Classical group

The classical groups are exactly the general linear groups over {{mathbb|R}}, {{mathbb|C}} and {{mathbb|H}} together with the automorphism groups of non-degenerate forms discussed below.{{harvnb|Rossmann|2002}} p. 94 These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is not used consistently in the interest of greater generality.

The complex classical groups are {{math|SL(n, {{mathbb|C}})}}, {{math|SO(n, {{mathbb|C}})}} and {{math|Sp(n, {{mathbb|C}})}}. A group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups. These are, in turn, {{math|SU(n)}}, {{math|SO(n)}} and {{math|Sp(n)}}. One characterization of the compact real form is in terms of the Lie algebra {{math|g}}. If {{math|g {{=}} u + iu}}, the complexification of {{math|u}}, and if the connected group {{math|K}} generated by {{math|{{mset|exp(X): X ∈ u}}}} is compact, then {{math|K}} is a compact real form.{{harvnb|Rossmann|2002}} p. 103

The classical groups can uniformly be characterized in a different way using real forms. The classical groups (here with the determinant 1 condition, but this is not necessary) are the following:

:The complex linear algebraic groups {{math|SL(n, {{mathbb|C}}), SO(n, {{mathbb|C}})}}, and {{math|Sp(n, {{mathbb|C}})}} together with their real forms.{{harvnb|Goodman|Wallach|2009}} See end of chapter 1

For instance, {{math|SO^∗(2n)}} is a real form of {{math|SO(2n, {{mathbb|C}})}}, {{math|SU(p, q)}} is a real form of {{math|SL(n, {{mathbb|C}})}}, and {{math|SL(n, {{mathbb|H}})}} is a real form of {{math|SL(2n, {{mathbb|C}})}}. Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization. The algebraic groups in question are Lie groups, but the "algebraic" qualifier is needed to get the right notion of "real form".

{{main|Bilinear form|Sesquilinear form}}

The classical groups are defined in terms of forms defined on {{math|Rⁿ}}, {{math|Cⁿ}}, and {{math|Hⁿ}}, where {{math|R}} and {{math|C}} are the fields of the real and complex numbers. The quaternions, {{math|H}}, do not constitute a field because multiplication does not commute; they form a division ring or a skew field or non-commutative field. However, it is still possible to define matrix quaternionic groups. For this reason, a vector space {{math|V}} is allowed to be defined over {{math|R}}, {{math|C}}, as well as {{math|H}} below. In the case of {{math|H}}, {{math|V}} is a right vector space to make possible the representation of the group action as matrix multiplication from the left, just as for {{math|R}} and {{math|C}}.{{harvnb|Rossmann|2002}}p. 93.

A form {{math|φ: V × V → F}} on some finite-dimensional right vector space over {{math|F {{=}} R, C}}, or {{math|H}} is bilinear if

:$\backslash varphi(x\backslash alpha,\; y\backslash beta)\; =\; \backslash alpha\backslash varphi(x,\; y)\backslash beta,\; \backslash quad\; \backslash forall\; x,y\; \backslash in\; V,\; \backslash forall\; \backslash alpha,\backslash beta\; \backslash in\; F.$ and if

:$\backslash varphi(x\_1+x\_2,y\_1+y\_2)=\backslash varphi(x\_1,y\_1)+\backslash varphi(x\_1,y\_2)+\backslash varphi(x\_2,y\_1)+\backslash varphi(x\_2,y\_2),\backslash quad\; \backslash forall\; x\_1,\; x\_2,\; y\_1,\; y\_2\; \backslash in\; V.$

It is called sesquilinear if

:$\backslash varphi(x\backslash alpha,\; y\backslash beta)\; =\; \backslash bar\{\backslash alpha\}\backslash varphi(x,\; y)\backslash beta,\; \backslash quad\; \backslash forall\; x,y\; \backslash in\; V,\; \backslash forall\; \backslash alpha,\backslash beta\; \backslash in\; F.$ and if

:$\backslash varphi(x\_1+x\_2,y\_1+y\_2)=\backslash varphi(x\_1,y\_1)+\backslash varphi(x\_1,y\_2)+\backslash varphi(x\_2,y\_1)+\backslash varphi(x\_2,y\_2),\; \backslash quad\; \backslash forall\; x\_1,\; x\_2,\; y\_1,\; y\_2\; \backslash in\; V.$

These conventions are chosen because they work in all cases considered. An automorphism of {{math|φ}} is a map {{math|Α}} in the set of linear operators on {{math|V}} such that

{{NumBlk|:|$\backslash varphi(Ax,\; Ay)\; =\; \backslash varphi(x,\; y),\; \backslash quad\; \backslash forall\; x,y\; \backslash in\; V.$|{{EquationRef|1}}}}

The set of all automorphisms of {{math|φ}} form a group, it is called the automorphism group of {{math|φ}}, denoted {{math|Aut(φ)}}. This leads to a preliminary definition of a classical group:

:A classical group is a group that preserves a bilinear or sesquilinear form on finite-dimensional vector spaces over {{math|R}}, {{math|C}} or {{math|H}}.

This definition has some redundancy. In the case of {{math|F {{=}} R}}, bilinear is equivalent to sesquilinear. In the case of {{math|F {{=}} H}}, there are no non-zero bilinear forms.{{harvnb|Rossmann|2002}} p. 105

A form is symmetric if

:$\backslash varphi(x,\; y)\; =\; \backslash varphi(y,\; x).$

It is skew-symmetric if

:$\backslash varphi(x,\; y)\; =\; -\backslash varphi(y,\; x).$

It is Hermitian if

:$\backslash varphi(x,\; y)\; =\; \backslash overline\{\backslash varphi(y,\; x)\}$

Finally, it is skew-Hermitian if

:$\backslash varphi(x,\; y)\; =\; -\backslash overline\{\backslash varphi(y,\; x)\}.$

A bilinear form {{math|φ}} is uniquely a sum of a symmetric form and a skew-symmetric form. A transformation preserving {{math|φ}} preserves both parts separately. The groups preserving symmetric and skew-symmetric forms can thus be studied separately. The same applies, mutatis mutandis, to Hermitian and skew-Hermitian forms. For this reason, for the purposes of classification, only purely symmetric, skew-symmetric, Hermitian, or skew-Hermitian forms are considered. The normal forms of the forms correspond to specific suitable choices of bases. These are bases giving the following normal forms in coordinates:

:$\backslash begin\{align\}$

\text{Bilinear symmetric form in (pseudo-)orthonormal basis:} \quad

\varphi(x, y) ={} &{\pm}\xi_1\eta_1 \pm \xi_2\eta_2 \pm \cdots \pm \xi_n\eta_n, & &(\mathbf R)\\

\text{Bilinear symmetric form in orthonormal basis:} \quad

\varphi(x, y) ={} &\xi_1\eta_1 + \xi_2\eta_2 + \cdots + \xi_n\eta_n, & &(\mathbf C)\\

\text{Bilinear skew-symmetric in symplectic basis:} \quad

\varphi(x, y) ={} &\xi_1\eta_{m + 1} + \xi_2\eta_{m + 2} + \cdots + \xi_m\eta_{2m = n} \\

&-\xi_{m + 1}\eta_1 - \xi_{m + 2}\eta_2 - \cdots - \xi_{2m = n}\eta_m, & &(\mathbf R, \mathbf C)\\

\text{Sesquilinear Hermitian:} \quad

\varphi(x, y) ={} &{\pm}\bar{\xi_1}\eta_1 \pm \bar{\xi_2}\eta_2 \pm \cdots \pm \bar{\xi_n}\eta_n, & &(\mathbf C, \mathbf H)\\

\text{Sesquilinear skew-Hermitian:} \quad

\varphi(x, y) ={} &\bar{\xi_1}\mathbf{j}\eta_1 + \bar{\xi_2}\mathbf{j}\eta_2 + \cdots + \bar{\xi_n}\mathbf{j}\eta_n, & &(\mathbf H)

\end{align}

The {{math|j}} in the skew-Hermitian form is the third basis element in the basis {{math|(1, i, j, k)}} for {{math|H}}. Proof of existence of these bases and Sylvester's law of inertia, the independence of the number of plus- and minus-signs, {{math|p}} and {{math|q}}, in the symmetric and Hermitian forms, as well as the presence or absence of the fields in each expression, can be found in {{harvtxt|Rossmann|2002}} or {{harvtxt|Goodman|Wallach|2009}}. The pair {{math|(p, q)}}, and sometimes {{math|p − q}}, is called the signature of the form.

Explanation of occurrence of the fields {{math|R, C, H}}: There are no nontrivial bilinear forms over {{math|H}}. In the symmetric bilinear case, only forms over {{math|R}} have a signature. In other words, a complex bilinear form with "signature" {{math|(p, q)}} can, by a change of basis, be reduced to a form where all signs are "{{math|+}}" in the above expression, whereas this is impossible in the real case, in which {{math|p − q}} is independent of the basis when put into this form. However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case. (The real case reduces to the symmetric case.) A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by {{mvar|i}}, so in this case, only {{math|H}} is interesting.

Image:Hermann Weyl ETH-Bib Portr 00890.jpg, the author of The Classical Groups. Weyl made substantial contributions to the representation theory of the classical groups.]]

The first section presents the general framework. The other sections exhaust the qualitatively different cases that arise as automorphism groups of bilinear and sesquilinear forms on finite-dimensional vector spaces over {{math|R}}, {{math|C}} and {{math|H}}.

Assume that {{math|φ}} is a non-degenerate form on a finite-dimensional vector space {{math|V}} over {{math|R, C}} or {{math|H}}. The automorphism group is defined, based on condition ({{EquationNote|1}}), as

:$\backslash mathrm\{Aut\}(\backslash varphi)\; =\; \backslash \{A\; \backslash in\; \backslash mathrm\{GL\}(V)\; :\; \backslash varphi(Ax,\; Ay)\; =\; \backslash varphi(x,\; y),\; \backslash quad\; \backslash forall\; x,y\; \backslash in\; V\backslash \}.$

Every {{math|A ∈ M_n(V)}} has an adjoint {{math|A^φ}} with respect to {{math|φ}} defined by

{{NumBlk|:|$\backslash varphi(Ax,\; y)\; =\; \backslash varphi(x,\; A^\backslash varphi\; y),\; \backslash qquad\; x,\; y\; \backslash in\; V.$|{{EquationRef|2}}}}

Using this definition in condition ({{EquationNote|1}}), the automorphism group is seen to be given by

{{NumBlk|:|$\backslash operatorname\{Aut\}(\backslash varphi)\; =\; \backslash \{A\; \backslash in\; \backslash operatorname\{GL\}(V):\; A^\backslash varphi\; A\; =\; 1\backslash \}.${{harvnb|Rossmann|2002}} p. 91|{{EquationRef|3}}}}

Fix a basis for {{math|V}}. In terms of this basis, put

:$\backslash varphi(x,\; y)\; =\; \backslash sum\; \backslash xi\_i\backslash varphi\_\{ij\}\backslash eta\_j$

where {{math|ξ_i, η_j}} are the components of {{math|x, y}}. This is appropriate for the bilinear forms. Sesquilinear forms have similar expressions and are treated separately later. In matrix notation one finds

:$\backslash varphi(x,\; y)\; =\; x^\{\backslash mathrm\; T\}\backslash Phi\; y$

and

{{NumBlk|:|$A^\backslash varphi\; =\; \backslash Phi^\{-1\}A^\{\backslash mathrm\; T\}\backslash Phi${{harvnb|Rossmann|2002}} p. 92|{{EquationRef|4}}}}

from ({{EquationNote|2}}) where {{math|Φ}} is the matrix {{math|(φ_ij)}}. The non-degeneracy condition means precisely that {{math|Φ}} is invertible, so the adjoint always exists. {{math|Aut(φ)}} expressed with this becomes

:$\backslash operatorname\{Aut\}(\backslash varphi)\; =\; \backslash left\backslash \{A\; \backslash in\; \backslash operatorname\{GL\}(V):\; \backslash Phi^\{-1\}A^\backslash mathrm\{T\}\backslash Phi\; A\; =\; 1\backslash right\backslash \}.$

The Lie algebra {{math|aut(φ)}} of the automorphism groups can be written down immediately. Abstractly, {{math|X ∈ aut(φ)}} if and only if

:$(e^\{tX\})^\backslash varphi\; e^\{tX\}\; =\; 1$

for all {{math|t}}, corresponding to the condition in ({{EquationNote|3}}) under the exponential mapping of Lie algebras, so that

:$\backslash mathfrak\{aut\}(\backslash varphi)\; =\; \backslash left\backslash \{X\; \backslash in\; M\_n(V):\; X^\backslash varphi\; =\; -X\backslash right\backslash \},$

or in a basis

{{NumBlk|:|$\backslash mathfrak\{aut\}(\backslash varphi)\; =\; \backslash left\backslash \{X\; \backslash in\; M\_n(V):\; \backslash Phi^\{-1\}X^\backslash mathrm\{T\}\backslash Phi\; =\; -X\backslash right\backslash \}$|{{EquationRef|5}}}}

as is seen using the power series expansion of the exponential mapping and the linearity of the involved operations. Conversely, suppose that {{math|X ∈ aut(φ)}}. Then, using the above result, {{math|φ(Xx, y) {{=}} φ(x, X^φy) {{=}} −φ(x, Xy)}}. Thus the Lie algebra can be characterized without reference to a basis, or the adjoint, as

:$\backslash mathfrak\{aut\}(\backslash varphi)\; =\; \backslash \{X\; \backslash in\; M\_n(V):\; \backslash varphi(Xx,\; y)\; =\; -\backslash varphi(x,\; Xy),\backslash quad\; \backslash forall\; x,y\; \backslash in\; V\backslash \}.$

The normal form for {{math|φ}} will be given for each classical group below. From that normal form, the matrix {{math|Φ}} can be read off directly. Consequently, expressions for the adjoint and the Lie algebras can be obtained using formulas ({{EquationNote|4}}) and ({{EquationNote|5}}). This is demonstrated below in most of the non-trivial cases.

When the form is symmetric, {{math|Aut(φ)}} is called {{math|O(φ)}}. When it is skew-symmetric then {{math|Aut(φ)}} is called {{math|Sp(φ)}}. This applies to the real and the complex cases. The quaternionic case is empty since no nonzero bilinear forms exists on quaternionic vector spaces.{{harvnb|Rossmann|2002}} p. 105

The real case breaks up into two cases, the symmetric and the antisymmetric forms that should be treated separately.

{{main|Orthogonal group|Indefinite orthogonal group}}

If {{math|φ}} is symmetric and the vector space is real, a basis may be chosen so that

:$\backslash varphi(x,\; y)\; =\; \backslash pm\; \backslash xi\_1\backslash eta\_1\; \backslash pm\; \backslash xi\_2\backslash eta\_2\; \backslash cdots\; \backslash pm\; \backslash xi\_n\backslash eta\_n.$

The number of plus and minus-signs is independent of the particular basis.{{harvnb|Rossmann|2002}} p. 107. In the case {{math|V {{=}} Rⁿ}} one writes {{math|O(φ) {{=}} O(p, q)}} where {{math|p}} is the number of plus signs and {{math|q}} is the number of minus-signs, {{math|p + q {{=}} n}}. If {{math|q {{=}} 0}} the notation is {{math|O(n)}}. The matrix {{math|Φ}} is in this case

:

after reordering the basis if necessary. The adjoint operation ({{EquationNote|4}}) then becomes

:

which reduces to the usual transpose when {{math|p}} or {{math|q}} is 0. The Lie algebra is found using equation ({{EquationNote|5}}) and a suitable ansatz (this is detailed for the case of {{math|Sp(m, R)}} below),

:

and the group according to ({{EquationNote|3}}) is given by

:$\backslash mathrm\{O\}(p,\; q)\; =\; \backslash \{g\; \backslash in\; \backslash mathrm\{GL\}(n,\; \backslash mathbb\{R\})|I\_\{p,q\}^\{-1\}g^\{\backslash mathrm\{T\}\}I\_\{p,q\}g\; =\; I\backslash \}.$

The groups {{math|O(p, q)}} and {{math|O(q, p)}} are isomorphic through the map

:

For example, the Lie algebra of the Lorentz group could be written as

:$\backslash mathfrak\{o\}(3,\; 1)\; =\; \backslash mathrm\{span\}\; \backslash left\backslash \{$

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

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

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

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

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

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

\right\}.

Naturally, it is possible to rearrange so that the {{math|q}}-block is the upper left (or any other block). Here the "time component" end up as the fourth coordinate in a physical interpretation, and not the first as may be more common.

{{main|Symplectic group}}

If {{math|φ}} is skew-symmetric and the vector space is real, there is a basis giving

:$\backslash varphi(x,\; y)\; =\; \backslash xi\_1\backslash eta\_\{m\; +\; 1\}\; +\; \backslash xi\_2\backslash eta\_\{m\; +\; 2\}\; \backslash cdots\; +\; \backslash xi\_m\backslash eta\_\{2m\; =\; n\}\; -\; \backslash xi\_\{m\; +\; 1\}\backslash eta\_1\; -\; \backslash xi\_\{m\; +\; 2\}\backslash eta\_2\; \backslash cdots\; -\; \backslash xi\_\{2m\; =\; n\}\backslash eta\_m,$

where {{math|n {{=}} 2m}}. For {{math|Aut(φ)}} one writes {{math|Sp(φ) {{=}} Sp(V)}} In case {{math|V {{=}} Rⁿ {{=}} R^2m}} one writes {{math|Sp(m, R)}} or {{math|Sp(2m, R)}}. From the normal form one reads off

:

By making the ansatz

:

where {{math|X, Y, Z, W}} are {{math|m}}-dimensional matrices and considering ({{EquationNote|5}}),

:

one finds the Lie algebra of {{math|Sp(m, R)}},

:

and the group is given by

:$\backslash mathrm\{Sp\}(m,\; \backslash mathbb\{R\})\; =\; \backslash \{g\; \backslash in\; M\_n(\backslash mathbb\{R\})|g^\{\backslash mathrm\{T\}\}J\_mg\; =\; J\_m\backslash \}.$

Like in the real case, there are two cases, the symmetric and the antisymmetric case that each yield a family of classical groups.

{{main|Complex orthogonal group}}

If case {{math|φ}} is symmetric and the vector space is complex, a basis

:$\backslash varphi(x,\; y)\; =\; \backslash xi\_1\backslash eta\_1\; +\; \backslash xi\_1\backslash eta\_1\; \backslash cdots\; +\; \backslash xi\_n\backslash eta\_n$

with only plus-signs can be used. The automorphism group is in the case of {{math|V {{=}} Cⁿ}} called {{math|O(n, C)}}. The Lie algebra is simply a special case of that for {{math|o(p, q)}},

:$\backslash mathfrak\{o\}(n,\; \backslash mathbb\{C\})\; =\; \backslash mathfrak\{so\}(n,\; \backslash mathbb\{C\})\; =\; \backslash \{X|X^\{\backslash mathrm\{T\}\}\; =\; -X\backslash \},$

and the group is given by

:$\backslash mathrm\{O\}(n,\; \backslash mathbb\{C\})\; =\; \backslash \{g|g^\{\backslash mathrm\{T\}\}g\; =\; I\_n\backslash \}.$

In terms of classification of simple Lie algebras, the {{math|so(n)}} are split into two classes, those with {{math|n}} odd with root system {{math|B_n}} and {{math|n}} even with root system {{math|D_n}}.

{{main|Symplectic group}}

For {{math|φ}} skew-symmetric and the vector space complex, the same formula,

:$\backslash varphi(x,\; y)\; =\; \backslash xi\_1\backslash eta\_\{m\; +\; 1\}\; +\; \backslash xi\_2\backslash eta\_\{m\; +\; 2\}\; \backslash cdots\; +\; \backslash xi\_m\backslash eta\_\{2m\; =\; n\}\; -\; \backslash xi\_\{m\; +\; 1\}\backslash eta\_1\; -\; \backslash xi\_\{m\; +\; 2\}\backslash eta\_2\; \backslash cdots\; -\; \backslash xi\_\{2m\; =\; n\}\backslash eta\_m,$

applies as in the real case. For {{math|Aut(φ)}} one writes {{math|Sp(φ) {{=}} Sp(V)}}. In the case $V\; =\; \backslash mathbb\{C\}^n\; =\; \backslash mathbb\{C\}^\{2m\}$ one writes {{math|Sp(m, $\backslash mathbb\{C\}$)}} or {{math|Sp(2m, $\backslash mathbb\{C\}$)}}. The Lie algebra parallels that of {{math|sp(m, $\backslash mathbb\{R\}$)}},

:

and the group is given by

:$\backslash mathrm\{Sp\}(m,\; \backslash mathbb\{C\})\; =\; \backslash \{g\; \backslash in\; M\_n(\backslash mathbb\{C\})|g^\{\backslash mathrm\{T\}\}J\_mg\; =\; J\_m\backslash \}.$

In the sesquilinear case, one makes a slightly different approach for the form in terms of a basis,

:$\backslash varphi(x,\; y)\; =\; \backslash sum\; \backslash bar\{\backslash xi\}\_i\backslash varphi\_\{ij\}\backslash eta\_j.$

The other expressions that get modified are

:$\backslash varphi(x,\; y)\; =\; x^*\backslash Phi\; y,\; \backslash qquad\; A^\backslash varphi\; =\; \backslash Phi^\{-1\}A^*\backslash Phi,${{harvnb|Rossmann|2002}} p. 93

:$\backslash operatorname\{Aut\}(\backslash varphi)\; =\; \backslash \{A\; \backslash in\; \backslash operatorname\{GL\}(V):\; \backslash Phi^\{-1\}A^*\backslash Phi\; A\; =\; 1\backslash \},$

{{NumBlk|:|$\backslash mathfrak\{aut\}(\backslash varphi)\; =\; \backslash \{X\; \backslash in\; M\_n(V):\; \backslash Phi^\{-1\}X^*\backslash Phi\; =\; -X\backslash \}.$|{{EquationRef|6}}}}

The real case, of course, provides nothing new. The complex and the quaternionic case will be considered below.

From a qualitative point of view, consideration of skew-Hermitian forms (up to isomorphism) provide no new groups; multiplication by {{math|i}} renders a skew-Hermitian form Hermitian, and vice versa. Thus only the Hermitian case needs to be considered.

{{main|Unitary group}}

A non-degenerate hermitian form has the normal form

:$\backslash varphi(x,\; y)\; =\; \backslash pm\; \backslash bar\{\backslash xi\_1\}\backslash eta\_1\; \backslash pm\; \backslash bar\{\backslash xi\_2\}\backslash eta\_2\; \backslash cdots\; \backslash pm\; \backslash bar\{\backslash xi\_n\}\backslash eta\_n.$

As in the bilinear case, the signature (p, q) is independent of the basis. The automorphism group is denoted {{math|U(V)}}, or, in the case of {{math|V {{=}} Cⁿ}}, {{math|U(p, q)}}. If {{math|q {{=}} 0}} the notation is {{math|U(n)}}. In this case, {{math|Φ}} takes the form

:

and the Lie algebra is given by

:

The group is given by

:$\backslash mathrm\{U\}(p,\; q)\; =\; \backslash \{g|I\_\{p,q\}^\{-1\}g^*I\_\{p,q\}g\; =\; I\backslash \}.$

:where g is a general n x n complex matrix and $g^\{*\}$ is defined as the conjugate transpose of g, what physicists call $g^\{\backslash dagger\}$.

As a comparison, a Unitary matrix U(n) is defined as

$\backslash mathrm\{U\}(n)\; =\; \backslash \{g|g^*g\; =\; I\backslash \}.$

We note that $\backslash mathrm\{U\}(n)$ is the same as $\backslash mathrm\{U\}(n,0)$

The space {{math|Hⁿ}} is considered as a right vector space over {{math|H}}. This way, {{math|A(vh) {{=}} (Av)h}} for a quaternion {{math|h}}, a quaternion column vector {{math|v}} and quaternion matrix {{math|A}}. If {{math|Hⁿ}} were a left vector space over {{math|H}}, then matrix multiplication from the right on row vectors would be required to maintain linearity. This does not correspond to the usual linear operation of a group on a vector space when a basis is given, which is matrix multiplication from the left on column vectors. Thus {{math|V}} is henceforth a right vector space over {{math|H}}. Even so, care must be taken due to the non-commutative nature of {{math|H}}. The (mostly obvious) details are skipped because complex representations will be used.

When dealing with quaternionic groups it is convenient to represent quaternions using complex {{nowrap|2×2-matrices}},

{{NumBlk|:|{{harvnb|Rossmann|2002}} p. 95.|{{EquationRef|7}}}}

With this representation, quaternionic multiplication becomes matrix multiplication and quaternionic conjugation becomes taking the Hermitian adjoint. Moreover, if a quaternion according to the complex encoding {{math|q {{=}} x + jy}} is given as a column vector {{math|(x, y)^T}}, then multiplication from the left by a matrix representation of a quaternion produces a new column vector representing the correct quaternion. This representation differs slightly from a more common representation found in the quaternion article. The more common convention would force multiplication from the right on a row matrix to achieve the same thing.

Incidentally, the representation above makes it clear that the group of unit quaternions ({{math|α{{overline|α}} + β{{overline|β}} {{=}} 1 {{=}} det Q}}) is isomorphic to {{math|SU(2)}}.

Quaternionic {{math|n×n}}-matrices can, by obvious extension, be represented by {{math|2n×2n}} block-matrices of complex numbers.{{harvnb|Rossmann|2002}} p. 94. If one agrees to represent a quaternionic {{nowrap|n×1}} column vector by a {{nowrap|2n×1}} column vector with complex numbers according to the encoding of above, with the upper {{math|n}} numbers being the {{math|α_i}} and the lower {{math|n}} the {{math|β_i}}, then a quaternionic {{math|n×n}}-matrix becomes a complex {{math|2n×2n}}-matrix exactly of the form given above, but now with α and β {{math|n×n}}-matrices. More formally

{{NumBlk|:||{{EquationRef|8}}}}

A matrix {{math|T ∈ GL(2n, C)}} has the form displayed in ({{EquationNote|8}}) if and only if {{math|J_n{{overline|T}} {{=}} TJ_n}}. With these identifications,

:

The space {{math|M_n(H) ⊂ M_2n(C)}} is a real algebra, but it is not a complex subspace of {{math|M_2n(C)}}. Multiplication (from the left) by {{math|i}} in {{math|M_n(H)}} using entry-wise quaternionic multiplication and then mapping to the image in {{math|M_2n(C)}} yields a different result than multiplying entry-wise by {{math|i}} directly in {{math|M_2n(C)}}. The quaternionic multiplication rules give {{math|i(X + jY) {{=}} (iX) + j(−iY)}} where the new {{math|X}} and {{math|Y}} are inside the parentheses.

The action of the quaternionic matrices on quaternionic vectors is now represented by complex quantities, but otherwise it is the same as for "ordinary" matrices and vectors. The quaternionic groups are thus embedded in {{math|M_2n(C)}} where {{math|n}} is the dimension of the quaternionic matrices.

The determinant of a quaternionic matrix is defined in this representation as being the ordinary complex determinant of its representative matrix. The non-commutative nature of quaternionic multiplication would, in the quaternionic representation of matrices, be ambiguous. The way {{math|M_n(H)}} is embedded in {{math|M_2n(C)}} is not unique, but all such embeddings are related through {{math|g ↦ AgA⁻¹, g ∈ GL(2n, C)}} for {{math|A ∈ O(2n, C)}}, leaving the determinant unaffected.{{harvnb|Goodman|Wallach|2009}} Exercise 14, Section 1.1. The name of {{math|SL(n, H)}} in this complex guise is {{math|SU^∗(2n)}}.

As opposed to in the case of {{math|C}}, both the Hermitian and the skew-Hermitian case bring in something new when {{math|H}} is considered, so these cases are considered separately.

Under the identification above,

:$\backslash mathrm\{GL\}(n,\; \backslash mathbb\{H\})\; =\; \backslash \{g\; \backslash in\; \backslash mathrm\{GL\}(2n,\; \backslash mathbb\{C\})|Jg\; =\; \backslash overline\{g\}J,\; \backslash mathrm\{det\}\backslash quad\; g\; \backslash ne\; 0\backslash \}\; \backslash equiv\; \backslash mathrm\{U\}^*(2n).$

Its Lie algebra {{math|gl(n, H)}} is the set of all matrices in the image of the mapping {{math|M_n(H) ↔ M_2n(C)}} of above,

:

The quaternionic special linear group is given by

:$\backslash mathrm\{SL\}(n,\; \backslash mathbb\{H\})\; =\; \backslash \{g\; \backslash in\; \backslash mathrm\{GL\}(n,\; \backslash mathbb\{H\})|\backslash mathrm\{det\}\backslash \; g\; =\; 1\backslash \}\; \backslash equiv\; \backslash mathrm\{SU\}^*(2n),$

where the determinant is taken on the matrices in {{math|C²ⁿ}}. Alternatively, one can define this as the kernel of the Dieudonné determinant $\backslash mathrm\{GL\}(n,\; \backslash mathbb\{H\})\; \backslash rightarrow\; \backslash mathbb\; H^*/[\backslash mathbb\; H^*,\; \backslash mathbb\; H^*]\; \backslash simeq\; \backslash mathbb\{R\}\_\{>\; 0\}^*$. The Lie algebra is

:

As above in the complex case, the normal form is

:$\backslash varphi(x,\; y)\; =\; \backslash pm\; \backslash bar\{\backslash xi\_1\}\backslash eta\_1\; \backslash pm\; \backslash bar\{\backslash xi\_2\}\backslash eta\_2\; \backslash cdots\; \backslash pm\; \backslash bar\{\backslash xi\_n\}\backslash eta\_n$

and the number of plus-signs is independent of basis. When {{math|V {{=}} Hⁿ}} with this form, {{math|Sp(φ) {{=}} Sp(p, q)}}. The reason for the notation is that the group can be represented, using the above prescription, as a subgroup of {{math|Sp(n, C)}} preserving a complex-hermitian form of signature {{math|(2p, 2q)}}{{harvnb|Rossmann|2002}} p. 94. If {{math|p}} or {{math|q {{=}} 0}} the group is denoted {{math|U(n, H)}}. It is sometimes called the hyperunitary group.

In quaternionic notation,

:

meaning that quaternionic matrices of the form

{{NumBlk|:|$$

\mathcal{Q} = \begin{pmatrix} \mathcal{X}_{p \times p} & \mathcal{Z}_{p \times q} \\ \mathcal{Z}^* & \mathcal{Y}_{q \times q} \end{pmatrix}, \quad

\mathcal{X}^* = -\mathcal{X}, \quad

\mathcal{Y}^* = -\mathcal{Y}

|{{EquationRef|9}}}}

will satisfy

:$\backslash Phi^\{-1\}\backslash mathcal\{Q\}^*\backslash Phi\; =\; -\backslash mathcal\{Q\},$

see the section about {{math|u(p, q)}}. Caution needs to be exercised when dealing with quaternionic matrix multiplication, but here only {{math|I}} and {{math|-I}} are involved and these commute with every quaternion matrix. Now apply prescription ({{EquationNote|8}}) to each block,

:$$

\mathcal{X} = \begin{pmatrix} X_{1 (p \times p)} & -\overline{X}_2 \\ X_2 & \overline{X}_1 \end{pmatrix}, \quad

\mathcal{Y} = \begin{pmatrix} Y_{1 (q \times q)} & -\overline{Y}_2 \\ Y_2 & \overline{Y}_1 \end{pmatrix}, \quad

\mathcal{Z} = \begin{pmatrix} Z_{1 (p \times q)} & -\overline{Z}_2 \\ Z_2 & \overline{Z}_1 \end{pmatrix},

and the relations in ({{EquationNote|9}}) will be satisfied if

:$X\_1^*\; =\; -X\_1,\; \backslash quad\; Y\_1^*\; =\; -Y\_1.$

The Lie algebra becomes

:$$

\mathfrak{sp}(p, q) = \left\{\left.

\begin{pmatrix}

\begin{bmatrix} X_{1 (p \times p)} & -\overline{X}_2 \\ X_2 & \overline{X}_1 \end{bmatrix} &

\begin{bmatrix} Z_{1 (p \times q)} & -\overline{Z}_2 \\ Z_2 & \overline{Z}_1 \end{bmatrix} \\

\begin{bmatrix} Z_{1 (p \times q)} & -\overline{Z}_2 \\ Z_2 & \overline{Z}_1 \end{bmatrix}^* &

\begin{bmatrix} Y_{1 (q \times q)} & -\overline{Y}_2 \\ Y_2 & \overline{Y}_1 \end{bmatrix}

\end{pmatrix}

\right|

X_1^* = -X_1,\quad Y_1^* = -Y_1

\right\}.

The group is given by

:$$

\mathrm{Sp}(p, q) =

\left\{g \in \mathrm{GL}( n, \mathbb{H}) \mid I_{p,q}^{-1} g^* I_{p,q}g = I_{p + q}\right\} =

\left\{g \in \mathrm{GL}(2n, \mathbb{C}) \mid K_{p,q}^{-1} g^* K_{p,q}g = I_{2(p + q)},\quad

K = \operatorname{diag}\left(I_{p,q}, I_{p,q}\right)\right\}.

Returning to the normal form of {{math|φ(w, z)}} for {{math|Sp(p, q)}}, make the substitutions {{math|w → u + jv}} and {{math|z → x + jy}} with {{math|u, v, x, y ∈ Cⁿ}}. Then

:$$

\varphi(w, z) =

\begin{bmatrix} u^* & v^* \end{bmatrix}K_{p, q}\begin{bmatrix} x \\ y \end{bmatrix} +

j\begin{bmatrix} u & -v \end{bmatrix}K_{p, q}\begin{bmatrix} y \\ x \end{bmatrix} =

\varphi_1(w, z) + \mathbf{j}\varphi_2(w, z), \quad

K_{p, q} = \mathrm{diag}\left(I_{p, q}, I_{p, q}\right)

viewed as a {{math|H}}-valued form on {{math|C²ⁿ}}.{{harvnb|Goodman|Wallach|2009}}Exercise 11, Chapter 1. Thus the elements of {{math|Sp(p, q)}}, viewed as linear transformations of {{math|C²ⁿ}}, preserve both a Hermitian form of signature {{math|(2p, 2q)}} and a non-degenerate skew-symmetric form. Both forms take purely complex values and due to the prefactor of {{math|j}} of the second form, they are separately conserved. This means that

:$\backslash mathrm\{Sp\}(p,\; q)\; =\; \backslash mathrm\{U\}\backslash left(\backslash mathbb\{C\}^\{2n\},\; \backslash varphi\_1\backslash right)\; \backslash cap\; \backslash mathrm\{Sp\}\backslash left(\backslash mathbb\{C\}^\{2n\},\; \backslash varphi\_2\backslash right)$

and this explains both the name of the group and the notation.

The normal form for a skew-hermitian form is given by

:$\backslash varphi(x,\; y)\; =\; \backslash bar\{\backslash xi\_1\}\backslash mathbf\{j\}\backslash eta\_1\; +\; \backslash bar\{\backslash xi\_2\}\backslash mathbf\{j\}\backslash eta\_2\; \backslash cdots\; +\; \backslash bar\{\backslash xi\_n\}\backslash mathbf\{j\}\backslash eta\_n,$

where {{math|j}} is the third basis quaternion in the ordered listing {{math|(1, i, j, k)}}. In this case, {{math|Aut(φ) {{=}} O^∗(2n)}} may be realized, using the complex matrix encoding of above, as a subgroup of {{math|O(2n, C)}} which preserves a non-degenerate complex skew-hermitian form of signature {{math|(n, n)}}.{{harvnb|Rossmann|2002}} p. 94. From the normal form one sees that in quaternionic notation

:$\backslash Phi\; =$

\left(\begin{smallmatrix}

\mathbf{j} & 0 & \cdots & 0 \\ 0 & \mathbf{j} & \cdots & \vdots \\

\vdots & & \ddots & & \\ 0 & \cdots & 0 & \mathbf{j}

\end{smallmatrix}\right) \equiv

\mathrm{j}_n

and from ({{EquationNote|6}}) follows that

{{NumBlk|:|$-\backslash Phi\; V^*\backslash Phi\; =\; -V\; \backslash Leftrightarrow\; V^*\; =\; \backslash mathrm\{j\}\_n\; V\backslash mathrm\{j\}\_n.$|{{EquationRef|9}}}}

for {{math|V ∈ o(2n)}}. Now put

:

according to prescription ({{EquationNote|8}}). The same prescription yields for {{math|Φ}},

:

Now the last condition in ({{EquationNote|9}}) in complex notation reads

:$$

\left(\begin{matrix} X & -\overline{Y} \\ Y & \overline{X} \end{matrix}\right)^* =

\left(\begin{matrix} 0 & -I_n \\ I_n & 0 \end{matrix}\right)

\left(\begin{matrix} X & -\overline{Y} \\ Y & \overline{X} \end{matrix}\right)

\left(\begin{matrix} 0 & -I_n \\ I_n & 0 \end{matrix}\right) \Leftrightarrow

X^\mathrm{T} = -X, \quad \overline{Y}^\mathrm{T} = Y.

The Lie algebra becomes

:

and the group is given by

:$$

\mathrm{O}^*(2n) =

\left\{g \in \mathrm{GL}(n, \mathbb{H}) \mid \mathrm{j}_n^{-1}g^*\mathrm{j}_n g = I_n\right\} =

\left\{g \in \mathrm{GL}(2n, \mathbb{C}) \mid J_{n}^{-1}g^* J_n g = I_{2n}\right\}.

The group {{math|SO^∗(2n)}} can be characterized as

:$\backslash mathrm\{O\}^*(2n)\; =\; \backslash left\backslash \{g\; \backslash in\; \backslash mathrm\{O\}(2n,\; \backslash mathbb\{C\})\; \backslash mid\; \backslash theta\backslash left(\backslash overline\{g\}\backslash right)\; =\; g\backslash right\backslash \},${{harvnb|Goodman|Wallach|2009}} p.11.

where the map {{math|θ: GL(2n, C) → GL(2n, C)}} is defined by {{math|g ↦ −J_2ngJ_2n}}.

Also, the form determining the group can be viewed as a {{math|H}}-valued form on {{math|C²ⁿ}}.{{harvnb|Goodman|Wallach|2009}} Exercise 12 Chapter 1. Make the substitutions {{math|x → w₁ + iw₂}} and {{math|y → z₁ + iz₂}} in the expression for the form. Then

:$\backslash varphi(x,\; y)\; =\; \backslash overline\{w\}\_2\; I\_n\; z\_1\; -\; \backslash overline\{w\}\_1\; I\_n\; z\_2\; +\; \backslash mathbf\{j\}(w\_1\; I\_n\; z\_1\; +\; w\_2\; I\_n\; z\_2)\; =\; \backslash overline\{\backslash varphi\_1(w,\; z)\}\; +\; \backslash mathbf\{j\}\backslash varphi\_2(w,\; z).$

The form {{math|φ₁}} is Hermitian (while the first form on the left hand side is skew-Hermitian) of signature {{math|(n, n)}}. The signature is made evident by a change of basis from {{math|(e, f)}} to {{math|((e + if)/{{sqrt|2}}, (e − if)/{{sqrt|2}})}} where {{math|e, f}} are the first and last {{math|n}} basis vectors respectively. The second form, {{math|φ₂}} is symmetric positive definite. Thus, due to the factor {{math|j}}, {{math|O^∗(2n)}} preserves both separately and it may be concluded that

:$\backslash mathrm\{O\}^*(2n)\; =\; \backslash mathrm\{O\}(2n,\; \backslash mathbb\{C\})\; \backslash cap\; \backslash mathrm\{U\}\backslash left(\backslash mathbb\{C\}^\{2n\},\; \backslash varphi\_1\backslash right),$

and the notation "O" is explained.

Classical groups, more broadly considered in algebra, provide particularly interesting matrix groups. When the field F of coefficients of the matrix group is either real number or complex numbers, these groups are just the classical Lie groups. When the ground field is a finite field, then the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups. Also, one may consider classical groups over a unital associative algebra R over F; where R = H (an algebra over reals) represents an important case. For the sake of generality the article will refer to groups over R, where R may be the ground field F itself.

Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 over the ground field, and most of them have associated "projective" quotients, which are the quotients by the center of the group. For orthogonal groups in characteristic 2 "S" has a different meaning.

The word "general" in front of a group name usually means that the group is allowed to multiply some sort of form by a constant, rather than leaving it fixed. The subscript n usually indicates the dimension of the module on which the group is acting; it is a vector space if R = F. Caveat: this notation clashes somewhat with the n of Dynkin diagrams, which is the rank.

The general linear group GL_n(R) is the group of all R-linear automorphisms of Rⁿ. There is a subgroup: the special linear group SL_n(R), and their quotients: the projective general linear group PGL_n(R) = GL_n(R)/Z(GL_n(R)) and the projective special linear group PSL_n(R) = SL_n(R)/Z(SL_n(R)). The projective special linear group PSL_n(F) over a field F is simple for n ≥ 2, except for the two cases when n = 2 and the field has order{{clarify|date=February 2014}} 2 or 3.

The unitary group U_n(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SU_n(R) and their quotients the projective unitary group PU_n(R) = U_n(R)/Z(U_n(R)) and the projective special unitary group PSU_n(R) = SU_n(R)/Z(SU_n(R))

The symplectic group Sp_2n(R) preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp_2n(R). The general symplectic group GSp_2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp_2n(F_q) over a finite field is simple for n ≥ 1, except for the cases of PSp₂ over the fields of two and three elements.

The orthogonal group O_n(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SO_n(R) and quotients, the projective orthogonal group PO_n(R), and the projective special orthogonal group PSO_n(R). In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1.

There is a nameless group often denoted by Ω_n(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩ_n(R), PΩ_n(R), PSΩ_n(R). (For positive definite quadratic forms over the reals, the group Ω happens to be the same as the orthogonal group, but in general it is smaller.) There is also a double cover of Ω_n(R), called the pin group Pin_n(R), and it has a subgroup called the spin group Spin_n(R). The general orthogonal group GO_n(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.

{{details|Group of Lie type#Notation issues}}

Contrasting with the classical Lie groups are the exceptional Lie groups, G₂, F₄, E₆, E₇, E₈, which share their abstract properties, but not their familiarity.Wybourne, B. G. (1974). Classical Groups for Physicists, Wiley-Interscience. {{ISBN|0471965057}}. These were only discovered around 1890 in the classification of the simple Lie algebras over the complex numbers by Wilhelm Killing and Élie Cartan.

{{reflist|2}}

• E. Artin (1957) [https://archive.org/details/geometricalgebra033556mbp/page/n115/mode/2up?view=theater Geometric Algebra, chapters III, IV, & V] via Internet Archive
• {{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=La géométrie des groupes classiques | url=https://books.google.com/books?id=AfYZAQAAIAAJ | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 5 | isbn=978-0-387-05391-2 | mr=0072144 | year=1955}}
• {{citation|last1=Goodman|first1=Roe|last2=Wallach|first2=Nolan R.|title=Symmetry, Representations, and Invariants|year=2009|publisher=Springer-Verlag|series=Graduate texts in mathematics|volume=255|isbn=978-0-387-79851-6}}
• {{cite book|authorlink=A. W. Knapp|last=Knapp|first=A. W.|title=Lie groups beyond an introduction|isbn=0-8176-4259-5|publisher=Birkhäuser|series=Progress in Mathematics|volume=120|edition=2nd|year=2002|location=Boston·Basel·Berlin}}
• {{Springer |id=C/c022410|title=Classical group|author=V. L. Popov}}
• {{citation|last=Rossmann|first= Wulf|title=Lie Groups - An Introduction Through Linear Groups|publisher=Oxford Science Publications|year=2002|series=Oxford Graduate Texts in Mathematics|isbn=0-19-859683-9|postscript=}}
• {{Citation |last1=Weyl |first1=Hermann |author1-link=Hermann Weyl |title=The Classical Groups. Their Invariants and Representations |url=https://books.google.com/books?isbn=0691057567 |publisher=Princeton University Press |isbn=978-0-691-05756-9 |mr=0000255 |year=1939}}

Category:Lie groups