; G\_2).$

These span a vector space of forms which have the right properties.

Returning to the derivations at hand and the condition

:$L\_i(d\_k(\backslash lambda^l\; \backslash otimes\; G\_1),\; \backslash lambda^m\; \backslash otimes\; G\_2)\; =\; -L\_i(\backslash lambda^l\; \backslash otimes\; G\_1,\; d\_k(\backslash lambda^m\; \backslash otimes\; G\_2)),$

one sees, using the definitions, that

:$l\backslash delta\_\{k+l+m,i\}\; =\; -m\backslash delta\_\{k+l+m,i\},$

or, with {{math|n {{=}} l + m}},

:$n\backslash delta\_\{k+n,i\}\; =\; 0.$

This (and the antisymmetry condition) holds if {{math|k {{=}} i}}, in particular it holds when {{math|k {{=}} i {{=}} 0}}.

Thus choose {{math|L {{=}} L₀}} and {{math|d {{=}} d₀}}. With these choices, the premises in the corollary are satisfied. The 2-cocycle {{mvar|φ}} defined by

:$\backslash varphi(P(\backslash lambda)\; \backslash otimes\; G\_1),\; Q(\backslash lambda)\; \backslash otimes\; G\_2))\; =\; L(\backslash lambda\backslash frac\{dP\}\{d\backslash lambda\}\; \backslash otimes\; G\_1,\; Q(\backslash lambda)\; \backslash otimes\; G\_2)$

is finally employed to define a central extension of {{math|g}},

:$\backslash mathfrak\; e\; =\; \backslash mathfrak\; g\; \backslash oplus\; \backslash mathbb\; CC,$

with Lie bracket

:$[P(\backslash lambda)\; \backslash otimes\; G\_1\; +\; \backslash mu\; C,\; Q(\backslash lambda)\; \backslash otimes\; G\_2\; +\; \backslash nu\; C]\; =\; P(\backslash lambda)Q(\backslash lambda)\backslash otimes[G\_1,\; G\_2]\; +\; \backslash varphi(P(\backslash lambda)\; \backslash otimes\; G\_1,Q(\backslash lambda)\; \backslash otimes\; G\_2)C.$

For basis elements, suitably normalized and with antisymmetric structure constants, one has

:

&= \lambda^{l+m}\otimes {C_{ij}}^kG_k + L(\lambda \frac{d\lambda^l}{d\lambda} \otimes G_i, \lambda^m \otimes G_j)C\\

&=\lambda^{l+m}\otimes {C_{ij}}^kG_k + lL(\lambda^l \otimes G_i, \lambda^m \otimes G_j)C\\

&=\lambda^{l+m}\otimes {C_{ij}}^kG_k + l\delta_{l+m, 0}K(G_i, G_j)C\\

&=\lambda^{l+m}\otimes {C_{ij}}^kG_k + l\delta_{l+m, 0}{C_{ik}}^m{C_{jm}}^kC = \lambda^{l+m}\otimes {C_{ij}}^kG_k + l\delta_{l+m, 0}\delta^{ij}C.

\end{align}

This is a universal central extension of the polynomial loop algebra.{{harvnb|Kac|1990}} Exercise 7.8.

;A note on terminology

In physics terminology, the algebra of above might pass for a Kac–Moody algebra, whilst it will probably not in mathematics terminology. An additional dimension, an extension by a derivation is required for this. Nonetheless, if, in a physical application, the eigenvalues of {{math|g₀}} or its representative are interpreted as (ordinary) quantum numbers, the additional superscript on the generators is referred to as the level. It is an additional quantum number. An additional operator whose eigenvalues are precisely the levels is introduced further below.

File:Murray Gell-Mann - World Economic Forum Annual Meeting 2012.jpg, 1969 Nobel Laureate in physics, initiated the field of current algebra in the 1960s. It exploits known local symmetries even without knowledge of the underlying dynamics to extract predictions, e.g. the Adler–Weisberger sum rule.]]

{{main|Current algebra}}

As an application of a central extension of polynomial loop algebra, a current algebra of a quantum field theory is considered (background). Suppose one has a current algebra, with the interesting commutator being

{{NumBlk2|:|$[J\_a^0(t,\; \backslash mathbf\; x),\; J\_b^i(t,\; \backslash mathbf\; y)]\; =\; i\{C\_\{ab\}\}^cJ\_c^i(t,\; \backslash mathbf\; x)\backslash delta(\backslash mathbf\; x\; -\; \backslash mathbf\; y)\; +\; S\_\{ab\}^\{ij\}\backslash partial\_j\backslash delta(\backslash mathbf\; x\; -\; \backslash mathbf\; y)\; +\; ...\; ,$|CA10}}

with a Schwinger term. To construct this algebra mathematically, let {{math|g}} be the centrally extended polynomial loop algebra of the previous section with

:$[\backslash lambda^l\; \backslash otimes\; G\_i\; +\; \backslash mu\; C,\; \backslash lambda^m\; \backslash otimes\; G\_j\; +\; \backslash nu\; C]\; =\; \backslash lambda^\{l+m\}\backslash otimes\; \{C\_\{ij\}\}^kG\_k\; +\; l\backslash delta\_\{l+m,\; 0\}\backslash delta\_\{ij\}C$

as one of the commutation relations, or, with a switch of notation ({{math|l→m, m→n, i→a, j→b, λ^m⊗G_a→T^m_a}}) with a factor of {{math|i}} under the physics convention,

:$[T^m\_a,\; T^n\_b]\; =\; i\{C\_\{ab\}\}^cT^\{m+n\}\_c\; +\; m\backslash delta\_\{m+n,\; 0\}\backslash delta\_\{ab\}C.$

Define using elements of {{math|g}},

:$J\_a(x)\; =\; \backslash frac\{\backslash hbar\}\{L\}\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}e^\{\backslash frac\{2\backslash pi\; inx\}\{L\}\}T\_a^\{-n\},\; x\; \backslash in\; \backslash mathbb\; R.$

One notes that

:$J\_a(x+L)\; =\; J\_a(x)$

so that it is defined on a circle. Now compute the commutator,

:

&=\left(\frac{\hbar}{L}\right)^2 \sum_{m,n=-\infty}^{\infty} e^{\frac{2\pi inx}{L}} e^{\frac{2\pi imy}{L}} [T_a^{-n},T_b^{-m}].\end{align}

For simplicity, switch coordinates so that {{math|y → 0, x → x − y ≡ z}} and use the commutation relations,

:

&=\left(\frac{\hbar}{L}\right)^2\sum_{m=-\infty}^{\infty} e^{\frac{2\pi i(-m)z}{L}}\sum_{l=-\infty}^\infty ie^{\frac{2\pi i(l)z}{L}}{C_{ab}}^cT^{-l}_c + \left(\frac{\hbar}{L}\right)^2\sum_{m,n=-\infty}^{\infty}e^{\frac{2\pi inz}{L}}m\delta_{m+n, 0}\delta_{ab}C\\

&=\left(\frac{\hbar}{L}\right)\sum_{m=-\infty}^{\infty} e^{\frac{2\pi imz}{L}}i{C_{ab}}^cJ_c(z) - \left(\frac{\hbar}{L}\right)^2\sum_{n=-\infty}^{\infty}e^{\frac{2\pi inz}{L}}n\delta_{ab}C\end{align}

Now employ the Poisson summation formula,

:$\backslash frac\{1\}\{L\}\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; e^\{\backslash frac\{-2\backslash pi\; inz\}\{L\}\}\; =\; \backslash frac\{1\}\{L\}\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash delta(z\; +\; nL)\; =\; \backslash delta(z)$

for {{mvar|z}} in the interval {{math|(0, L)}} and differentiate it to yield

:$-\backslash frac\{2\backslash pi\; i\}\{L^2\}\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; ne^\{\backslash frac\{-2\backslash pi\; inz\}\{L\}\}\; =\; \backslash delta\text{'}(z),$

and finally

:$[J\_a(x-y),J\_b(0)]\; =\; i\backslash hbar\; \{C\_\{ab\}\}^cJ\_c(x-y)\backslash delta(x-y)\; +\; \backslash frac\{i\backslash hbar^2\}\{2\backslash pi\}\backslash delta\_\{ab\}C\backslash delta\text{'}(x-y),$

or

:$[J\_a(x),J\_b(y)]\; =\; i\backslash hbar\; \{C\_\{ab\}\}^cJ\_c(x)\backslash delta(x-y)\; +\; \backslash frac\{i\backslash hbar^2\}\{2\backslash pi\}\backslash delta\_\{ab\}C\backslash delta\text{'}(x-y),$

since the delta functions arguments only ensure that the arguments of the left and right arguments of the commutator are equal (formally {{math|δ(z) {{=}} δ(z − 0) ↦ δ((x −y) − 0) {{=}} δ(x −y)}}).

By comparison with {{EquationNote|CA10}}, this is a current algebra in two spacetime dimensions, including a Schwinger term, with the space dimension curled up into a circle. In the classical setting of quantum field theory, this is perhaps of little use, but with the advent of string theory where fields live on world sheets of strings, and spatial dimensions are curled up, there may be relevant applications.

File:Moody Baake.jpg (left), Fellow of the Royal Society of Canada, is a Canadian mathematician at University of Alberta. He is co-discoverer of the Kac–Moody algebra together with Victor Kac, Fellow of the American Mathematical Society, a Russian mathematician working at MIT.]]

The derivation {{math|d₀}} used in the construction of the 2-cocycle {{mvar|φ}} in the previous section can be extended to a derivation {{mvar|D}} on the centrally extended polynomial loop algebra, here denoted by {{math|g}} in order to realize a Kac–Moody algebra{{harvnb|Kac|1990}}{{harvnb|Bäuerle|de Kerf|1990}} (background). Simply set

:$D(P(\backslash lambda)\; \backslash otimes\; G\; +\; \backslash mu\; C)\; =\; \backslash lambda\; \backslash frac\{dP(\backslash lambda)\}\{d\backslash lambda\}\; \backslash otimes\; G.$

Next, define as a vector space

:$\backslash mathfrak\; e\; =\; \backslash mathbb\; Cd\; +\; \backslash mathfrak\; g.$

The Lie bracket on {{math|e}} is, according to the standard construction with a derivation, given on a basis by

:

&= \lambda^{m+n} \otimes [G_1, G_2] + m\delta_{m+n,0}K(G_1, G_2)C + \nu n\lambda^n \otimes G_1 - \nu'm \lambda^m \otimes G_2.\end{align}

For convenience, define

:$G\_i^m\; \backslash leftrightarrow\; \backslash lambda^m\; \backslash otimes\; G\_i.$

In addition, assume the basis on the underlying finite-dimensional simple Lie algebra has been chosen so that the structure coefficients are antisymmetric in all indices and that the basis is appropriately normalized. Then one immediately through the definitions verifies the following commutation relations.

:

{}[C,G_i^m] &= 0, \quad 1 \le i, j, N,\quad m,n \in \mathbb Z\\

{}[D, G_i^m] &= mG_i^m\\

{}[D,C] &= 0.\end{align}

These are precisely the short-hand description of an untwisted affine Kac–Moody algebra. To recapitulate, begin with a finite-dimensional simple Lie algebra. Define a space of formal Laurent polynomials with coefficients in the finite-dimensional simple Lie algebra. With the support of a symmetric non-degenerate alternating bilinear form and a derivation, a 2-cocycle is defined, subsequently used in the standard prescription for a central extension by a 2-cocycle. Extend the derivation to this new space, use the standard prescription for a split extension by a derivation and an untwisted affine Kac–Moody algebra obtains.

{{main|Virasoro algebra}}

The purpose is to construct the Virasoro algebra (named after Miguel Angel Virasoro)Miguel Angel Virasoro, born 1940 is an Argentine physicist. The Virasoro algebra, named after him, was first published in {{harvtxt|Virasoro|1970}} as a central extension by a 2-cocycle {{mvar|φ}} of the Witt algebra {{math|W}} (background). For details see Schottenloher.{{harvnb|Schottenloher|2008|loc=Thm. 5.1, pp. 79}} The Jacobi identity for 2-cocycles yields

{{NumBlk2|:|$(l-m)\backslash eta\_\{l+m,n\}\; +\; (m-n)\backslash eta\_\{m+n,l\}\; +\; (n-l)\backslash eta\_\{n+l,m\}\; =\; 0,\; \backslash quad\; \backslash eta\_\{ij\}\; =\; \backslash varphi(d\_i,\; d\_j).$|V10}}

Letting {{math|$l\; =\; 0$}} and using antisymmetry of {{mvar|η}} one obtains

:$(m+p)\backslash eta\_\{mp\}=(m-p)\backslash eta\_\{m+p,0\}.$

In the extension, the commutation relations for the element {{math|d₀}} are

:$[d\_0\; +\; \backslash mu\; C,\; d\_m\; +\; \backslash nu\; C]\_\backslash varphi\; =\; -md\_m\; +\; \backslash eta\_\{0m\}C\; =\; -m(d\_m\; -\; \backslash frac\{\backslash eta\_\{0m\}\}\{m\}C).$

It is desirable to get rid of the central charge on the right hand side. To do this define

:$f:W\; \backslash to\; \backslash mathbb\; C;\; d\_m\; \backslash to\; \backslash frac\{\backslash varphi(d\_0,d\_m)\}\{m\}\; =\; \backslash frac\{\backslash eta\_\{0m\}\}\{m\}.$

Then, using {{math|f}} as a 1-cochain,

:$\backslash eta\text{'}\_\{0n\}\; =\; \backslash varphi\text{'}(d\_0,\; d\_n)\; =\; \backslash varphi(d\_0,\; d\_n)\; +\; \backslash delta\; f([d\_0,\; d\_n])\; =\; \backslash varphi(d\_0,\; d\_n)\; -n\; \backslash frac\{\backslash eta^\{0n\}\}\{n\}=\; 0,$

so with this 2-cocycle, equivalent to the previous one, one hasThe same effect can be obtained by a change of basis in {{math|W}}.

:$[d\_0\; +\; \backslash mu\; C,\; d\_m\; +\; \backslash nu\; C]\_\{\backslash varphi\text{'}\}\; =\; -md\_m.$

With this new 2-cocycle (skip the prime) the condition becomes

:$(n+p)\backslash eta\_\{mp\}\; =\; (n-p)\backslash eta\_\{m+p,0\}=0,$

and thus

:$\backslash eta\_\{mp\}=a(m)\backslash delta\_\{m.\; -p\},\; \backslash quad\; a(-m)\; =\; -a(m),$

where the last condition is due to the antisymmetry of the Lie bracket. With this, and with {{math|l + m + p {{=}} 0}} (cutting out a "plane" in {{math|$\backslash mathbb\{Z\}^3$}}), ({{EquationNote|V10}}) yields

:$(2m+p)a(p)\; +\; (m-p)a(m+p)\; +\; (m+2p)a(m)\; =\; 0,$

that with {{math|p {{=}} 1}} (cutting out a "line" in {{math|$\backslash mathbb\{Z\}^2$}}) becomes

:$(m-1)a(m+1)\; -\; (m+2)a(m)\; +\; (2m+1)a(1)\; =\; 0.$

This is a difference equation generally solved by

:$a(m)\; =\; \backslash alpha\; m\; +\; \backslash beta\; m^3.$

The commutator in the extension on elements of {{math|W}} is then

:$[d\_l,\; d\_m]\; =\; (l-m)d\_\{l+m\}\; +\; (\backslash alpha\; m\; +\; \backslash beta\; m^3)\backslash delta\_\{l,-m\}C.$

With {{mvar|β {{=}} 0}} it is possible to change basis (or modify the 2-cocycle by a 2-coboundary) so that

:$[d\text{'}\_l,\; d\text{'}\_m]\; =\; (l-m)d\_\{l+m\},$

with the central charge absent altogether, and the extension is hence trivial. (This was not (generally) the case with the previous modification, where only {{math|d₀}} obtained the original relations.) With {{mvar|β ≠ 0}} the following change of basis,

:$d\text{'}\_l\; =\; d\_l\; +\; \backslash delta\_\{0l\}\backslash frac\{\backslash alpha+\backslash gamma\}\{2\}C,$

the commutation relations take the form

:$[d\text{'}\_l,\; d\text{'}\_m]\; =\; (l-m)d\text{'}\_\{l+m\}\; +\; (\backslash gamma\; m\; +\; \backslash beta\; m^3)\backslash delta\_\{l,-m\}C,$

showing that the part linear in {{mvar|m}} is trivial. It also shows that {{math|H²(W, $\backslash mathbb\{C\}$)}} is one-dimensional (corresponding to the choice of {{mvar|β}}). The conventional choice is to take {{math|α {{=}} −β {{=}} {{frac|1|12}}}} and still retaining freedom by absorbing an arbitrary factor in the arbitrary object {{math|C}}. The Virasoro algebra {{math|V}} is then

:$\backslash mathcal\; V\; =\; \backslash mathcal\; W\; +\; \backslash mathbb\; C\; C,$

with commutation relations

{{Equation box 1

|indent =:

|title=

|equation = $[d\_l\; +\; \backslash mu\; C,\; d\_m\; +\; \backslash nu\; C]\; =\; (l-m)d\_\{l+m\}\; +\; \backslash frac\{(m\; -\; m^3)\}\{12\}\backslash delta\_\{l,-m\}C.$

|cellpadding= 6

|border

|border colour = #0073CF

|bgcolor=#F9FFF7}}

{{main|Bosonic string theory}}

The relativistic classical open string (background) is subject to quantization. This roughly amounts to taking the position and the momentum of the string and promoting them to operators on the space of states of open strings. Since strings are extended objects, this results in a continuum of operators depending on the parameter {{mvar|σ}}. The following commutation relations are postulated in the Heisenberg picture.{{harvnb|Zwiebach|2004|loc=Chapter 12}}

:

{}[x_0^-(\tau),p^+(\tau)] &= -i.\end{align}

All other commutators vanish.

Because of the continuum of operators, and because of the delta functions, it is desirable to express these relations instead in terms of the quantized versions of the Virasoro modes, the Virasoro operators. These are calculated to satisfy

:$[\backslash alpha\_m^I,\; \backslash alpha\_n^J]\; =\; m\backslash eta^\{IJ\}\backslash delta\_\{m+n,0\}$

They are interpreted as creation and annihilation operators acting on Hilbert space, increasing or decreasing the quantum of their respective modes. If the index is negative, the operator is a creation operator, otherwise it is an annihilation operator. (If it is zero, it is proportional to the total momentum operator.) Since the light cone plus and minus modes were expressed in terms of the transverse Virasoro modes, one must consider the commutation relations between the Virasoro operators. These were classically defined (then modes) as

:$L\_n\; =\; \backslash frac\{1\}\{2\}\backslash sum\_\{p\; \backslash in\; \backslash mathbb\; Z\}\backslash alpha\_\{n-p\}^I\backslash alpha\_p^I.$

Since, in the quantized theory, the alphas are operators, the ordering of the factors matter. In view of the commutation relation between the mode operators, it will only matter for the operator {{math|L₀}} (for which {{math|m + n {{=}} 0}}). {{math|L₀}} is chosen normal ordered,

:$L\_0\; =\; \backslash frac\{1\}\{2\}\backslash alpha\_0^I\backslash alpha\_0^I\; +\; \backslash sum\_\{p=1\}^\backslash infty\; \backslash alpha\_\{-p\}^I\backslash alpha\_p^I,$

= \alpha' p^Ip^I + \sum_{p=1}^\infty p \alpha_{p}^{I\dagger}\alpha_p^I + c

where {{mvar|c}} is a possible ordering constant. One obtains after a somewhat lengthy calculation{{harvnb|Zwiebach|2004|pp=219–228}} the relations

:$[L\_m,\; L\_n]\; =\; (m-n)L\_\{m+n\},\; \backslash quad\; m+n\backslash ne\; 0.$

If one would allow for {{math|m + n {{=}} 0}} above, then one has precisely the commutation relations of the Witt algebra. Instead one has

:$[L\_m,\; L\_n]\; =\; (m-n)L\_\{m+n\}\; +\; \backslash frac\{D-2\}\{12\}(m^3-m)\backslash delta\_\{m+n,0\},\backslash quad\; \backslash forall\; m,n\; \backslash in\; \backslash mathbb\; Z.$

upon identification of the generic central term as {{math|(D − 2)}} times the identity operator, this is the Virasoro algebra, the universal central extension of the Witt algebra.

The operator {{math|L₀}} enters the theory as the Hamiltonian, modulo an additive constant. Moreover, the Virasoro operators enter into the definition of the Lorentz generators of the theory. It is perhaps the most important algebra in string theory.{{harvnb|Zwiebach|2004|p=227}} The consistency of the Lorentz generators, by the way, fixes the spacetime dimensionality to 26. While this theory presented here (for relative simplicity of exposition) is unphysical, or at the very least incomplete (it has, for instance, no fermions) the Virasoro algebra arises in the same way in the more viable superstring theory and M-theory.

{{main|Group extension}}

A projective representation {{math|Π(G)}} of a Lie group {{mvar|G}} (background) can be used to define a so-called group extension {{math|G_ex}}.

In quantum mechanics, Wigner's theorem asserts that if {{math|G}} is a symmetry group, then it will be represented projectively on Hilbert space by unitary or antiunitary operators. This is often dealt with by passing to the universal covering group of {{math|G}} and take it as the symmetry group. This works nicely for the rotation group {{math|SO(3)}} and the Lorentz group {{math|O(3, 1)}}, but it does not work when the symmetry group is the Galilean group. In this case one has to pass to its central extension, the Bargmann group,{{harvnb|Bargmann|1954}} which is the symmetry group of the Schrödinger equation. Likewise, if {{math|G {{=}} $\backslash mathbb\{R\}^2$}}, the group of translations in position and momentum space, one has to pass to its central extension, the Heisenberg group.{{harvnb|Tuynman|Wiegerinck|1987}}

Let {{mvar|ω}} be the 2-cocycle on {{math|G}} induced by {{math|Π}}. DefineIf the 2-cocycle takes its values in the abelian group {{math|U(1)}}, i. e. it is a phase factor, which will always be the case in the contezt of Wigner's theorem, then {{math|$\backslash mathbb\{C\}^*$}} may be replaced with {{math|U(1)}} in the construction.

:$G\_\{\backslash mathrm\; \{ex\}\}\; =\; \backslash mathbb\; C^*\; \backslash times\; G\; =\; \backslash \{(\backslash lambda,g)|\backslash lambda\; \backslash in\; \backslash mathbb\; C,\; g\; \backslash in\; G\backslash \}$

as a set and let the multiplication be defined by

:$(\backslash lambda\_1,g\_1)(\backslash lambda\_2,g\_2)\; =\; (\backslash lambda\_1\backslash lambda\_2\backslash omega(g\_1,g\_2),g\_1g\_2).$

Associativity holds since {{mvar|ω}} is a 2-cocycle on {{math|G}}. One has for the unit element

:$(1,e)(\backslash lambda,g)\; =\; (\backslash lambda\backslash omega(e,g),g)\; =\; (\backslash lambda,g)\; =\; (\backslash lambda,g)(1,e),$

and for the inverse

:$(\backslash lambda,g)^\{-1\}\; =\; \backslash left(\backslash frac\{1\}\{\backslash lambda\; \backslash omega(g,\; g^\{-1\})\},\; g^\{-1\}\backslash right).$

The set {{math|($\backslash mathbb\{C\}^*$, e)}} is an abelian subgroup of {{math|G_ex}}. This means that {{math|G_ex}} is not semisimple. The center of {{mvar|G}}, {{math|Z(G) {{=}} {z ∈ G{{!}}zg {{=}} gz ∀g ∈ G}}} includes this subgroup. The center may be larger.

At the level of Lie algebras it can be shown that the Lie algebra {{math|g_ex}} of {{math|G_ex}} is given by

:$\backslash mathfrak\{g\}\_\{\backslash mathrm\{ex\}\}\; =\; \backslash mathbb\; CC\; \backslash oplus\; \backslash mathfrak\; g,$

as a vector space and endowed with the Lie bracket

:$[\backslash mu\; C\; +\; G\_1,\backslash nu\; C\; +\; G\_2]\; =\; [G\_1,\; G\_2]\; +\; \backslash eta(G\_1,\; G\_2)C.$

Here {{mvar|η}} is a 2-cocycle on {{math|g}}. This 2-cocycle can be obtained from {{mvar|ω}} albeit in a highly nontrivial way.{{harvnb|Bäuerle|de Kerf|ten Kroode|1997|loc=Chapter 18.}} The reference states the fact and that it is difficult to show. No further references are given. Expressions on a slightly different form can be found though in {{harvtxt|Tuynman|Wiegerinck|1987}} and {{harvtxt|Bargmann|1954}}.

Now by using the projective representation {{math|Π}} one may define a map {{math|Π_ex}} by

:$\backslash Pi\_\{\backslash mathrm\{ex\}\}((\backslash lambda,g))\; =\; \backslash lambda\backslash Pi(g).$

It has the properties

:$\backslash Pi\_\{\backslash mathrm\{ex\}\}((\backslash lambda\_1,g\_1))\backslash Pi\_\{\backslash mathrm\{ex\}\}((\backslash lambda\_2,g\_2))=\backslash lambda\_1\backslash lambda\_2\backslash Pi(g\_1)\backslash Pi(g\_2)=\backslash lambda\_1\backslash lambda\_2\backslash omega(g\_1,g\_2)\backslash Pi(g\_1g\_2)=\backslash Pi\_\{\backslash mathrm\{ex\}\}(\backslash lambda\_1\backslash lambda\_2\backslash omega(g\_1,g\_2),g\_1g\_2)\; =\; \backslash Pi\_\{\backslash mathrm\{ex\}\}((\backslash lambda\_1,\; g\_1)(\backslash lambda\_2,g\_2)),$

so {{math|Π_ex(G_ex)}} is a bona fide representation of {{math|G_ex}}.

In the context of Wigner's theorem, the situation may be depicted as such (replace {{math|$\backslash mathbb\{C\}^*$}} by {{math|U(1)}}); let {{math|SH}} denote the unit sphere in Hilbert space {{mvar|H}}, and let {{math|(·,·)}} be its inner product. Let {{math|PH}} denote ray space and {{math|[·,·]}} the ray product. Let moreover a wiggly arrow denote a group action. Then the diagram

File:Lie algebra extension figure 4.svg

commutes, i.e.

:$\backslash pi\_2\; \backslash circ\; \backslash Pi\_\{\backslash mathrm\{ex\}\}((\backslash lambda,\; g))(\backslash psi)\; =\; \backslash Pi\; \backslash circ\; \backslash pi(g)(\backslash pi\_1(\backslash psi)),\; \backslash quad\; \backslash psi\; \backslash in\; S\backslash mathcal\; H.$

Moreover, in the same way that {{math|G}} is a symmetry of {{math|PH}} preserving {{math|[·,·]}}, {{math|G_ex}} is a symmetry of {{math|SH}} preserving {{math|(·,·)}}. The fibers of {{math|π₂}} are all circles. These circles are left invariant under the action of {{math|U(1)}}. The action of {{math|U(1)}} on these fibers is transitive with no fixed point. The conclusion is that {{math|SH}} is a principal fiber bundle over {{math|PH}} with structure group {{math|U(1)}}.

In order to adequately discuss extensions, structure that goes beyond the defining properties of a Lie algebra is needed. Rudimentary facts about these are collected here for quick reference.

A derivation {{mvar|δ}} on a Lie algebra {{math|g}} is a map

:$\backslash delta:\; \backslash mathfrak\; g\; \backslash rightarrow\; \backslash mathfrak\; g$

such that the Leibniz rule

:$\backslash delta[G\_1,\; G\_2]\; =\; [\backslash delta\; G\_1,\; G\_2]\; +\; [G\_1,\; \backslash delta\; G\_2]$

holds. The set of derivations on a Lie algebra {{math|g}} is denoted {{math|der g}}. It is itself a Lie algebra under the Lie bracket

:$[\backslash delta\_1,\; \backslash delta\_2]\; =\; \backslash delta\_1\; \backslash circ\; \backslash delta\_2\; -\; \backslash delta\_2\; \backslash circ\; \backslash delta\_1.$

It is the Lie algebra of the group {{math|Aut g}} of automorphisms of {{math|g}}.{{harvnb|Rossmann|2002|loc=Section 2.2}} One has to show

:$\backslash delta[G\_1,\; G\_1]\; =\; [\backslash delta\; G\_1,\; G\_2]\; +\; [G\_1,\; \backslash delta\; G\_2]\; \backslash Leftrightarrow\; e^\{t\backslash delta\}[G\_1,G\_2]\; =\; [e^\{t\backslash delta\}G\_1,\; e^\{t\backslash delta\}G\_2],\; \backslash quad\; \backslash forall\; t\; \backslash in\; \backslash mathbb\; R.$

If the rhs holds, differentiate and set {{math|t {{=}} 0}} implying that the lhs holds. If the lhs holds {{math|(A)}}, write the rhs as

:$[G\_1,G\_2]\backslash ;\; \backslash overset\{?\}\{=\}\backslash ;\; e^\{-t\backslash delta\}[e^\{t\backslash delta\}G\_1,\; e^\{t\backslash delta\}G\_2],$

and differentiate the rhs of this expression. It is, using {{math|(A)}}, identically zero. Hence the rhs of this expression is independent of {{mvar|t}} and equals its value for {{math|t {{=}} 0}}, which is the lhs of this expression.

If {{math|G ∈ g}}, then {{math|ad_G}}, acting by {{math|ad_G1(G₂) {{=}} [G₁, G₂]}}, is a derivation. The set {{math|ad_G: G ∈ g}} is the set of inner derivations on {{math|g}}. For finite-dimensional simple Lie algebras all derivations are inner derivations.{{harvnb|Humphreys|1972}}

{{main|Semidirect product}}

Consider two Lie groups {{mvar|G}} and {{mvar|H}} and {{math|Aut H}}, the automorphism group of {{mvar|H}}. The latter is the group of isomorphisms of {{mvar|H}}. If there is a Lie group homomorphism {{math|Φ:G → Aut H}}, then for each {{math|g ∈ G}} there is a {{math|Φ(g) ≡ Φ_g ∈ Aut H}} with the property {{math|Φ_gg' {{=}} Φ_gΦ_g', g,g' ∈ G}}. Denote with {{mvar|E}} the set {{math|H × G}} and define multiplication by

{{NumBlk2|:|$(h,\; g)(h\text{'},\; g\text{'})\; =\; (h\backslash phi\_g(h\text{'}),\; gg\text{'}),\; \backslash quad\; g,\; g\text{'}\; \backslash in\; G,\; h,\; h\text{'}\; \backslash in\; H.$|4}}

Then {{math|E}} is a group with identity {{math|(e_H, e_G)}} and the inverse is given by {{math|(h, g)⁻¹ {{=}} (Φ_g⁻¹(h⁻¹), g⁻¹)}}. Using the expression for the inverse and equation ({{EquationNote|4}}) it is seen that {{mvar|H}} is normal in {{mvar|E}}. Denote the group with this semidirect product as {{math|E {{=}} H ⊗_S G}}.

Conversely, if {{math|E {{=}} H ⊗_S G}} is a given semidirect product expression of the group {{math|E}}, then by definition {{math|H}} is normal in {{math|E}} and {{math|C_g ∈ Aut H}} for each {{math|g ∈ G}} where {{math|C_g (h) ≡ ghg⁻¹}} and the map {{math|Φ:g ↦ C_g}} is a homomorphism.

Now make use of the Lie correspondence. The maps {{math|Φ_g:H → H, g ∈ G}} each induce, at the level of Lie algebras, a map {{math|Ψ_g:h → h}}. This map is computed by

{{NumBlk2|:|$\backslash Psi\_g(G)\; =\; \backslash left\; .\backslash frac\{d\}\{dt\}\backslash Phi\_g(e^\{tG\})\backslash right|\_\{t\; =\; 0\},\; \backslash quad\; G\; \backslash in\; \backslash mathfrak\; g,\; g\; \backslash in\; G.$|5}}

For instance, if {{math|G}} and {{math|H}} are both subgroups of a larger group {{mvar|E}} and {{math|Φ_g {{=}} ghg⁻¹}}, then

{{NumBlk2|:|$\backslash Psi\_g(G)\; =\; \backslash left\; .\backslash frac\{d\}\{dt\}ge^\{tG\}g^\{-1\}\backslash right|\_\{t\; =\; 0\}\; =\; gGg^\{-1\}\; =\; \backslash mathrm\{Ad\}\_g(G),$|{{#invoke:DecodeEncode|encode|s=5'}}}}

and one recognizes {{math|Ψ}} as the adjoint action {{math|Ad}} of {{mvar|E}} on {{math|h}} restricted to {{mvar|G}}. Now {{math|Ψ:G → Aut h}} [ {{math|⊂ GL(h)}} if {{math|h}} is finite-dimensional] is a homomorphism,To see this, apply formula ({{EquationNote|4}}) to {{math|Ψ_gg'}}, recall that {{math|Φ}} is a homomorphism, and use {{math|Φ_g(e^G) {{=}} e^Ψ_g(G)}} a couple of times. and appealing once more to the Lie correspondence, there is a unique Lie algebra homomorphism {{math|ψ:g → Lie(Aut h) {{=}} Der h ⊂ gl(h)}}.The fact that the Lie algebra of {{math|Aut h)}} is {{math|Der h}}, the set of all derivations of {{math|h}} (itself being a Lie algebra under the obvious bracket), can be found in {{harvnb|Rossmann|2002|p=51}} This map is (formally) given by

{{NumBlk2|:|$\backslash psi\_G\; =\; \backslash left\; .\backslash frac\{d\}\{dt\}\backslash Psi\_\{e^\{tG\}\}\backslash right|\_\{t=0\},\backslash quad\; G\; \backslash in\; \backslash mathfrak\; g$|6}}

for example, if {{math|Ψ {{=}} Ad}}, then (formally)

{{NumBlk2|:|$\backslash psi\_G\; =\; \backslash left\; .\backslash frac\{d\}\{dt\}\backslash mathrm\{Ad\}\_\{e^\{tG\}\}\backslash right|\_\{t=0\}\; =\; \backslash left\; .\backslash frac\{d\}\{dt\}e^\{\backslash mathrm\{ad\}\_\{tG\}\}\backslash right|\_\{t=0\}\; =\; \backslash mathrm\{ad\}\_G,$|{{#invoke:DecodeEncode|encode|s=6'}}}}

where a relationship between {{math|Ad}} and the adjoint action {{math|ad}} rigorously proved in here is used.

Lie algebra

The Lie algebra is, as a vector space, {{math|e {{=}} h ⊕ g}}. This is clear since {{math|GH}} generates {{math|E}} and {{math|G ∩ H {{=}} (e_H, e_G)}}. The Lie bracket is given by{{harvnb|Knapp|2002}}

:$[H\_1\; +\; G\_1,H\_2\; +\; G\_2]\_\backslash mathfrak\; e\; =\; [H\_1,\; H\_2]\_\backslash mathfrak\; h\; +\; \backslash psi\_\{G\_1\}(H\_2)\; -\backslash psi\_\{G\_2\}(H\_1)\; +\; [G\_1,\; G\_2]\_\backslash mathfrak\; g.$

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Computation of Lie bracket}}

To compute the Lie bracket, begin with a surface in {{math|E}} parametrized by {{math|s}} and {{math|t}}. Elements of {{math|h}} in {{math|e {{=}} h ⊕ g}} are decorated with a bar, and likewise for {{math|g}}.

:$$

\begin{align}

e^{e^{t\overline{G}}s\overline{H}e^{-t\overline{G}}} &= e^{t\overline{G}}e^{s\overline{H}}e^{-t\overline{G}}=(1,e^{tG})(e^{sH},1)(1,e^{-tG})\\

&=(\phi_{e^{tG}}(e^{sH}), e^{tG})(1,e^{-tG}) = (\phi_{e^{tG}}(e^{sH})\phi_{e^{tG}}(1),1)\\

&= (\phi_{e^{tG}}(e^{sH}),1)

\end{align}

One has

:$$

\frac{d}{ds} \left. e^{Ad_{e^{t\overline{G}}}s\overline{H}}\right|_{s=0} = Ad_{e^{t\overline{G}}}\overline{H}

and

:$$

\frac{d}{ds} \left. (\phi_{e^{tG}}(e^{sH}),1)\right|_{s = 0} = (\Psi_{e^{tG}}(H), 0)

by {{EquationNote|5}} and thus

:$$

Ad_{e^{t\overline{G}}}\overline{H} = (\Psi_{e^{tG}}(H), 0).

Now differentiate this relationship with respect to {{mvar|t}} and evaluate at {{math|t {{=}} 0}}:

:$$

\frac{d}{dt} \left .e^{t\overline{G}}\overline{H}e^{-t\overline{G}}\right|_{t=0} = [\overline{G}, \overline{H}]

and

:$$

\frac{d}{dt} \left .(\Psi_{e^{tG}}(H), 0)\right|_{t=0} = (\psi_G(H), 0)

by {{EquationNote|6}} and thus

:$[H\_1\; +\; G\_1,H\_2\; +\; G\_2]\_\backslash mathfrak\; e\; =\; [H\_1,\; H\_2]\_\backslash mathfrak\; h\; +\; [G\_1,\; H\_2]\; +\; [H\_1,\; G\_2]\; +\; [G\_1,\; G\_2]\_\backslash mathfrak\; g\; =\; [H\_1,\; H\_2]\_\backslash mathfrak\; h\; +\; \backslash psi\_\{G\_1\}(H\_2)\; -\backslash psi\_\{G\_2\}(H\_1)\; +\; [G\_1,\; G\_2]\_\backslash mathfrak\; g.$

{{Hidden end}}

{{main|Algebraic topology|Cohomology|Lie algebra cohomology}}

For the present purposes, consideration of a limited portion of the theory Lie algebra cohomology suffices. The definitions are not the most general possible, or even the most common ones, but the objects they refer to are authentic instances of more the general definitions.

2-cocycles

The objects of primary interest are the 2-cocycles on {{math|g}}, defined as bilinear alternating functions,

:$$

\phi:\mathfrak g \times \mathfrak g \rightarrow F,

that are alternating,

:$$

\phi(G_1, G_2) = -\phi(G_2, G_1),

and having a property resembling the Jacobi identity called the Jacobi identity for 2-cycles,

:$$

\phi(G_1, [G_2, G_3]) + \phi(G_2, [G_3, G_1]) + \phi(G_3, [G_1, G_2]) = 0.

The set of all 2-cocycles on {{math|g}} is denoted {{math|Z²(g, F)}}.

2-cocycles from 1-cochains

Some 2-cocycles can be obtained from 1-cochains. A 1-cochain on {{math|g}} is simply a linear map,

:$$

f:\mathfrak g \rightarrow F

The set of all such maps is denoted {{math|C¹(g, F)}} and, of course (in at least the finite-dimensional case) {{math|C¹(g, F) ≅ g*}}. Using a 1-cochain {{mvar|f}}, a 2-cocycle {{math|δf}} may be defined by

:$$

\delta f(G_1, G_2) = f([G_1, G_2]).

The alternating property is immediate and the Jacobi identity for 2-cocycles is (as usual) shown by writing it out and using the definition and properties of the ingredients (here the Jacobi identity on {{math|g}} and the linearity of {{mvar|f}}). The linear map {{math|δ:C¹(g, F) → Z²(g, F)}} is called the coboundary operator (here restricted to {{math|C¹(g, F)}}).

The second cohomology group

Denote the image of {{math|C¹(g, F)}} of {{mvar|δ}} by {{math|B²(g, F)}}. The quotient

:$$

H^2(\mathfrak g, \mathbb F) = Z^2(\mathfrak g, \mathbb F)/B^2(\mathfrak g, \mathbb F)

is called the second cohomology group of {{math|g}}. Elements of {{math|H²(g, F)}} are equivalence classes of 2-cocycles and two

2-cocycles {{math|φ₁}} and {{math|φ₂}} are called equivalent cocycles if they differ by a 2-coboundary, i.e. if {{math|φ₁ {{=}} φ₂ + δf}} for some {{math|f ∈ C¹(g, F)}}. Equivalent

2-cocycles are called cohomologous. The equivalence class of {{math|φ ∈ Z²(g, F)}} is denoted {{math|[φ] ∈ H²}}.

These notions generalize in several directions. For this, see the main articles.

{{main|Structure constants}}

Let {{math|B}} be a Hamel basis for {{math|g}}. Then each {{math|G ∈ g}} has a unique expression as

:$G\; =\; \backslash sum\_\{\backslash alpha\; \backslash in\; A\}c\_\backslash alpha\; G\_\backslash alpha,\; \backslash quad\; c\_\backslash alpha\; \backslash in\; F,\; G\_\backslash alpha\; \backslash in\; B$

for some indexing set {{mvar|A}} of suitable size. In this expansion, only finitely many {{math|c_α}} are nonzero. In the sequel it is (for simplicity) assumed that the basis is countable, and Latin letters are used for the indices and the indexing set can be taken to be {{math|$\backslash mathbb\{N\}^*$ {{=}} 1, 2, ...}}. One immediately has

:$[G\_i,\; G\_j]\; =\; \{C\_\{ij\}\}^k\; G\_k$

for the basis elements, where the summation symbol has been rationalized away, the summation convention applies. The placement of the indices in the structure constants (up or down) is immaterial. The following theorem is useful:

Theorem:There is a basis such that the structure constants are antisymmetric in all indices if and only if the Lie algebra is a direct sum of simple compact Lie algebras and {{math|u(1)}} Lie algebras. This is the case if and only if there is a real positive definite metric {{mvar|g}} on {{math|g}} satisfying the invariance condition

:$g\_\{\backslash alpha\backslash beta\}\{C^\backslash beta\}\_\{\backslash gamma\backslash delta\}=-g\_\{\backslash gamma\backslash beta\}\{C^\backslash beta\}\_\{\backslash alpha\backslash delta\}.$

in any basis. This last condition is necessary on physical grounds for non-Abelian gauge theories in quantum field theory. Thus one can produce an infinite list of possible gauge theories using the Cartan catalog of simple Lie algebras on their compact form (i.e., {{math|sl(n, $\backslash mathbb\{C\}$) → su(n)}}, etc. One such gauge theory is the {{math|U(1) × SU(2) × SU(3)}} gauge theory of the standard model with Lie algebra {{math|u(1) ⊕ su(2) ⊕ su(3)}}.{{harvnb|Weinberg|1996|loc=Appendix A, Ch 15.}}

{{main|Killing form}}

The Killing form is a symmetric bilinear form on {{math|g}} defined by

:$K(G\_1,\; G\_2)\; =\; \backslash mathrm\{trace\}\; (\backslash mathrm\{ad\}\_\{G\_1\}\backslash mathrm\{ad\}\_\{G\_2\}).$

Here {{math|ad_G}} is viewed as a matrix operating on the vector space {{math|g}}. The key fact needed is that if {{math|g}} is semisimple, then, by Cartan's criterion, {{mvar|K}} is non-degenerate. In such a case {{mvar|K}} may be used to identify {{math|g}} and {{math|g^∗}}. If {{math|λ ∈ g^∗}}, then there is a {{math|ν(λ) {{=}} G_λ ∈ g}} such that

:$\backslash langle\; \backslash lambda,\; G\; \backslash rangle\; =\; K(G\_\backslash lambda,\; G)\; \backslash quad\; \backslash forall\; G\; \backslash in\; \backslash mathfrak\; g.$

This resembles the Riesz representation theorem and the proof is virtually the same. The Killing form has the property

:$K([G\_1,\; G\_2],\; G\_3)\; =\; K(G\_1,\; [G\_2,\; G\_3]),$

which is referred to as associativity. By defining {{math|g_αβ {{=}} K[G_α,G_β]}} and expanding the inner brackets in terms of structure constants, one finds that the Killing form satisfies the invariance condition of above.

{{main|Loop algebra|Loop group}}

A loop group is taken as a group of smooth maps from the unit circle {{math|S¹}} into a Lie group {{math|G}} with the group structure defined by the group structure on {{math|G}}. The Lie algebra of a loop group is then a vector space of mappings from {{math|S¹}} into the Lie algebra {{math|g}} of {{math|G}}. Any subalgebra of such a Lie algebra is referred to as a loop algebra. Attention here is focused on polynomial loop algebras of the form

:$\backslash \{h:\; S^1\; \backslash to\; \backslash mathfrak\; g|h(\backslash lambda)\; =\; \backslash sum\; \backslash lambda^n\; G\_n,\; n\; \backslash in\; \backslash mathbb\; Z,\; \backslash lambda\; =\; e^\{i\backslash theta\}\; \backslash in\; S^1,\; G\_n\; \backslash in\; \backslash mathfrak\; g\backslash \}.$

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Derivation of the Lie algebra}}

To see this, consider elements {{math|H(λ)}} near the identity in {{math|G}} for {{math|H}} in the loop group, expressed in a basis {{math|{G_k}}} for {{math|g}}

:$$

H(\lambda) = e^{h^k(\lambda)G_k} = e_G + h^k(\lambda)G_k + \ldots ,

where the {{math|h^k(λ)}} are real and small and the implicit sum is over the dimension {{math|K}} of {{math|g}}.

Now write

:$$

h^k(\lambda) = \sum_{n=-\infty}^\infty \theta^k_{-n}\lambda^n

to obtain

:$$

e^{h^k(\lambda)G_k} = 1_G + \sum_{n=-\infty}^\infty \theta^k_{-n}\lambda^nG_k + \ldots .

Thus the functions

:$$

h:S^1 \to \mathfrak g; h(\lambda) = \sum_{n=-\infty}^\infty \sum_{k=1}^K\theta^k_{-n}\lambda^nG_k \equiv \sum_{n=-\infty}^\infty \lambda^nG_n

constitute the Lie algebra.

{{Hidden end}}

A little thought confirms that these are loops in {{math|g}} as {{mvar|θ}} goes from {{math|0}} to {{math|2π}}. The operations are the ones defined pointwise by the operations in {{math|g}}. This algebra is isomorphic with the algebra

:$C[\backslash lambda,\; \backslash lambda^\{-1\}]\; \backslash otimes\; \backslash mathfrak\; g,$

where {{math|C[λ, λ⁻¹]}} is the algebra of Laurent polynomials,

:$\backslash sum\; \backslash lambda^k\; G\_k\; \backslash leftrightarrow\; \backslash sum\; \backslash lambda^k\; \backslash otimes\; G\_k.$

The Lie bracket is

:$[P(\backslash lambda)\; \backslash otimes\; G\_1,\; Q(\backslash lambda)\; \backslash otimes\; G\_2]\; =\; P(\backslash lambda)Q(\backslash lambda)\; \backslash otimes\; [G\_1,\; G\_2].$

In this latter view the elements can be considered as polynomials with (constant!) coefficients in {{math|g}}. In terms of a basis and structure constants,

:$[\backslash lambda^m\; \backslash otimes\; G\_i,\; \backslash lambda^n\; \backslash otimes\; G\_j]\; =\; \{C\_\{ij\}\}^k\backslash lambda^\{m+n\}\; \backslash otimes\; G\_k.$

It is also common to have a different notation,

:$\backslash lambda^m\; \backslash otimes\; G\_i\; \backslash cong\; \backslash lambda^mG\_i\; \backslash leftrightarrow\; T^m\_i(\backslash lambda)\; \backslash equiv\; T^m\_i,$

where the omission of {{mvar|λ}} should be kept in mind to avoid confusion; the elements really are functions {{math|S¹ → g}}. The Lie bracket is then

{{Equation box 1

|indent =

|title=

|equation = :$[T^m\_i,\; T^n\_j]\; =\; \{C\_\{ij\}\}^kT^\{m+n\}\_k,$

|cellpadding= 6

|border

|border colour = #0073CF

|bgcolor=#F9FFF7}}

which is recognizable as one of the commutation relations in an untwisted affine Kac–Moody algebra, to be introduced later, without the central term. With {{math|m {{=}} n {{=}} 0}}, a subalgebra isomorphic to {{math|g}} is obtained. It generates (as seen by tracing backwards in the definitions) the set of constant maps from {{math|S¹}} into {{mvar|G}}, which is obviously isomorphic with {{math|G}} when {{math|exp}} is onto (which is the case when {{mvar|G}} is compact. If {{math|G}} is compact, then a basis {{math|(G_k)}} for {{math|g}} may be chosen such that the {{math|G_k}} are skew-Hermitian. As a consequence,

:$T\_i^\{n\backslash dagger\}\; =\; (\backslash lambda^nG\_i)^\{\backslash dagger\}\; =\; -\backslash lambda^\{-n\}G\_i\; =\; -T\_i^\{-n\}.$

Such a representation is called unitary because the representatives

:$H(\backslash lambda)\; =\; e^\{\backslash theta\_\{n\}^k\; T\_k^\{-n\}\}\; \backslash in\; G$

are unitary. Here, the minus on the lower index of {{mvar|T}} is conventional, the summation convention applies, and the {{mvar|λ}} is (by the definition) buried in the {{math|T}}s in the right hand side.

Current algebras arise in quantum field theories as a consequence of global gauge symmetry. Conserved currents occur in classical field theories whenever the Lagrangian respects a continuous symmetry. This is the content of Noether's theorem. Most (perhaps all) modern quantum field theories can be formulated in terms of classical Lagrangians (prior to quantization), so Noether's theorem applies in the quantum case as well. Upon quantization, the conserved currents are promoted to position dependent operators on Hilbert space. These operators are subject to commutation relations, generally forming an infinite-dimensional Lie algebra. A model illustrating this is presented below.

To enhance the flavor of physics, factors of {{math|i}} will appear here and there as opposed to in the mathematical conventions.Roughly, the whole Lie algebra is multiplied by {{mvar|i}}, there is an {{mvar|i}} occurring in the definition of the structure constants and the exponent in the exponential map (Lie theory) acquires a factor of (minus) {{mvar|i}}. the main reason for this convention is that physicists like their Lie algebra elements to be Hermitian (as opposed to skew-Hermitian) in order for them to have real eigenvalues and hence be candidates for observables.

Consider a column vector {{math|Φ}} of scalar fields {{math|(Φ₁, Φ₂, ..., Φ_N)}}. Let the Lagrangian density be

:$\backslash mathcal\; L\; =\; \backslash partial\_\backslash mu\; \backslash phi^\backslash dagger\backslash partial^\backslash mu\backslash phi\; -\; m^2\backslash phi^\backslash dagger\backslash phi.$

This Lagrangian is invariant under the transformationSince {{math|U {{=}} −iΣα^aT_a}} and {{math|U^†}} are constant, they may be pulled out of partial derivatives. The {{math|U}} and {{math|U^†}} then combine in {{math|U^†U {{=}} I}} by unitarity.

:$\backslash phi\; \backslash mapsto\; e^\{-i\backslash sum\_\{a=1\}^r\backslash alpha^aF\_a\}\backslash phi,$

where {{math|{F₁, F₁, ..., F_r}}} are generators of either {{math|U(N)}} or a closed subgroup thereof, satisfying

:$[F\_a,\; F\_b]\; =\; i\{C\_\{ab\}\}^cF\_c.$

Noether's theorem asserts the existence of {{math|r}} conserved currents,

:$J\_a^\backslash mu\; =\; -\backslash pi^\backslash mu\; iF\_a\backslash phi,\; \backslash quad\; \backslash pi^\{k\backslash mu\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\; L\}\{\backslash partial\; (\backslash partial\_\backslash mu\; \backslash phi\_k)\},$

where {{math|π^k0 ≡ π^k}} is the momentum canonically conjugate to {{math|Φ_k}}.

The reason these currents are said to be conserved is because

:$\backslash partial\_\backslash mu\; J^\backslash mu\_a\; =\; 0,$

and consequently

:$Q\_a(t)\; =\; \backslash int\; J^0\_a\; d^3x\; =\; \backslash mathrm\{const\}\; \backslash equiv\; Q\_a,$

the charge associated to the charge density {{math|J_a⁰}} is constant in time.This follows from Gauss law is based on the assumption of a sufficiently rapid fall-off of the fields at infinity. This (so far classical) theory is quantized promoting the fields and their conjugates to operators on Hilbert space and by postulating (bosonic quantization) the commutation relations{{harvnb|Greiner|Reinhardt|1996}}There are alternative routes to quantization, e.g. one postulates the existence of creation and annihilation operators for all particle types with certain exchange symmetries based on which statistics, Bose–Einstein or Fermi–Dirac, the particles obey, in which case the above are derived for scalar bosonic fields using mostly Lorentz invariance and the demand for the unitarity of the S-matrix. In fact, all operators on Hilbert space can be built out of creation and annihilation operators. See e.g. {{harvtxt|Weinberg|2002}}, chapters 2–5.

:

{}[\phi_k(t, x), \phi_l(t, x)]&= [\pi^k(t, x), \pi^l(t, x)] = 0.\end{align}

The currents accordingly become operatorsThis step is ambiguous, since the classical fields commute whereas the operators don't. Here it is pretended that this problem doesn't exist. In reality, it is never serious as long as one is consistent. They satisfy, using the above postulated relations, the definitions and integration over space, the commutation relations

:

{}[Q_a, Q_b] &= i{Q_{ab}}^cQ_c\\

{}[Q_a, J_b^\mu(t, \mathbf x)] &= i{C_{ab}}^cJ_c^\mu(t, \mathbf x),\end{align}

where the speed of light and the reduced Planck constant have been set to unity. The last commutation relation does not follow from the postulated commutation relations (these are fixed only for {{math|π^k0}}, not for {{math|π^k1, π^k2, π^k3}}), except for {{math|μ {{=}} 0}} For {{math|μ {{=}} 1, 2, 3}} the Lorentz transformation behavior is used to deduce the conclusion. The next commutator to consider is

:$[J\_a^0(t,\; \backslash mathbf\; x),\; J\_b^i(t,\; \backslash mathbf\; y)]\; =\; i\{C\_\{ab\}\}^cJ\_c^i(t,\; \backslash mathbf\; x)\backslash delta(\backslash mathbf\; x\; -\; \backslash mathbf\; y)\; +\; S\_\{ab\}^\{ij\}\backslash partial\_j\backslash delta(\backslash mathbf\; x\; -\; \backslash mathbf\; y)\; +\; ...\; .$

The presence of the delta functions and their derivatives is explained by the requirement of microcausality that implies that the commutator vanishes when {{math|x ≠ y}}. Thus the commutator must be a distribution supported at {{math|x {{=}} y}}.{{harvnb|Bäuerle|de Kerf|1990}} Section 17.5. The first term is fixed due to the requirement that the equation should, when integrated over {{math|X}}, reduce to the last equation before it. The following terms are the Schwinger terms. They integrate to zero, but it can be shown quite generally{{harvnb|Bäuerle|de Kerf|1990|pp=383–386}} that they must be nonzero.

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Existence of Schwinger terms}}

Consider a conserved current

{{NumBlk2|:|$\backslash partial\_0J^0\; +\; \backslash partial\_i\; J^i=0,\; \backslash quad\; \backslash langle\; 0|J^i|0\backslash rangle=0,\; \backslash quad\; J^\{0\backslash dagger\}\; J^0\; =\; J^0J^\{0\backslash dagger\}\; =\; I.$|S10}}

with a generic Schwinger term

:$[J^0(t,\backslash mathbf\; x),J^i(t,\backslash mathbf\; y)]\; =\; i\backslash delta(\backslash mathbf\; x\; -\; \backslash mathbf\; y)J^i(t,\backslash mathbf\; x)\; +\; C^i(\backslash mathbf\; x,\; \backslash mathbf\; y).$

By taking the vacuum expectation value (VEV),

:$\backslash langle\; 0|C^i(\backslash mathbf\; x,\; \backslash mathbf\; y)|0\backslash rangle\; =\; \backslash langle\; 0|[J^0(t,\backslash mathbf\; x),J^i(t,\backslash mathbf\; y)]|0\backslash rangle,$

one finds

:

where {{EquationNote|S10}} and Heisenberg's equation of motion have been used as well as {{math|H{{!}}0⟩ {{=}} 0}} and its conjugate.

Multiply this equation by {{math|f(x)f(y)}} and integrate with respect to {{math|x}} and {{math|y}} over all space, using integration by parts, and one finds

:$-i\backslash int\backslash int\; d\backslash mathbf\; x\; d\backslash mathbf\; y\backslash langle\; 0|C^i(\backslash mathbf\; x,\; \backslash mathbf\; y)|0\backslash rangle\; f(\backslash mathbf\; x)\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y^i\}f(\backslash mathbf\; x)\; =\; 2\backslash langle\; 0|FHF|\backslash rangle,\; \backslash quad\; F\; =\; \backslash int\; J^0(\backslash mathbf\; x)f(\backslash mathbf\; x).$

Now insert a complete set of states, {{math|{{!}}n⟩}}

:$\backslash langle\; 0|FHF|\backslash rangle\; =\; \backslash sum\_\{mn\}\backslash langle\; 0|F|m\backslash rangle\backslash langle\; m|H|n\backslash rangle\backslash langle\; n|F|0\backslash rangle=\backslash sum\_\{mn\}\backslash langle\; 0|F|m\backslash rangle\; E\_n\backslash delta\_\{mn\}\backslash langle\; n|F|0\backslash rangle\; )\; \backslash sum\_\{n\; \backslash ne\; 0\}|\backslash langle\; 0|F|n\backslash rangle|^2E\_n\; >\; 0\; \backslash Rightarrow\; C^i(\backslash mathbf\; x,\; \backslash mathbf\; y)\; \backslash ne\; 0.$

Here hermiticity of {{mvar|F}} and the fact that not all matrix elements of {{mvar|F}} between the vacuum state and the states from a complete set can be zero.

{{Hidden end}}

{{main|Kac–Moody algebra}}

Let {{math|g}} be an {{math|N}}-dimensional complex simple Lie algebra with a dedicated suitable normalized basis such that the structure constants are antisymmetric in all indices with commutation relations

:$[G\_i,G\_j]\; =\; \{C\_\{ij\}\}^kG\_k,\; \backslash quad\; 1\; \backslash le\; i,\; j,\; N.$

An untwisted affine Kac–Moody algebra {{math|{{overline|g}}}} is obtained by copying the basis for each {{math|n ∈ $\backslash mathbb\{Z\}$}} (regarding the copies as distinct), setting

:$\backslash overline\{\backslash mathfrak\; g\}\; =\; FC\; \backslash oplus\; FD\; \backslash oplus\; \backslash bigoplus\_\{1\; \backslash le\; i\; \backslash le\; \backslash N,m\; \backslash in\; \backslash mathbb\; Z\}\; FG^i\_m$

as a vector space and assigning the commutation relations

:

{}[C,G_i^m] &= 0, \quad 1 \le i, j, N,\quad m,n \in \mathbb Z\\

{}[D, G_i^m] &= mG_i^m\\

{}[D,C] &= 0.\end{align}

If {{math|C {{=}} D {{=}} 0}}, then the subalgebra spanned by the {{math|G^m_i}} is obviously identical to the polynomial loop algebra of above.

File:Ernst Witt.jpeg (1911–1991), German mathematician. The Witt algebras, studied by him over finite fields in the 1930s, were first examined in the complex case by Cartan in 1909.]]

{{main|Witt algebra}}

The Witt algebra, named after Ernst Witt, is the complexification of the Lie algebra {{math|VectS¹}} of smooth vector fields on the circle {{math|S¹}}. In coordinates, such vector fields may be written

:$X\; =\; f(\backslash varphi)\backslash frac\{d\}\{d\backslash varphi\},$

and the Lie bracket is the Lie bracket of vector fields, on {{math|S¹}} simply given by

:$[X,\; Y]\; =\; \backslash left[f\backslash frac\{d\}\{d\backslash varphi\},\; g\backslash frac\{d\}\{d\backslash varphi\}\backslash right]\; =\; \backslash left(f\backslash frac\{dg\}\{d\backslash varphi\}\; -\; g\backslash frac\{df\}\{d\backslash varphi\}\backslash right)\backslash frac\{d\}\{d\backslash varphi\}.$

The algebra is denoted {{math|W {{=}} VectS¹ + iVectS¹}}.

A basis for {{mvar|W}} is given by the set

:$\backslash \{d\_n,\; n\; \backslash in\; \backslash mathbb\; Z\backslash \}\; =\; \backslash left\backslash \{\backslash left\; .ie^\{in\backslash varphi\}\backslash frac\{d\}\{d\backslash varphi\}\; =\; -z^\{n+1\}\backslash frac\{d\}\{dz\}\backslash right|n\; \backslash in\; \backslash mathbb\; Z\; \backslash right\backslash \}.$

This basis satisfies

{{Equation box 1

|indent =

|title=

|equation = :$[d\_l,\; d\_m]\; =\; (l-m)d\_\{l+m\}\; \backslash equiv\; \{C\_\{lm\}\}^nd\_n\; =\; (l-m)\backslash delta\_\{l+m\}^nd\_n,\backslash quad\; l,m,n\; \backslash in\; \backslash mathbb\; Z.$

|cellpadding= 6

|border

|border colour = #0073CF

|bgcolor=#F9FFF7}}

This Lie algebra has a useful central extension, the Virasoro algebra. It has {{math|3-}}dimensional subalgebras isomorphic with {{math|su(1, 1)}} and {{math|sl(2, $\backslash mathbb\{R\}$)}}. For each {{math|n ≠ 0}}, the set {{math|{d₀, d_−n, d_n}}} spans a subalgebra isomorphic to {{math|su(1, 1) ≅ sl(2, $\backslash mathbb\{R\}$)}}.

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Relationship to {{math|sl(2, $\backslash mathbb\{R\}$)}} and {{math|su(1, 1)}}}}

For {{math|m, n ∈ {−1, 0, 1}}} one has

:$[d\_0,\; d\_\{-1\}]\; =\; d\_\{-1\},\; \backslash quad\; [d\_0,\; d\_\{1\}]\; =\; -d\_\{1\},\backslash quad\; [d\_1,\; d\_\{-1\}]\; =\; 2d\_0.$

These are the commutation relations of {{math|sl(2, $\backslash mathbb\{R\}$)}} with

:

\quad d_{-1} \leftrightarrow X = \left(\begin{smallmatrix} 0 & 1\\ 0 & 0\end{smallmatrix}\right),

\quad d_1 \leftrightarrow Y = \left(\begin{smallmatrix} 0 & 0\\ 1 & 0\end{smallmatrix}\right), \quad H, X, Y \in \mathfrak{sl}(2, \mathbb R).

The groups {{math|SU(1, 1)}} and {{math|SL(2, $\backslash mathbb\{R\}$)}} are isomorphic under the map{{harvnb|Rossmann|2002|loc=Section 4.2}}

:

and the same map holds at the level of Lie algebras due to the properties of the exponential map.

A basis for {{math|su(1, 1)}} is given, see classical group, by

:

\quad U_1 = \left(\begin{smallmatrix} 0 & -i\\ i & 0\end{smallmatrix}\right),

\quad U_2 = \left(\begin{smallmatrix} i & 0\\ 0 & -i\end{smallmatrix}\right)

Now compute

:

=\left(\begin{smallmatrix} 0 & 1\\ 1 & 0\end{smallmatrix}\right) = U_0,\\

X_{\mathfrak{su}(1,1)} &= \left(\begin{smallmatrix} 1 & -i\\ 1 & i\end{smallmatrix}\right)X\left(\begin{smallmatrix} 1 & -i\\ 1 & i\end{smallmatrix}\right)^{-1}

=\frac{1}{2}\left(\begin{smallmatrix} i & -i\\ i & -i\end{smallmatrix}\right) = \frac{1}{2}(U_1+U_2),\\

Y_{\mathfrak{su}(1,1)} &= \left(\begin{smallmatrix} 1 & -i\\ 1 & i\end{smallmatrix}\right)Y\left(\begin{smallmatrix} 1 & -i\\ 1 & i\end{smallmatrix}\right)^{-1}

=\frac{1}{2}\left(\begin{smallmatrix} -i & -i\\ i & i\end{smallmatrix}\right) = \frac{1}{2}(U_1-U_2).

\end{align}

The map preserves brackets and there are thus Lie algebra isomorphisms between the subalgebra of {{mvar|W}} spanned by {{math|{d₀, d₋₁, d₁}}} with real coefficients, {{math|sl(2, $\backslash mathbb\{R\}$)}} and {{math|su(1, 1)}}. The same holds for any subalgebra spanned by {{math|{d₀, d_−n, d_n}, n ≠ 0}}, this follows from a simple rescaling of the elements (on either side of the isomorphisms).

{{Hidden end}}

{{main|Wigner's theorem|Projective representation|Group cohomology}}

If {{mvar|M}} is a matrix Lie group, then elements {{mvar|X}} of its Lie algebra m can be given by

:$X\; =\; \backslash frac\{d\}\{dt\}\backslash left\; .(g(t))\backslash right|\_\{t=0\},$

where {{mvar|g}} is a differentiable path in {{mvar|M}} that goes through the identity element at {{math|t {{=}} 0}}. Commutators of elements of the Lie algebra can be computed as{{cite book |last1=Hall |first1=Brian |title=Lie Groups, Lie Algebras, and Representations - An Elementary Introduction |date=2015 |publisher=Springer |location=Switzerland |isbn=978-3-319-13466-6 |page=57 |edition=2nd}}

:$[X\_1,\; X\_2]\; =\; \backslash left\; .\backslash frac\{d\}\{dt\}\backslash right|\_\{t\; =\; 0\}\backslash left\; .\backslash frac\{d\}\{ds\}\; \backslash right|\_\{s\; =\; 0\}\; e^\{tX\_1\}e^\{sX\_2\}e^\{-tX\_1\}.$

Likewise, given a group representation {{math|U(M)}}, its Lie algebra {{math|u(m)}} is computed by

:

&= \left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}U(e^{tX_1}e^{sX_2}e^{-tX_1})\end{align},

where $Y\_1=\backslash left\; .\backslash frac\{d\}\{dt\}\backslash right|\_\{t\; =\; 0\}U(e^\{tX\_1\})$ and $Y\_2=\backslash left\; .\backslash frac\{d\}\{ds\}\backslash right|\_\{s\; =\; 0\}U(e^\{sX\_2\})$.

Then there is a Lie algebra isomorphism between {{math|m}} and {{math|u(m)}} sending bases to bases, so that {{math|u}} is a faithful representation of {{math|m}}.

If however {{math|U(G)}} is an admissible set of representatives of a projective unitary representation, i.e. a unitary representation up to a phase factor, then the Lie algebra, as computed from the group representation, is not isomorphic to {{math|m}}. For {{math|U}}, the multiplication rule reads

:$U(g\_1)U(g\_2)\; =\; \backslash omega(g\_1,\; g\_2)U(g\_1g\_2)\; =\; e^\{i\backslash xi(g\_1,\; g\_2)\}U(g\_1g\_2).$

The function {{mvar|ω}},often required to be smooth, satisfies

:

\omega(g_1, g_2g_3)\omega(g_2,g_3) &= \omega(g_1,g_2)\omega(g_1g_2,g_3)\\

\omega(g,g^{-1})&=\omega(g^{-1},g).\end{align}

It is called a 2-cocycle on {{mvar|M}}.

From the above equalities, $(U(g))^\{-1\}=\backslash frac\{1\}\{\backslash omega(g,g^\{-1\})\}U(g^\{-1\})$, so one has

:$\backslash begin\{align\}[]\; [Y\_1,\; Y\_2]$

&= \left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}U(e^{tX_1})U(e^{sX_2})(U(e^{tX_1}))^{-1}\\

&= \left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}\frac{1}{\omega(e^{tX_1},e^{-tX_1})}U(e^{tX_1})U(e^{sX_2})U(e^{-tX_1})\\

&=\left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}\frac{\omega(e^{tX_1},e^{sX_2})\omega(e^{tX_1}e^{sX_2}, e^{-tX_1})}{\omega(e^{tX_1},e^{-tX_1})}U(e^{tX_1}e^{sX_2}e^{-tX_1})\\

&\equiv \left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}\Omega(e^{tX_1},e^{sX_2})U(e^{tX_1}e^{sX_2}e^{-tX_1})\\

&= \left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}U(e^{tX_1}e^{sX_2}e^{-tX_1})+ \left .\frac{d}{dt}\right|_{t = 0}\left .\frac{d}{ds} \right|_{s = 0}\Omega(e^{tX_1},e^{sX_2})I,\end{align}

because both {{math|Ω}} and {{mvar|U}} evaluate to the identity at {{math|t {{=}} 0}}. For an explanation of the phase factors {{mvar|ξ}}, see Wigner's theorem. The commutation relations in {{math|m}} for a basis,

:$[X\_i,X\_j]\; =\; \{C\_\{ij\}^k\}X\_k$

become in {{math|u}}

:$[Y\_i,Y\_j]\; =\; \{C\_\{ij\}^k\}Y\_k\; +\; D\_\{ij\}I,$

so in order for {{math|u}} to be closed under the bracket (and hence have a chance of actually being a Lie algebra) a central charge {{math|I}} must be included.

{{main|Bosonic string theory}}

A classical relativistic string traces out a world sheet in spacetime, just like a point particle traces out a world line. This world sheet can locally be parametrized using two parameters {{mvar|σ}} and {{mvar|τ}}. Points {{math|x^μ}} in spacetime can, in the range of the parametrization, be written {{math|x^μ {{=}} x^μ(σ, τ)}}. One uses a capital {{mvar|X}} to denote points in spacetime actually being on the world sheet of the string. Thus the string parametrization is given by {{math|(σ, τ) ↦(X⁰(σ, τ), X¹(σ, τ), X²(σ, τ), X³(σ, τ))}}. The inverse of the parametrization provides a local coordinate system on the world sheet in the sense of manifolds.

The equations of motion of a classical relativistic string derived in the Lagrangian formalism from the Nambu–Goto action are{{harvnb|Zwiebach|2004}} Equation 6.53 (supported by 6.49, 6.50).

:$\backslash frac\{\backslash partial\; \backslash mathcal\; P\_\backslash mu^\backslash tau\}\{\backslash partial\; \backslash tau\}\; +\; \backslash frac\{\backslash partial\; \backslash mathcal\; P\_\backslash mu^\backslash sigma\}\{\backslash partial\; \backslash sigma\}\; =\; 0,\; \backslash quad$

\mathcal P_\mu^\tau = -\frac{T_0}{c}\frac{(\dot X \cdot X')X'_\mu - (X')^2\dot X_\mu}{\sqrt{(\dot X \cdot X')^2 - (\dot X)^2(X')^2}},\quad

\mathcal P_\mu^\sigma = -\frac{T_0}{c}\frac{(\dot X \cdot X')X'_\mu - (\dot X)^2 X'_\mu}{\sqrt{(\dot X \cdot X')^2 - (\dot X)^2(X')^2}}.

A dot over a quantity denotes differentiation with respect to {{mvar|τ}} and a prime differentiation with respect to {{mvar|σ}}. A dot between quantities denotes the relativistic inner product.

These rather formidable equations simplify considerably with a clever choice of parametrization called the light cone gauge. In this gauge, the equations of motion become

:$\backslash ddot\; X^\backslash mu\; -\; \{X^\backslash mu\}\text{'}\text{'}\; =\; 0,$

the ordinary wave equation. The price to be paid is that the light cone gauge imposes constraints,

:$\backslash dot\; X^\backslash mu\; \backslash cdot\; \{X^\backslash mu\}\text{'}\; =\; 0,\; \backslash quad\; (\backslash dot\; X)^2\; +\; (X\text{'})^2\; =\; 0,$

so that one cannot simply take arbitrary solutions of the wave equation to represent the strings. The strings considered here are open strings, i.e. they don't close up on themselves. This means that the Neumann boundary conditions have to be imposed on the endpoints. With this, the general solution of the wave equation (excluding constraints) is given by

:$X^\backslash mu(\backslash sigma,\; \backslash tau)\; =\; x\_0^\backslash mu\; +\; 2\backslash alpha\text{'}p\_0^\backslash mu\backslash tau\; -\; i\backslash sqrt\{2\backslash alpha\text{'}\}\backslash sum\_\{n=1\}\backslash left(\; a\_n^\{\backslash mu*\}e^\{in\backslash tau\}\; -\; a\_n^\{\backslash mu\}e^\{-in\backslash tau\}\backslash right)\backslash frac\{\backslash cos\; n\backslash sigma\}\{\backslash sqrt\; n\},$

where {{math|α'}} is the slope parameter of the string (related to the string tension). The quantities {{math|x₀}} and {{math|p₀}} are (roughly) string position from the initial condition and string momentum. If all the {{math|α{{supsub|μ|n}}}} are zero, the solution represents the motion of a classical point particle.

This is rewritten, first defining

:$\backslash alpha\_0^\backslash mu\; =\; \backslash sqrt\{2\backslash alpha\text{'}\}a\_\; \backslash mu,\backslash quad\; \backslash alpha\_n^\backslash mu\; =\; a\_n^\backslash mu\backslash sqrt\{n\},\; \backslash quad\; \backslash alpha\_\{-n\}^\backslash mu\; =\; a\_n^\{\backslash mu*\}\backslash sqrt\{n\},$

and then writing

:$X^\backslash mu(\backslash sigma,\; \backslash tau)\; =\; x\_0^\backslash mu\; +\; \backslash sqrt\{2\backslash alpha\text{'}\}\backslash alpha\_0^\backslash mu\; \backslash tau\; +\; i\backslash sqrt\{2\backslash alpha\text{'}\}\backslash sum\_\{n\backslash ne\; 0\}\backslash frac\{1\}\{n\}\backslash alpha\_n^\{\backslash mu\}e^\{-in\backslash tau\}\backslash cos\; n\backslash sigma.$

In order to satisfy the constraints, one passes to light cone coordinates. For {{math|I {{=}} 2, 3, ...d}}, where {{math|d}} is the number of space dimensions, set

:$\backslash begin\{align\}$

X^I(\sigma, \tau) &= x_0^I + \sqrt{2\alpha'}\alpha_0^I \tau + i\sqrt{2\alpha'}\sum_{n \ne 0}\frac{1}{n}\alpha_n^{I}e^{-in\tau}\cos n\sigma,\\

X^+(\sigma, \tau) &= \sqrt{2\alpha'}\alpha_0^+ \tau,\\

X^-(\sigma, \tau) &= x_0^- + \sqrt{2\alpha'}\alpha_0^- \tau + i\sqrt{2\alpha'}\sum_{n \ne 0}\frac{1}{n}\alpha_n^{-}e^{-in\tau}\cos n\sigma. \end{align}

Not all {{math|α_n^μ, n ∈ $\backslash mathbb\{Z\}$, μ ∈ {+, −, 2, 3, ..., d}}} are independent. Some are zero (hence missing in the equations above), and the "minus coefficients" satisfy

:$\backslash sqrt\{2\backslash alpha\text{'}\}\backslash alpha\_n^-\; =\; \backslash frac\{1\}\{2p^+\}\backslash sum\_\{p\; \backslash in\; \backslash mathbb\; Z\}\backslash alpha\_\{n-p\}^I\backslash alpha\_p^I.$

The quantity on the left is given a name,

:$\backslash sqrt\{2\backslash alpha\text{'}\}\backslash alpha\_n^-\; \backslash equiv\; \backslash frac\{1\}\{p^+\}L\_n,\backslash quad\; L\_n\; =\; \backslash frac\{1\}\{2\}\backslash sum\_\{p\; \backslash in\; \backslash mathbb\; Z\}\backslash alpha\_\{n-p\}^I\backslash alpha\_p^I,$

the transverse Virasoro mode.

When the theory is quantized, the alphas, and hence the {{math|L_n}} become operators.

• Group cohomology
• Group contraction (Inönu–Wigner contraction)
• Group extension
• Lie algebra cohomology

{{reflist|group=nb}}

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