Invariant Form of the Generators of Semisimple Lie and Quantum Algebras in Their Arbitrary Finite-Dimensional Representation

An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of matrix equations. The result is presented in the form of N _l×N _l matrices, where N _l is the dimension of the corresponding representation determined by the invariant Weyl formula.     

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.     

Presenting degenerate Ringel-Hall algebras of cyclic quivers

Using the generators labelled by simple and sincere semisimple modules for the Ringel-Hall algebra H_q(n) of a cyclic quiver Δ(n), we give a presentation for the degenerate algebra H₀(n). This is achieved by establishing a presentation for the generic extension monoid algebra of Δ(n). As an application, we show that both the degenerate Ringel-Hall algebra and the degenerate quantum affine sl_n admit multiplicative bases.     

Generalized Serre relations for Lie algebras associated with positive unit forms

Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebra and a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre’s Theorem [J.-P. Serre, Complex Semisimple Lie Algebras (G.A. Jones, Trans.), Springer-Verlag, New York, 1987] gives then a representation of the given Lie algebra in generators and relations in terms of the Cartan matrix.In this work, we generalize Serre’s Theorem to give an explicit representation in generators and relations for any simply laced semisimple Lie algebra in terms of a positive quasi-Cartan matrix. Such a quasi-Cartan matrix expresses the dual of the Killing form for a Z-base of roots. Here, by a Z-base of roots, we mean a set of linearly independent roots which generate all roots as linear combinations with integral coefficients.     

Ramified partition algebras

For each natural number n, poset T, and |T|–tuple of scalars Q, we introduce the ramified partition algebra P _n ^(T) (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T|=1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representation theory in case ${{T=\underline{{2}}:=({1,2},\leq)}}$. We show that ${{P_n^{(\underline{{2}})$ (Q) is quasi–hereditary over field k when Q ₁ Q ₂ is invertible in k and k is such that certain finite group algebras over k are semisimple (e.g. when k is algebraically closed, characteristic zero). Under these conditions we determine an index set for simple modules of ${{P_n^{(\underline{{2}})$ (Q), and construct standard modules with this index set. We show that there are unboundedly many choices of Q such that ${{P_n^{(\underline{{2}})$ (Q) is not semisimple for sufficiently large n, but that it is generically semisimple for all n. We construct tensor space representations of certain non–semisimple specializations of ${{P_n^{(\underline{{2}})$ (Q), and show how to use these to build clock model transfer matrices [24] in arbitrary physical dimensions. Sadly Ahmed died before this work was completed. His memory lives on.     

The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E _n for n = 6; 7; 8 are shown to be semisimple and free over the integral domain \mathbbZ[ d^±1,l^±1,m ]

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

The centralizer algebra of the diagonal action of the group <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(ℂ) in a mixed tensor space

We consider the walled Brauer algebra Br _{k, l}(n) introduced by V. Turaev and K. Koike. We prove that it is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large n ∈ ℕ, to the centralizer algebra of the diagonal action of the group GL_n(ℂ) in a mixed tensor space. We also give the presentation of the algebra Br _{k, l}(n) by generators and relations. For a generic value of the parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for this family of algebras and give realizations for the irreducible representations. We also give a new, more natural proof of the formulas for the characters of the walled Brauer algebras. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 170–198.     

Hecke algebras with independent parameters

We study the Hecke algebra \({\mathcal {H}}({\mathbf {q}})\) over an arbitrary field \({\mathbb {F}}\) of a Coxeter system (W, S) with independent parameters \({\mathbf {q}}=(q_s\in {\mathbb {F}}:s\in S)\) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of \({\mathcal {H}}({\mathbf {q}})\) could be as small as 1. We construct a basis for \({\mathcal {H}}({\mathbf {q}})\) when (W, S) is simply laced. We also characterize when \({\mathcal {H}}({\mathbf {q}})\) is commutative, which happens only if the Coxeter diagram of (W, S) is simply laced and bipartite. In particular, for type A, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.     

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.     

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

We construct n distinct weight bases, which we call extremal bases, for the adjoint representation of each simple Lie algebra 𝔤 of rank n: One construction for each simple root. We explicitly describe actions of the Chevalley generators on the basis elements. We show that these extremal bases are distinguished by their “supporting graphs” in three ways. (In general, the supporting graph of a weight basis for a representation of a semisimple Lie algebra is a directed graph with colored edges that describe the supports of the actions of the Chevalley generators on the elements of the basis.) We show that each extremal basis constructed is essentially the only basis with its supporting graph (i.e., each extremal basis is solitary), and that each supporting graph is a modular lattice. Each extremal basis is shown to be edge-minimizing: Its supporting graph has the minimum number of edges. The extremal bases are shown to be the only edge-minimizing as well as the only modular lattice weight bases (up to scalar multiples) for the adjoint representation of 𝔤. The supporting graph for an extremal basis is shown to be a distributive lattice if and only if the associated simple root corresponds to an end node for a “branchless” simple Lie algebra, i.e., type A, B, C, F, or G. For each extremal basis, basis elements for the Cartan subalgebra are explicitly expressed in terms of the h _i Chevalley generators.     

Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).     

Endomorphism algebras and q-traces

For a braided vector space (V,σ) with braiding σ of Hecke type, we introduce three associative algebra structures on the space of graded endomorphisms of the quantum symmetric algebra S_σ(V). We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when V is the fundamental representation of U_qsl_N+1 and σ is the action of the R-matrix, this trace is a scalar multiple of the quantum trace of type A.     

Spectral representations for abelian automorphism groups of von Neumann algebras

Let $\text{A}$ be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as ¹-automorphisms of $\text{A}$. Let M(σ) be the Banach algebra of bounded linear operators on $\text{A}$ generated by ∝ σ_tdμ(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) $\text{A}$ has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σ_t is inner. It is shown that the Banach algebra L(σ) generated by $\text{∝ ?(t)σ}{}_{\mathrm{t}}\text{dt (? ? L}{}^1\text{(G))}$ is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L(σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on $\text{A}$. Also the spectral subspaces of σ are given in terms of projections.     

Group elements of some semisimple finite-dimensional Hopf algebras

V. A. Artamonov and I. A. Chubarov proved a criterion under which an element of some semisimple finite-dimensional Hopf algebra is group-like. The studied Hopf algebra has only one nonone- dimensional irreducible representation. Let n be a dimension of this representation. It is shown in this paper that for odd prime n the set of group-like elements of these algebras is a cyclic group of order 2n.     

Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus g = 1, 2,.. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators L _2q , q = ?1, 0, 1, 2,.., of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.     

Adapted algebras for the Berenstein-Zelevinsky conjecture

Let G be a simply connected semisimple complex Lie group and fix a maximal unipotent subgroup U^- of G. Let q be an indeterminate and let B* denote the dual canonical basis (cf. [19]) of the quantized algebra Cq[U^-] of regular functions on U^-. Following [20], fix a Z^N_≧0-parametrization of this basis, where N = dim U^-. In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of B* q-commute if and only if they are multiplicative, i.e., their product is an element of B* up to a power of q. To any reduced decomposition w₀ of the longest element of the Weyl group of g, we associate a subalgebra A_w0, called adapted algebra, of Cq[U^-] such that (1) A_w0 is a q-polynomial algebra which equals Cq[U^-] up to localization, (2) A_w0 is spanned by a subset of B*, (3) the Berenstein–Zelevinsky conjecture is true on A_w0. Then we test the conjecture when one element belongs to the q-center of Cq[U^-].     

Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra B _n (q,r) by lifting bases for cell modules of B _n−1(q,r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke algebra of the symmetric group, acts triangularly. The triangular action of this Abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B–M–W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori–Hecke algebra of the symmetric group. Research supported by Japan Society for Promotion of Science.     

Presenting generalized Schur algebras in types B, C, D

We give explicit presentations by generators and relations of certain generalized Schur algebras (associated with tensor powers of the natural representation) in types B, C, D. This extends previous results in type A obtained by two of the authors. The presentation is compatible with the Serre presentation of the corresponding universal enveloping algebra. In types C, D this gives a presentation of the corresponding classical Schur algebra (the image of the representation on a tensor power) since the classical Schur algebra coincides with the generalized Schur algebra in those types. This coincidence between the generalized and classical Schur algebra fails in type B, in general.