Kac–Moody algebra
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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.
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[edit] Definition
A Kac–Moody algebra is given by the following:
- An n by n generalized Cartan matrix C = (cij) of rank r.
- A vector space
over the complex numbers of dimension 2n − r. - A set of n linearly independent elements
of and a set of n linearly independent elements 
of the dual space, such that 
. The are known as coroots, while the are known as roots.
The Kac–Moody algebra is the Lie algebra 
defined by generators ei and fi and the elements of and relations
![[e_i,f_i] = \alpha_i.\](../../../math/c/7/6/c763e8c0d45dbf04f180425e2486e000.png)
![[e_i,f_j] = 0\](../../../math/4/0/2/4020645bc014a888b239695f646f56bb.png)
for 
![[e_i,x]=\alpha_i^*(x)e_i](../../../math/c/a/7/ca705cffaedea39812493b85557713a9.png)
, for 
![[f_i,x]=-\alpha_i^*(x)f_i](../../../math/8/b/a/8ba5f8695fa256c10bf80530880d8b94.png)
, for ![[x,x'] = 0\](../../../math/e/f/a/efa6930200909465c880c8514c69fd3c.png)
for 
Where ![\textrm{ad}: \mathfrak{g}\to\textrm{End}(\mathfrak{g}),\textrm{ad}(x)(y)=[x,y]](../../../math/5/7/a/57ac9c358dc0b541c3343cc2665d217d.png)
is the adjoint representation of .
A real (possibly infinite-dimensional) Lie algebra is also considered a Kac–Moody algebra if its complexification is a Kac–Moody algebra.
[edit] Interpretation
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](../../../math/4/0/c/40c56808980838923d45537644bb2da0.png)
where ω is an element of 
, 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, ei has weight α*i and fi 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 for 
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
