Kac–Moody algebra

The initial construction by Élie Cartan and Wilhelm Killing of finite dimensional simple Lie algebras from the Cartan integers was type dependent. In 1966 Jean-Pierre Serre showed that relations of Claude Chevalley and Harish-Chandra,{{cite journal |last=Harish-Chandra |title=On some applications of the universal enveloping algebra of a semisimple Lie algebra |journal=Trans. Amer. Math. Soc. |volume=70 |issue=1 |year=1951 |pages=28–96 |doi= 10.1090/S0002-9947-1951-0044515-0|jstor=1990524 |doi-access=free }} with simplifications by Nathan Jacobson,{{cite book |last=Jacobson |first=N. |title=Lie algebras |series=Interscience Tracts in Pure and Applied Mathematics |volume=10 |publisher=Interscience Publishers (a division of John Wiley & Sons) |location=New York-London |year=1962 }} give a defining presentation for the Lie algebra.{{cite book |last=Serre |first=J.-P. |title=Algèbres de Lie semi-simples complexes |language=fr |publisher=W. A. Benjamin |location=New York-Amsterdam |year=1966 }} One could thus describe a simple Lie algebra in terms of generators and relations using data from the matrix of Cartan integers, which is naturally positive definite.

"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." – A. J. ColemanColeman, A. John, "The Greatest Mathematical Paper of All Time," [https://www.math.umd.edu/~jda/744/coleman.pdf The Mathematical Intelligencer], vol. 11, no. 3, pp. 29–38.

In his 1967 thesis, Robert Moody considered Lie algebras whose Cartan matrix is no longer positive definite.{{cite journal |last=Moody |first=R. V. |title=Lie algebras associated with generalized cartan matrices |journal=Bull. Amer. Math. Soc. |volume=73 |issue= 2|year=1967 |pages=217–222 |url=https://www.ams.org/journals/bull/1967-73-02/S0002-9904-1967-11688-4/S0002-9904-1967-11688-4.pdf |doi=10.1090/S0002-9904-1967-11688-4 |doi-access=free }}Moody 1968, A new class of Lie algebras This still gave rise to a Lie algebra, but one which is now infinite dimensional. Simultaneously, Z-graded Lie algebras were being studied in Moscow where I. L. Kantor introduced and studied a general class of Lie algebras including what eventually became known as Kac–Moody algebras.{{cite journal |last=Kantor |first=I. L. |title=Graded Lie algebras |language=ru |journal=Trudy Sem. Vektor. Tenzor. Anal. |volume=15 |year=1970 |pages=227–266 }} Victor Kac was also studying simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite dimensional Lie algebras evolved. An account of the subject, which also includes works of many others is given in (Kac 1990).Kac, 1990 See also (Seligman 1987).{{cite journal |last=Seligman |first=George B. |title=Book Review: Infinite dimensional Lie algebras |journal=Bull. Amer. Math. Soc. |series=N.S. |volume=16 |year=1987 |issue=1 |pages=144–150 |doi=10.1090/S0273-0979-1987-15492-9 |doi-access=free }}

Given an n×n generalized Cartan matrix $C\; =\; \backslash begin\{pmatrix\}\; c\_\{ij\}\backslash end\{pmatrix\}$, one can construct a Lie algebra $\backslash mathfrak\{g\}\text{'}(C)$ defined by generators $e\_i$, $h\_i$, and $f\_i\; \backslash left(i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}\backslash right)$ and relations given by:

• $\backslash left[h\_i,\; h\_j\backslash right]\; =\; 0\backslash$ for all $i,\; j\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}$;
• $\backslash left[h\_i,\; e\_j\backslash right]\; =\; c\_\{ij\}e\_j$;
• $\backslash left[h\_i,\; f\_j\backslash right]\; =\; -c\_\{ij\}f\_j$;
• $\backslash left[e\_i,\; f\_j\backslash right]\; =\; \backslash delta\_\{ij\}h\_i$, where $\backslash delta\_\{ij\}$ is the Kronecker delta;
• If $i\; \backslash neq\; j$ (so $c\_\{ij\}\; \backslash leq\; 0$) then $\backslash textrm\{ad\}(e\_i)^\{1-c\_\{ij\}\}(e\_j)\; =\; 0$ and $\backslash operatorname\{ad\}(f\_i)^\{1-c\_\{ij\}\}(f\_j)\; =\; 0$, where $\backslash operatorname\{ad\}:\; \backslash mathfrak\{g\}\backslash to\backslash operatorname\{End\}(\backslash mathfrak\{g\}),\backslash operatorname\{ad\}(x)(y)\; =\; [x,\; y],$ is the adjoint representation of $\backslash mathfrak\{g\}$.

Under a "symmetrizability" assumption, $\backslash mathfrak\{g\}\text{'}(C)$ identifies with the derived subalgebra $\backslash mathfrak\{g\}\text{'}(C)\; =\; [\backslash mathfrak\{g\}(C),\; \backslash mathfrak\{g\}(C)]$ of the affine Kac-Moody algebra $\backslash mathfrak\{g\}(C)$ defined below.Kac 1990, Infinite dimensional Lie Algebras, Third Edition

Assume we are given an $n\; \backslash times\; n$ generalized Cartan matrix {{nowrap|1=C = (c_ij)}} of rank r. For every such $C$, there exists a unique up to isomorphism realization of $C$, i.e. a triple $(\backslash mathfrak\{h\},\; \backslash \{\backslash alpha\_i\backslash \}\_\{i\; =\; 1\}^n,\; \backslash \{\backslash alpha\_i^\backslash vee\backslash \}\_\{i\; =\; 1\}^n,$) where $\backslash mathfrak\{h\}$ is a complex vector space, $\backslash \{\backslash alpha\_i^\backslash vee\backslash \}\_\{i\; =\; 1\}^n$ is a subset of elements of $\backslash mathfrak\{h\}$, and $\backslash \{\backslash alpha\_i\backslash \}\_\{i\; =\; 1\}^n$ is a subset of the dual space $\backslash mathfrak\{h\}^*$ satisfying the following three conditions:Kac 1990, Infinite dimensional Lie algebras, Proposition 1.1

1. The vector space $\backslash mathfrak\{h\}$ has dimension 2n − r
2. The sets $\backslash \{\backslash alpha\_i\backslash \}\_\{i\; =\; 1\}^n$ and $\backslash \{\backslash alpha\_i^\backslash vee\backslash \}\_\{i\; =\; 1\}^n$ are linearly independent and
3. For every $1\; \backslash leq\; i,\; j\; \backslash leq\; n,\; \backslash alpha\_i\backslash left(\backslash alpha\_j^\backslash vee\backslash right)\; =\; c\_\{ji\}$.

The $\backslash alpha\_i$ are analogue to the simple roots of a semi-simple Lie algebra, and the $\backslash alpha\_i^\backslash vee$ to the simple coroots.

Then we define the Kac-Moody algebra associated to $C$ as the Lie algebra $\backslash mathfrak\{g\}\; :=\; \backslash mathfrak\{g\}(C)$ defined by generators $e\_i$ and $f\_i\; \backslash left(i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}\backslash right)$ and the elements of $\backslash mathfrak\{h\}$ and relations

• $\backslash left[h,\; h\text{'}\backslash right]\; =\; 0\backslash$ for $h,h\text{'}\; \backslash in\; \backslash mathfrak\{h\}$;
• $\backslash left[h,\; e\_i\backslash right]\; =\; \backslash alpha\_i(h)e\_i$, for $h\; \backslash in\; \backslash mathfrak\{h\}$;
• $\backslash left[h,\; f\_i\backslash right]\; =\; -\backslash alpha\_i(h)f\_i$, for $h\; \backslash in\; \backslash mathfrak\{h\}$;
• $\backslash left[e\_i,\; f\_j\backslash right]\; =\; \backslash delta\_\{ij\}\backslash alpha\_i^\backslash vee$, where $\backslash delta\_\{ij\}$ is the Kronecker delta;
• If $i\; \backslash neq\; j$ (so $c\_\{ij\}\; \backslash leq\; 0$) then $\backslash textrm\{ad\}(e\_i)^\{1-c\_\{ij\}\}(e\_j)\; =\; 0$ and $\backslash operatorname\{ad\}(f\_i)^\{1-c\_\{ij\}\}(f\_j)\; =\; 0$, where $\backslash operatorname\{ad\}:\; \backslash mathfrak\{g\}\backslash to\backslash operatorname\{End\}(\backslash mathfrak\{g\}),\backslash operatorname\{ad\}(x)(y)\; =\; [x,\; y],$ is the adjoint representation of $\backslash mathfrak\{g\}$.

A real (possibly infinite-dimensional) Lie algebra is also considered a Kac–Moody algebra if its complexification is a Kac–Moody algebra.

$\backslash mathfrak\{h\}$ is the analogue of a Cartan subalgebra for the Kac–Moody algebra $\backslash mathfrak\{g\}$.

If $x\backslash neq\; 0$ is an element of $\backslash mathfrak\{g\}$ such that

:$\backslash forall\; h\backslash in\backslash mathfrak\{h\},\; [h,\; x]\; =\; \backslash lambda(h)x$

for some $\backslash lambda\backslash in\backslash mathfrak\{h\}^*\backslash backslash\backslash \{0\backslash \}$, then $x$ is called a root vector and $\backslash lambda$ is a root of $\backslash mathfrak\{g\}$. (The zero functional is not considered a root by convention.) The set of all roots of $\backslash mathfrak\{g\}$ is often denoted by $\backslash Delta$ and sometimes by $R$. For a given root $\backslash lambda$, one denotes by $\backslash mathfrak\{g\}\_\backslash lambda$ the root space of $\backslash lambda$; that is,

:$\backslash mathfrak\{g\}\_\backslash lambda\; =\; \backslash \{x\backslash in\backslash mathfrak\{g\}:\backslash forall\; h\backslash in\backslash mathfrak\{h\},\; [h,x]\; =\; \backslash lambda(h)x\backslash \}$.

It follows from the defining relations of $\backslash mathfrak\{g\}$ that $e\_i\backslash in\backslash mathfrak\{g\}\_\{\backslash alpha\_i\}$ and $f\_i\backslash in\backslash mathfrak\{g\}\_\{-\backslash alpha\_i\}$. Also, if $x\_1\backslash in\backslash mathfrak\{g\}\_\{\backslash lambda\_1\}$ and $x\_2\backslash in\backslash mathfrak\{g\}\_\{\backslash lambda\_2\}$, then $\backslash left[x\_1,\; x\_2\backslash right]\backslash in\backslash mathfrak\{g\}\_\{\backslash lambda\_1+\backslash lambda\_2\}$ by the Jacobi identity.

A fundamental result of the theory is that any Kac–Moody algebra can be decomposed into the direct sum of $\backslash mathfrak\{h\}$ and its root spaces, that is

:$\backslash mathfrak\{g\}\; =\; \backslash mathfrak\{h\}\backslash oplus\backslash bigoplus\_\{\backslash lambda\backslash in\backslash Delta\}\; \backslash mathfrak\{g\}\_\backslash lambda$,

and that every root $\backslash lambda$ can be written as $\backslash lambda\; =\; \backslash sum\_\{i=1\}^n\; z\_i\backslash alpha\_i$ with all the $z\_i$ being integers of the same sign.

Properties of a Kac–Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix C. In order to classify Kac–Moody algebras, it is enough to consider the case of an indecomposable matrix C, that is, assume that there is no decomposition of the set of indices I into a disjoint union of non-empty subsets I₁ and I₂ such that C_ij = 0 for all i in I₁ and j in I₂. Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra:

: $\backslash mathfrak\{g\}(C)\; \backslash simeq\; \backslash mathfrak\{g\}\backslash left(C\_1\backslash right)\; \backslash oplus\; \backslash mathfrak\{g\}\backslash left(C\_2\backslash right),$

where the two Kac–Moody algebras in the right hand side are associated with the submatrices of C corresponding to the index sets I₁ and I₂.

An important subclass of Kac–Moody algebras corresponds to symmetrizable generalized Cartan matrices C, which can be decomposed as DS, where D is a diagonal matrix with positive integer entries and S is a symmetric matrix. Under the assumptions that C is symmetrizable and indecomposable, the Kac–Moody algebras are divided into three classes:

• A positive definite matrix S gives rise to a finite-dimensional simple Lie algebra.
• A positive semidefinite matrix S gives rise to an infinite-dimensional Kac–Moody algebra of affine type, or an affine Lie algebra.
• An indefinite matrix S gives rise to a Kac–Moody algebra of indefinite type.
• Since the diagonal entries of C and S are positive, S cannot be negative definite or negative semidefinite.

Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified. They correspond to Dynkin diagrams and affine Dynkin diagrams. Little is known about the Kac–Moody algebras of indefinite type, although the groups corresponding to these Kac–Moody algebras were constructed over arbitrary fields by Jacques Tits.{{cite journal |last=Tits |first=J. |title =Uniqueness and presentation of Kac–Moody groups over fields|journal=Journal of Algebra |volume=105 |issue=2 |pages=542–573 |year=1987 |doi=10.1016/0021-8693(87)90214-6|doi-access=free }}

Among the Kac–Moody algebras of indefinite type, most work has focused on those hyperbolic type, for which the matrix S is indefinite, but for each proper subset of I, the corresponding submatrix is positive definite or positive semidefinite. Hyperbolic Kac–Moody algebras have rank at most 10, and they have been completely classified.{{cite journal |last1=Carbone |first1=L. |last2=Chung |first2=S. |last3=Cobbs |first3=C. |last4=McRae |first4=R. |last5=Nandi |first5=D. |last6=Naqvi |first6=Y. |last7=Penta |first7=D. |title=Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits |journal=J. Phys. A: Math. Theor. |volume=43 |issue=15 |pages=155–209 |year=2010 |doi=10.1088/1751-8113/43/15/155209 |arxiv=1003.0564 |bibcode=2010JPhA...43o5209C |s2cid=16946456 }} There are infinitely many of rank 2, and 238 of ranks between 3 and 10.

• Weyl–Kac character formula
• Generalized Kac–Moody algebra
• Integrable module
• Monstrous moonshine

{{Reflist}}

{{refbegin|2}}

• {{cite journal |last1=Berman |first1=Stephen |last2=Parshall |first2=Karen Hunger |date=13 January 2002 |title=Victor Kac and Robert Moody: Their Paths to Kac-Moody Lie Algebras |journal=The Mathematical Intelligencer |volume=24 |pages=50–60 |doi=10.1007/BF03025312|s2cid=120670625 }}
• Robert V. Moody, A new class of Lie algebras, Journal of Algebra, 10 (1968), 211–230. {{doi|10.1016/0021-8693(68)90096-3}} {{MR|0229687}}
• Victor Kac, Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press (1990) {{ISBN|0-521-46693-8}} [https://books.google.com/books?id=kuEjSb9teJwC&dq=Victor%20G.%20Kac&pg=PP1]
• Antony Wassermann, [https://arxiv.org/abs/1004.1287 Lecture notes on Kac–Moody and Virasoro algebras]
• {{springer|id=K/k055050|author=|title=Kac–Moody algebra}}
• Victor G. Kac, Simple irreducible graded Lie algebras of finite growth Math. USSR Izv., 2 (1968) pp. 1271–1311, Izv. Akad. Nauk USSR Ser. Mat., 32 (1968) pp. 1923–1967
• {{cite book|editor-last1=Sthanumoorthy|editor-first1=N.|editor-last2=Misra|editor-first2=Kailash C.|title=Kac-Moody Lie Algebras and Related Topics|date=2004|publisher=AMS|isbn=0-8218-3337-5}}
• Shrawan Kumar, Kac–Moody Groups, their Flag Varieties and Representation Theory, 1st edition, Birkhäuser (2002). {{ISBN|3-7643-4227-7}}.
• {{cite book|last=Zhe-xian|first=Wan|title=Introduction to Kac-Moody Algebra|date=1991|publisher=World Scientific|isbn=981-02-0224-5}}

{{refend}}

• [http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html SIGMA: Special Issue on Kac–Moody Algebras and Applications]

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Category:Lie algebras

Category:Moonshine theory