Kac–Moody algebra

From Wikipedia, the free encyclopedia

In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. Kac–Moody algebras have applications throughout mathematics and mathematical physics. They are named after the Russian-American mathematician Victor Kac and the Canadian mathematician Robert Moody.

1 Definition
2 Interpretation
3 Types of Kac–Moody algebras
4 References
5 See also

[edit] Definition

A Kac–Moody algebra is given by the following:

An n by n generalized Cartan matrix $C = (c i j)$ of rank r.
A vector space $\mathfrak{h}$ over the complex numbers of dimension 2n − 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 $i \neq j.$
$[e_i,x]=\alpha_i^*(x)e_i$ , 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.

[edit] 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.

[edit] 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.

[edit] References

A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras
V. Kac Infinite dimensional Lie algebras ISBN 0521466938
, "Kac–Moody algebra" SpringerLink Encyclopaedia of Mathematics (2001)
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

Category: Lie algebras

Kac–Moody algebra

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Interpretation

[edit] Types of Kac–Moody algebras

[edit] References

[edit] See also

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