Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

Binary lie algebras of small dimensions

We give a simple and shorter proof of the Gainov theorem in [1], which dealt with classifying non-Lie binary Lie algebras of dimension ≤4 over a field of characteristic ≠2. Concurrently, the case of characteristic 2 is treated, and we find out an exotic 4-dimensional non-Lie Mal'tsev algebra, which is a split extension of an irreducible 1-dimensional Mal'tsev module over a simple 3-dimensional Lie algebra. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 320–328, May–June, 1998.     

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

On the tensor product of a finite and an infinite dimensional representation

There are two main results in the paper. The first gives the infinitesimal character that can occur in the tensor product V V_λ of an irreducible finite dimensional representation V_λ and an irreducible infinite dimensional representation V of a semisimple Lie algebra . The statement is that the infinitesimal characters are x_{v + μi}, I = 1, 2,…, k, where μ_i are the weights of V_λand v is the “pseudo” highest weight of V.The second result proves that if V is a Harish-Chandra module (one which comes from a group representation), then V V_λ has a finite composition series. But then the irreducible components in the composition series have the infinitesimal characters given in the first results.     

A family of regular vertex operator algebras with two generators

For every m ∈ ℂ ∖ {0, −2} and every nonnegative integer k we define the vertex operator (super)algebra D _m,k having two generators and rank . If m is a positive integer then D _m,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that D _m,k is a regular vertex operator (super) algebra and find the number of inequivalent irreducible modules.      

A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.     

Flag Partial Differential Equations and Representations of Lie Algebras

Flag partial differential equations naturally appear in the problem of decomposing the polynomial algebra (symmetric tensor) over an irreducible module of a Lie algebra into the direct sum of its irreducible submodules. Many important linear partial differential equations in physics and geometry are also of flag type. In this paper, we use the grading technique in algebra to develop the methods of solving such equations. In particular, we find new special functions by which we are able to explicitly give the solutions of the initial value problems of a large family of constant-coefficient linear partial differential equations in terms of their coefficients. As applications to representations of Lie algebras, we find certain explicit irreducible polynomial representations of the Lie algebras $sl(n,\mathbb {F}),\;so(n,\mathbb {F})$ and the simple Lie algebra of type G ₂.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

Reduction of the adjoint representation of sl(2,1): generators of primitive ideals

We present a reduction of the adjoint representation of the Lie superalge-bra,sl(2,1) and a study of the quotient algebra B(^c,k)= u/u(C?c)+u(D?kc), where c,k are two complex numbers. Under some additional conditions, we prove that every irreducible infinite dimensional representation of B(^c,k) is faithful, and that B(^C,K) is a primitive algebra. We give explicitly a set of generators of primitive degenerate ideal of infinite codimension. Essentially we prove that any minimal primitive ideal of u(sl(2,1)) is generated, as a 2-sided ideal, by its intersection with the algebra of gg-iuvariants.     

Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(M_n), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(M_n) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(M_n).     

Class of representations of skew derivation Lie algebra over quantum torus

Let L be the skew derivation Lie algebra of the quantum torus ℂ_q. In this paper, we give a class of irreducible representations for L with infinite dimensional weight spaces.      

Algebras and differential equations with additive symmetry

The set of all m-ary algebra structures on a given vector space affords, by the change of basis action, a representation of the general linear group. The invariants of a given subgroup are identified with those algebras admitting that subgroup as algebra automorphisms. Any finite dimensional representation of the additive group as automorphisms is obtained as the exponential of a nilpotent derivation. The latter can be embedded in the Lie algebra sl(2) so that the maximal vectors in an irreducible decomposition of the set of algebras as an sl(2) module are the invariants of the given action of the additive group. Dimension formulas and explicit bases are computed for the space of algebras with certain additive group actions. Employing the equivalence of the categories of m-ary algebras and systems of autonomous mth order homogeneous differential equations, the algebraic results are connected to the construction of first integrals and semi-invariants.     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Multiplicity formulas for finite dimensional and generalized principal series representations

The article presents two results. (1) Let a be a reductive Lie algebra over ℂ and let b be a reductive subalgebra of a. The first result gives the formula for multiplicity with which a finite dimensional irreducible representation of b appears in a given finite dimensional irreducible representation of a in a general situation. This generalizes a known theorem due to Kostant in a special case. (2) LetG be a connected real semisimple Lie group andK a maximal compact subgroup ofG. The second result is a formula for multiplicity with which an irreducible representation ofK occurs in a generalized representation ofG arising not necessarily from fundamental Cartan subgroup ofG. This generalizes a result due to Enright and Wallach in a fundamental case.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

Test rank of an abelian product of a free Lie algebra and a free abelian Lie algebra

Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/g_k( F) ^￠F/\gamma _{k}\left( F\right) ^{^{\prime }}.     

Involutionen von Jordan-Algebren

In this paper we will give some algebraic results on certain Lie algebras defined by involutions of Jordan algebras. Most of the results will be used elsewhere for applications in analysis. Let A be the (-1)-eigenspace of an involution JId of a central simple Jordan algebra A of degree at least 3, let D be the Lie algebra of all inner derivations of A leaving A_{_} invariant, and let h be the Lie algebra D+L(A_{_}), where L denotes the regular representation of A. In the case where the 1-eigenspace A₊ of J is central simple too, we will show A₊=A_{_}A_{_} and prove that h is semi-simple and irreducible on A. If A₊ is not central simple, then A_{_}A_{_} has codimension 1 in A₊ and h is semi-simple, but irreducible on A_{_}A_{_}+A_{_} only if the characteristic of the groundfield does not divide the degree of A. At characteristic O we will view D as an extension of the derivationalgebra of A₊ and determine the structure of the kernel of this extension.     

An analysis of the multiplicity spaces in branching of symplectic groups

Branching of symplectic groups is not multiplicity free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra B{\mathcal{B}}. The algebra B{\mathcal{B}} is a graded algebra whose components encode the multiplicities of irreducible representations of Sp _2n–2 in irreducible representations of Sp _2n . Our first theorem states that the map taking an element of Sp _2n to its principal n × (n + 1) submatrix induces an isomorphism of B{\mathcal{B}} to a different branching algebra B^￠{\mathcal{B}^{\prime}}. The algebra B^￠{\mathcal{B}^{\prime}} encodes multiplicities of irreducible representations of GL _n–1 in certain irreducible representations of GL _n+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp _2n to Sp _2n–2 is canonically an irreducible module for the n-fold product of SL ₂. In particular, this induces a canonical decomposition of the multiplicity spaces into one-dimensional spaces, thereby resolving the multiplicities.