- Kac–Moody algebra
In

mathematics , a**Kac–Moody algebra**is aLie algebra , usually infinite-dimensional, that can be defined by generators and relations through ageneralized Cartan matrix . Kac–Moody algebras are named afterVictor Kac andRobert Moody , who independently discovered them.These algebras form a generalization of finite-dimensionalsemisimple Lie algebra s, and many properties related to structure of the Lie algebra, itsroot system , irreducible representations, connection toflag manifold s have natural analogues in the Kac-Moody setting. A class of Kac-Moody algebras calledis of particular importance in mathematics andaffine Lie algebra stheoretical physics , especiallyconformal field theory and the theory ofexactly solvable model s. Kac discovered an elegant proof of certain combinatorial identities,Macdonald identities , which is based on the representation theory of affine Kac-Moody algebras. Garland and Lepowski demonstrated thatRogers-Ramanujan identities can be derived in a similar fashion.**Definition**A Kac–Moody algebra is given by the following:

# An n by ngeneralized Cartan matrix $C\; =\; (c\_\{ij\})$ of rank "r".

# Avector space $mathfrak\{h\}$ over thecomplex number s of dimension 2"n" − "r".

# A set of "n"linearly independent elements $alpha\_i$ of $mathfrak\{h\}$ and a set of "n" linearly independent elements $alpha\_i^*$ of thedual space , such that $alpha\_i^*(alpha\_j)\; =\; c\_\{ij\}$. The $alpha\_i$ are known as**coroots**, while the $alpha\_i^*$ are known as**roots**.The Kac–Moody algebra is the Lie algebra $mathfrak\{g\}$ defined by

generator s $e\_i$ and $f\_i$ and the elements of $mathfrak\{h\}$ and relations

* $[e\_i,f\_i]\; =\; alpha\_i.$

* $[e\_i,f\_j]\; =\; 0$ for $i\; eq\; j.$

* $[e\_i,x]\; =alpha\_i^*(x)e\_i$, for $x\; in\; mathfrak\{h\}.$

* $[f\_i,x]\; =-alpha\_i^*(x)f\_i$, for $x\; in\; mathfrak\{h\}.$

* $[x,x\text{'}]\; =\; 0$ for $x,x\text{'}\; in\; mathfrak\{h\}.$

* $extrm\{ad\}(e\_i)^\{1-c\_\{ij(e\_j)\; =\; 0.$

* $extrm\{ad\}(f\_i)^\{1-c\_\{ij(f\_j)\; =\; 0.$Where $extrm\{ad\}:\; mathfrak\{g\}\; o\; extrm\{End\}(mathfrak\{g\}),\; extrm\{ad\}(x)(y)=\; [x,y]$ is the adjoint representation of $mathfrak\{g\}$.A real (possibly infinite-dimensional)

Lie algebra is also considered a Kac–Moody algebra if itscomplexification is a Kac–Moody algebra.**Interpretation**$mathfrak\{h\}$ is a

Cartan subalgebra of the Kac–Moody algebra.If "g" is an element of the Kac–Moody algebra such that

:$forall\; xin\; mathfrak\{h\},\; [g,x]\; =omega(x)g$

where ω is an element of $mathfrak\{h\}^*$, then "g" is said to have weight ω. The Kac–Moody algebra can be diagonalized into weight

eigenvector s. The Cartan subalgebra "h" has weight zero, "e"_{"i"}has weight α*_{"i"}and "f"_{"i"}has weight −α*_{"i"}. If theLie bracket of two weight eigenvectors is nonzero, then its weight is the sum of their weights. The condition $[e\_i,f\_j]\; =\; 0$ for $i\; eq\; j$ simply means the α*_{"i"}aresimple root s.**Types of Kac–Moody algebras**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", i.e. assume that there is no decomposition of the set of indices "I" into a disjoint union of non-empty subsets "I"

_{1}and "I"_{2}such that "C"_{"ij"}= 0 for all "i" in "I"_{1}and "j" in "I"_{2}. Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra:: $mathfrak\{g\}(C)simeqmathfrak\{g\}(C\_1)oplusmathfrak\{g\}(C\_2)$,

where the two Kac–Moody algebras in the right hand side are associated with the submatrices of "C" corresponding to the index sets "I"

_{1}and "I"_{2}.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 asymmetric 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-dimensionalsimple Lie algebra .

* Apositive semidefinite matrix "S" gives rise to an infinite-dimensional Kac–Moody algebra of**affine type**, or anaffine Lie algebra .

* Anindefinite 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 diagram s andaffine Dynkin diagram s. Very little is known about the Kac–Moody algebras of indefinite type. Among those, the main focus has been on the (generalized) Kac–Moody algebras of**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. Such matrices have rank at most 10 and have also been completely determined.**References**

*A. J. Wassermann, [*http://iml.univ-mrs.fr/~wasserm/ Lecture Notes on the Kac-Moody and Virasoro algebras*]

*V. Kac "Infinite dimensional Lie algebras" ISBN 0521466938

*springer|id=K/k055050|author=|title=Kac–Moody algebra

*V.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

*R.V. Moody, "A new class of Lie algebras" J. of Algebra , 10 (1968) pp. 211–230**ee also***

Weyl–Kac character formula

*Generalized Kac–Moody algebra

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