The Potts Models and a Generalisation of the Clifford Algebras

Journal of Algebra, 1997

It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.

1999

The Cayley-Hamilton-Newton theorem - which underlies the Newton identities and the Cayley-Hamilton identity - is reviewed, first, for the classical matrices with commuting entries, second, for two q-matrix algebras, the RTT-algebra and the RLRL-algebra. The Cayley-Hamilton-Newton identities for these q-algebras are related by the factorization map. A class of algebras M(R,F) is presented. The algebras M(R,F) include the RTT-algebra and

The talk is about general properties of the Lie algebras of triangular polynomial derivations, an explicit classification of their ideals, an explicit isomorphism criterion for their Lie factor algebras. For the Lie algebras of triangular polynomial derivations, their groups of automorphisms and their explicit generators are found. An analogue of the Dixmier Conjecture is proved for these Lie algebras: every Lie algebra monomorphism is an automorphism (but not every epimorphism is an isomorphism).

Journal of Algebra, 1999

Quantum algebras are generalizations of quasitriangular Hopf algebras and as such are used to construct invariants of 1᎐1 tangles and in some cases of knots and 3-manifolds. In this paper we develop a general description of quantum algebra structures whose underlying algebra is the Ž . algebra M k of all n = n matrices over a field k. An important ingredin ent in the structure of a quantum algebra in this case is a solution Ž . 2 R g M k to the quantum Yang᎐Baxter equation. n We use our general results and the classification of solutions to the Ž . Ž . Ž . quantum Yang᎐Baxter equation R g M k s M k m M k found in 4 2 2

Lecture Notes in Physics, 2015

Journal of Physics A: Mathematical and General, 1999

The Cayley-Hamilton-Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras M(R,F ) defined by a pair of compatible solutions of the Yang-Baxter equation. This class includes the RTTalgebras as well as the Reflection equation algebras. * On leave of absence from P. N. Lebedev

Letters in Mathematical Physics, 2008

We give a presentation of the endomorphism algebra End Uq(sl2) (V ⊗r ), where V is the 3-dimensional irreducible module for quantum sl 2 over the function field C(q 1 2 ). This will be as a quotient of the Birman-Wenzl-Murakami algebra BM W r (q) := BM W r (q -4 , q 2q -2 ) by an ideal generated by a single idempotent Φ q . Our presentation is in analogy with the case where V is replaced by the 2-dimensional irreducible U q (sl 2 )-module, the BMW algebra is replaced by the Hecke algebra H r (q) of type A r-1 , Φ q is replaced by the quantum alternator in H 3 (q), and the endomorphism algebra is the classical realisation of the Temperley-Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on V ⊗r are consequences of relations among the three R-matrices acting on V ⊗4 . The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.

arXiv preprint math/0508506, 2005

In our previous paper [GPS2] the Cayley-Hamilton identity for the GL(m|n) type quantum matrix algebra was obtained. Here we continue investigation of that identity. We derive it in three alternative forms and, most importantly, we obtain it in a factorized form. The factorization leads to a separation of the spectra of the quantum supermatrix into the "even" and "odd" parts. The latter, in turn, allows us to parameterize the characteristic subalgebra (which can also be called the subalgebra of spectral invariants) in terms of the supersymmetric polynomials in the eigenvalues of the quantum supermatrix. For our derivation we use two auxiliary results which may be of independent interest. First, we calculate the multiplication rule for the linear basis of the Schur functions s λ (M ) for the characteristic subalgebra of the Hecke type quantum matrix algebra. The structure constants in this basis are the Littlewood-Richardson coefficients. Second, we derive a series of bilinear relations in the graded ring Λ of Schur symmetric functions in countably many variables (see [Mac]). * gurevich@univ-valenciennes.fr †

Journal of Mathematical Physics, 1994

In his classic book on group representations and special functions 1991 Mathematics Subject Classi cation. 33D55, 33D45, 17B37, 81R50.

arXiv: Representation Theory, 2015

As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the quantum Schur superalgebra of type Q with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and then provide a classification of their irreducible representations over a certain extension of the field of complex rational functions.