Kac–Moody algebra

In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. Kac–Moody algebras are named after Victor Kac and Robert Moody, who independently discovered them.These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to structure of the Lie algebra, its root system, irreducible representations, connection to flag manifolds have natural analogues in the Kac-Moody setting. A class of Kac-Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models. 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 that Rogers-Ramanujan identities can be derived in a similar fashion.

Definition

A Kac–Moody algebra is given by the following:
# An n by n generalized Cartan matrix $C\; =\; (c\_\{ij\})$ of rank "r".
# A vector space $mathfrak\{h\}$ over the complex numbers 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 the dual 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 generators $e\_i$ 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 its complexification 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 eigenvectors. The Cartan subalgebra "h" has weight zero, "e"_"i" has weight α*_"i" and "f"_"i" has weight −α*_"i". If the Lie 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" are simple roots.

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 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. 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