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In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and 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, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated that Rogers–Ramanujan identities can be derived in a similar fashion.[1]

History of Kac–Moody algebras

The initial construction by Élie Cartan and Wilhelm Killing of finite dimensional simple Lie algebras from the Cartan integers was type dependent. In 1966 Jean-Pierre Serre showed that relations of Claude Chevalley and Harish-Chandra,[2] with simplifications by Nathan Jacobson,[3] give a defining presentation for the Lie algebra.[4] One could thus describe a simple Lie algebra in terms of generators and relations using data from the matrix of Cartan integers, which is naturally positive definite.

In his 1967 thesis, Robert Moody considered Lie algebras whose Cartan matrix is no longer positive definite.[5][6] This still gave rise to a Lie algebra, but one which is now infinite dimensional. Simultaneously, Z-graded Lie algebras were being studied in Moscow where I. L. Kantor introduced and studied a general class of Lie algebras including what eventually became known as Kac–Moody algebras.[7] Victor Kac was also studying simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite dimensional Lie algebras evolved. An account of the subject, which also includes works of many others is given in (Kac 1990).[8] See also (Seligman 1987).[9]
Definition

A Kac–Moody algebra is given by the following:

An n×n generalized Cartan matrix C = (cij) 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^\vee\ of \( \mathfrak{h} and a set of n linearly independent elements \( \alpha_i$$ of the dual space $$\mathfrak{h}^*$$ , such that $$\alpha_i(\alpha_j^\vee) = c_{ji}$$ . The $$\alpha_i$$ are analogue to the simple roots of a semi-simple Lie algebra, and the $$\alpha_i^\vee to the simple coroots. The Kac–Moody algebra is the Lie algebra \( \mathfrak{g}$$ defined by generators $$e_i$$ and $$f_i (i \in \{1,\ldots,n\})$$ and the elements of $$\mathfrak{h}$$ and relations

$$[h,h'] = 0\$$ for $$h,h' \in \mathfrak{h}$$ ;
$$[h,e_i] = \alpha_i(h)e_i$$ , for $$h \in \mathfrak{h}$$ ;
$$[h,f_i] = -\alpha_i(h)f_i$$, for $$h \in \mathfrak{h}$$ ;
$$[e_i,f_j] = \delta_{ij}\alpha_i^\vee$$ , where $$\delta_{ij}$$ is the Kronecker delta;
If $$i \neq j (so \( c_{ij} \leq 0)$$ then $$\textrm{ad}(e_i)^{1-c_{ij}}(e_j) = 0$$ and $$\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.
Root-space decomposition of a Kac–Moody algebra

$$\mathfrak{h}$$ is the analogue of a Cartan subalgebra for the Kac–Moody algebra \mathfrak{g}.

If $$x\neq 0$$ is an element of $$\mathfrak{g}$$ such that

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

for some $$\lambda\in\mathfrak{h}^*\backslash\{0\}$$ , then x is called a root vector and $$\lambda$$ is a root of $$\mathfrak{g}$$ . (The zero functional is not considered a root by convention.) The set of all roots of $$\mathfrak{g}$$ is often denoted by \Delta and sometimes by R. For a given root \lambda one denotes by $$\mathfrak{g}_\lambda$$ the root space of $$\lambda$$ , that is

$$\mathfrak{g}_\lambda = \{x\in\mathfrak{g}:\forall h\in\mathfrak{h}, [h,x] = \lambda(h)x\}$$ .

It follows from the defining relations of $$\mathfrak{g}$$ that $$e_i\in\mathfrak{g}_{\alpha_i}$$ and $$f_i\in\mathfrak{g}_{-\alpha_i}$$ . Also, if $$x_1\in\mathfrak{g}_{\lambda_1}$$ and $$x_2\in\mathfrak{g}_{\lambda_2}$$ , then $$[x_1,x_2]\in\mathfrak{g}_{\lambda_1+\lambda_2}$$ by the Jacobi identity.

A fundamental result of the theory is that any Kac–Moody algebra can be decomposed into the direct sum of $$\mathfrak{h}$$ and its root spaces, that is

$$\mathfrak{g} = \mathfrak{h}\oplus\bigoplus_{\lambda\in\Delta} \mathfrak{g}_\lambda,$$

and that every root $$\lambda$$ can be written as $$\lambda = \sum_{i=1}^n z_i\alpha_i$$ with all the $$z_i$$ being integers of the same sign.

Types of Kac–Moody algebras

Properties of a Kac–Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix C. In order to classify Kac–Moody algebras, it is enough to consider the case of an indecomposable matrix C, that is, assume that there is no decomposition of the set of indices I into a disjoint union of non-empty subsets I1 and I2 such that Cij = 0 for all i in I1 and j in I2. Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra:

$$\mathfrak{g}(C)\simeq\mathfrak{g}(C_1)\oplus\mathfrak{g}(C_2),$$

where the two Kac–Moody algebras in the right hand side are associated with the submatrices of C corresponding to the index sets I1 and I2.

An important subclass of Kac–Moody algebras corresponds to symmetrizable generalized Cartan matrices C, which can be decomposed as DS, where D is a diagonal matrix with positive integer entries and S is a symmetric matrix. Under the assumptions that C is symmetrizable and indecomposable, the Kac–Moody algebras are divided into three classes:

• A positive definite matrix S gives rise to a finite-dimensional simple Lie algebra.
• A positive semidefinite matrix S gives rise to an infinite-dimensional Kac–Moody algebra of affine type, or an affine Lie algebra.
• An indefinite matrix S gives rise to a Kac–Moody algebra of indefinite type.
• Since the diagonal entries of C and S are positive, S cannot be negative definite or negative semidefinite.

Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified. They correspond to Dynkin diagrams and affine Dynkin diagrams. Very little is known about the Kac–Moody algebras of indefinite type. Among those, the main focus has been on the (generalized) Kac–Moody algebras of hyperbolic type, for which the matrix S is indefinite, but for each proper subset of I, the corresponding submatrix is positive definite or positive semidefinite. Such matrices have rank at most 10 and have also been completely determined.[10]

Weyl–Kac character formula
Generalized Kac–Moody algebra

Notes

(?) Garland, H.; Lepowsky, J. (1976). "Lie algebra homology and the Macdonald-Kac formulas". Invent. Math. 34 (1): 37–76. doi:10.1007/BF01418970.
Harish-Chandra (1951). "On some applications of the universal enveloping algebra of a semisimple Lie algebra". Trans. Amer. Math. Soc. 70 (1): 28–28. doi:10.1090/S0002-9947-1951-0044515-0. JSTOR 1990524.
Jacobson, N. (1962). Lie algebras. Interscience Tracts in Pure and Applied Mathematics 10. New York-London: Interscience Publishers (a division of John Wiley & Sons).
Serre, J.-P. (1966). Algèbres de Lie semi-simples complexes (in French). New York-Amsterdam: W. A. Benjamin.
Moody, R. V. (1967). "Lie algebras associated with generalized cartan matrices" (PDF). Bull. Amer. Math. Soc. 73 (2): 217–222. doi:10.1090/S0002-9904-1967-11688-4.
Moody 1968, A new class of Lie algebras
Kantor, I. L. (1970). "Graded Lie algebras". Trudy Sem. Vektor. Tenzor. Anal. (in Russian) 15: 227–266.
Kac, 1990
Seligman, George B. (1987). "Book Review: Infinite dimensional Lie algebras". Bull. Amer. Math. Soc. N.S. 16 (1): 144–150. doi:10.1090/S0273-0979-1987-15492-9.

Carbone, L.; Chung, S.; Cobbs, C.; McRae, R.; Nandi, D.; Naqvi, Y.; Penta, D. (2010). "Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits". J. Phys. A: Math. Theor. 43 (15): 155209. arXiv:1003.0564. doi:10.1088/1751-8113/43/15/155209.

References

R.V. Moody, A new class of Lie algebras, J. of Algebra, 10 (1968) pp. 211–230
V. Kac, Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press (1990) ISBN 0-521-46693-8 [1]
A. J. Wassermann, Lecture notes on Kac–Moody and Virasoro algebras
Hazewinkel, Michiel, ed. (2001), "Kac–Moody algebra", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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
S. Kumar, Kac–Moody Groups, their Flag Varieties and Representation Theory, 1st edition, Birkhäuser (2002). ISBN 3-7643-4227-7.