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:

1. An n by n generalized Cartan matrix $C=(c_{ij})$ of rank r.
2. A vector space ${\mathfrak {h}}$ over the complex numbers of dimension 2n − r.
3. 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 $i\neq j.$
• $[e_{i},x]=\alpha _{i}^{*}(x)e_{i}$, for $x\in {\mathfrak {h}}.$
• $[f_{i},x]=-\alpha _{i}^{*}(x)f_{i}$, for $x\in {\mathfrak {h}}.$
• $[x,x']=0\$ for $x,x'\in {\mathfrak {h}}.$
• ${\textrm {ad}}(e_{i})^{1-c_{ij}}(e_{j})=0.$
• ${\textrm {ad}}(f_{i})^{1-c_{ij}}(f_{j})=0.$

Where ${\textrm {ad}}:{\mathfrak {g}}\to {\textrm {End}}({\mathfrak {g}}),{\textrm {ad}}(x)(y)=[x,y]$ 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

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

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 $i\neq j$ simply means the α*_i are simple roots.

Types of Kac–Moody algebras

C can be decomposed as DS where D is a positive diagonal matrix and S is a symmetric matrix.

• finite-dimensional simple Lie algebras (S is positive definite)
• affine (S is positive semidefinite)
• hyperbolic (S is indefinite)

S can never be negative definite or negative semidefinite because its diagonal entries are positive.

References

• A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras
• V. Kac Infinite dimensional Lie algebras ISBN 0521466938
• "Kac–Moody algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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

See also

• Weyl–Kac character formula
• Generalized Kac–Moody algebra