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 for the Russian-American mathematician Victor Kac and the Canadian mathematician Robert Moody.
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[edit] Definition
A Kac–Moody algebra is specified by the following pieces of data:
- 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 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 
![[e_i,[e_i,\ldots,[e_i,e_j]]] = \mathcal C_{e_i}^{1-c_{ij}}\;e_j = 0](../../../math/d/6/b/d6b80a3be771771b0ed1b33148363ab5.png)
for 
applications of 
![[f_i,[f_i,\ldots,[f_i,f_j]]] = \mathcal C_{f_i}^{1-c_{ij}}\;f_j = 0](../../../math/2/7/f/27f0c901462cd6ba724ce3100ea35065.png)
for applications of 
Where ![\mathcal C_{x}\;y = [x,y]](../../../math/f/d/2/fd2cfc23c7646e1f54549df82c31dd15.png)
is used as an alternate notation for the Commutator.
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.
