Kac-moody Algebra

These algebras form a generalization of finite-dimensional Semisimple Lie Algebra s, and many properties related to structure of the Lie algebra, its Root System , Irreducible Representations , connection to Flag Manifold s have natural analogues in the Kac-Moody setting. A class of Kac-Moody algebras called Affine Lie Algebra s is of particular importance in mathematics and Theoretical Physics , especially Conformal Field Theory and the theory of Exactly Solvable Model s. 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 Number s of dimension 2''n'' − ''r''.

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 Generator s

e_i

and

f_i

and the elements of

\mathfrak{h}

and relations

{Link without Title} = \alpha_i.\

{Link without Title} = 0\ for $i$

{Link without Title} =\alpha_i^---(x)e_i, for $x \in \mathfrak{h}.$

{Link without Title} =-\alpha_i^---(x)f_i, for $x \in \mathfrak{h}.$

{Link without Title} = 0\ for $x,x' \in \mathfrak{h}.$

$extrm{ad}(e_i)^{1-c_{ij}}(e_j) = 0.$

$extrm{ad}(f_i)^{1-c_{ij}}(f_j) = 0.$

extrm{ad}: \mathfrak{g}	o	extrm{End}(\mathfrak{g}),	extrm{ad}(x)(y)=

{Link without Title}

Adjoint Representation

\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

:

orall x\in \mathfrak{h}\,

{Link without Title} =\omega(x)g

, then ''g'' is said to have Weight ω. The Kac–Moody algebra can be diagonalized into weight Eigenvector s. 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 {Link without Title} = 0\ for $i$

_''i'' are Simple Root s.

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 Algebra s (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

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

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	CATEGORIES ABOUT KAC–MOODY ALGEBRA

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Kac-moody Algebra