Kac-Moody Superalgebras Research Articles

Shapovalov determinant for loop superalgebras

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra.

Classification of Finite-Growth General Kac–Moody Superalgebras

A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree. In this article we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case.

Dynkin diagrams of hyperbolic Kac Moody superalgebras

Hyperbolic Kac–Moody superalgebras are classified in terms of their Dynkin diagrams. These types of Kac–Moody superalgebras are those whose diagrams revert to either that of simple or affine superalgebras upon deletion of one of the vertices. It is found that the maximum rank of this type of algebra is 6.

Satake superdiagrams and Iwasawa decomposition of some hyperbolic Kac Moody superalgebras

The involutive automorphisms of hyperbolic Kac–Moody superalgebras are computed from the Satake superdiagrams corresponding to these algebras. These are then used to furnish a general treatment of the Iwasawa decomposition of these algebras. In particular, we consider (1)(2, 1, α) and Â(1)(0, 1) as representative examples for the purpose of illustration.

On the BRST Cohomology of Superstrings with/without Pure Spinors

We replace our earlier condition that physical states of the superstring have non-negative grading by the requirement that they are analytic in a new real commuting constant t which we associate with the central charge of the underlying Kac-Moody superalgebra. The analogy with the twisted N=2 SYM theory suggests that our covariant superstring is a twisted version of another formulation with an equivariant cohomology. We prove that our vertex operators correspond in one-to-one fashion to the vertex operators in Berkovits' approach based on pure spinors. Also the zero-momentum cohomology is equal in both cases. Finally, we apply the methods of equivariant cohomology to the superstring, and obtain the same BRST charge as obtained earlier by relaxing the pure spinor constraints.

Automorphic Forms, Fake Monster Algebras, and Hyperbolic Reflection Groups

We construct two families of automorphic forms related to twisted fake monster algebras and calculate their Fourier expansions. This gives a new proof of their denominator identities and shows that they define automorphic forms of singular weight. We also obtain new infinite product identities which are the denominator identities of generalized Kac–Moody superalgebras. Finally we describe the reflection groups of the root lattices of these algebras.

Twisting the Fake Monster Superalgebra

We calculate the twisted denominator identities of the fake monster superalgebra corresponding to elements of 2.Aut(E8) of odd order. They give many new infinite product identities and new examples of generalized Kac–Moody superalgebras. We also obtain generalizations of Jacobi's identity on theta functions.

Peterson-type dimension formulas for graded Lie superalgebras

Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

FREUDENTHAL-TYPE MULTIPLICITY FORMULAS FOR GENERALIZED KAC-MOODY SUPERALGEBRAS

In this paper we show that the Peterson root multiplicity formula and the Freudenthal weight multiplicity formula can be extended to the case of generalized Kac-Moody superalgebras. Applying these to some modular functions, we derive interesting relations among the Fourier coefficients. In particular, it will be shown that the Fourier coefficients of the elliptic modular function j − 744 can be determined only by the first three ones.

Crystal Bases for Kac–Moody Superalgebras

In this paper, we introduce the notion of crystal bases of Kac–Moody superalgebras. We prove the existence of the crystal bases for integrable modules following Kashiwara's grand loop argument and we also prove the tensor product rule for these bases. We simplify his argument without introducing the notion of the boson structure on the U−q(g).

Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras

Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this paper, we show that the Euler-Poincar\'e principle yields the generalized denominator identity for ${\frak L}$ and derive a closed form formula for the supertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a) \in \Gamma\times A$. We discuss the applications of our supertrace formula to various classes of infinite dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac-Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $GL(n) \times GL(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and the irreducible highest weight modules over a generalized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest weight modules over the orbit Lie superalgebra $\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.

Construction of Kac–Moody superalgebras as minimal graded Lie superalgebras and weight multiplicities for Kac–Moody superalgebras

We construct a Kac–Moody superalgebra ℒ as the minimal graded Lie superalgebra with local part V*⊕ge⊕V, where 𝔤 is a “smaller” Lie superalgebra inside ℒ, V is an irreducible highest weight 𝔤-module, and V* is the contragredient of V. We show that the weight multiplicities of irreducible highest weight modules over Kac–Moody superalgebras of finite type and affine type [more precisely, Kac–Moody superalgebras of type B(0,r), B(1)(0,r), A(4)(2r,0), A(2)(2r−1,0), and C(2)(r+1)] are given by polynomials in the rank r. The degree of these weight multiplicity polynomials are less than or equal to the depth of weights.

The Fake Monster Superalgebra

We show that the physical states of a 10 dimensional superstring moving on a torus form a generalized Kac–Moody superalgebra. This gives the first explicit realizations of these algebras. For a special torus the denominator function of this algebra is an automorphic form so that we can determine the simple roots. We call this algebra the fake monster superalgebra.

Involutive automorphisms and Iwasawa decomposition of affine Kac-Moody superalgebras

The involutive automorphisms of affine Kac-Moody superalgebras are computed from Satake superdiagrams corresponding to these algebras. These are then used to furnish a general treatment of the Iwasawa decomposition of these algebras. In particular, we consider A (1) (0,1) and B (1) (1,1) as representative examples for the purpose of illustration.

Quasi-Hopf superalgebras and elliptic quantum supergroups

We introduce the quasi-Hopf superalgebras which are Z2-graded versions of Drinfeld’s quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is, the face type Bq,λ(G) and the vertex type Aq,p[sl(n|∧n)] (and Aq,p[gl(n|∧n)]), where 𝒢 is any Kac–Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for Uq[sl(n|∧n) in a nonstandard system of simple roots, all of which are fermionic.

Gröbner–Shirshov Bases for Lie Superalgebras and Their Universal Enveloping Algebras

We show that a set of monic polynomials in a free Lie superalgebra is a Gröbner–Shirshov basis for a Lie superalgebra if and only if it is a Gröbner–Shirshov basis for its universal enveloping algebra. We investigate the structure of Gröbner–Shirshov bases for Kac–Moody superalgebras and give explicit constructions of Gröbner–Shirshov bases for classical Lie superalgebras.

Graded Lie Superalgebras and the Superdimension Formula

In this paper, we investigate the structure of graded Lie superalgebras L=⊕(α,a)∈Γ×AL(α,a), where Γ is a countable abelian semigroup and A is a countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie superalgebra L, we derive a superdimension formula for the homogeneous subspaces L(α,a)(α∈Γ,a∈A), which enables us to study the structure of graded Lie superalgebras in a unified way. We discuss the applications of our superdimension formula to free Lie superalgebras, generalized Kac–Moody superalgebras, and Monstrous Lie superalgebras. In particular, the product identities for normalized formal power series are interpreted as the denominator identities for free Lie superalgebras. We also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of Monstrous Lie superalgebras.

Automorphic forms with singularities on Grassmannians

We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some of the applications are as follows. We construct families of holomorphic automorphic forms which can be written as infinite products, which give many new examples of generalized Kac-Moody superalgebras. We extend the Shimura and Maass-Gritsenko correspondences to modular forms with singularities. We prove some congruences satisfied by the theta functions of positive definite lattices, and find a sufficient condition for a Lorentzian lattice to have a reflection group with a finite volume fundamental domain. We give some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.

Gauge symmetry in background charge conformal field theory

We present a mechanism to construct four-dimensional charged massless Ramond states using the discrete states of a fivebrane Liouville internal conformal field theory. This conformal field theory has background charge, and admits an inner product which allows positive norm states. A connection among supergravity soliton solutions, Liouville conformal field theory, non-critical string theory and their gauge symmetry properties is given. A generalized construction of the SU(2) super Kac-Moody algebra mixing with the N = 1 super Virasoro algebra is analyzed. How these Ramond states evade the DKV no-go theorem is explained.

Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras

We find automorphic form corrections which are generalized Lorentzian Kac-Moody superalgebras without odd real simple roots for two elliptic Lorentzian Kac-Moody algebras of rank 3 with a lattice Weyl vector, and calculate multiplicities of their simple and arbitrary roots. These Kac-Moody algebras are defined by hyperbolic symmetrized generalized Cartan matrices [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of rank 3. Both these algebras have elliptic type (i.e., their Weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces) and have a lattice Weyl vector. The correcting automorphic forms are Siegel modular forms. The form corresponding to G 1 is the classical Siegel cusp form of weight 5 which is the product of ten even theta-constants. In particular we find an infinite product formula for this modular form.