Lie Superalgebra Gl Research Articles

CLASSICAL YANG-BAXTER EQUATIONS AND QUANTUM gl(p, q)-GAUDIN MODEL

A completely integrable Gaudin model associated with the Lie superalgebra gl(p, q) is presented, and its integrability is demonstrated in a straightforward way.

AN ALGEBRAIC APPROACH TO VERTEX MODELS AND TRANSFER MATRIX SPECTRA

We analyse the transfer matrix spectra of vertex models associated with the Lie superalgebras gl(P|M) and sl(P|M) using the representation theory of the Hecke algebra Hn(q). We develop a terminology for discussing the Bethe ansatz computation of the spectrum from this perspective. Using representations coming from these vertex models we develop some new methods for dealing with the analysis of Hecke algebras in any specialisation, including roots of unity. We also discuss the construction and spectrum of fusion models from the viewpoint of representation theory, begining a classification of the spectrum and identifying some sectors with trivial spectrum. In particular we show that the spectrum of the sl(P|M) λ-fusion model is trivial if λP+1>M.

Construction of Graded Covariant GL ( m / n ) Modules Using Tableaux

Irreducible covariant tensor modules for the Lie supergroups GL(m/n) and the Lie superalgebras gl(m/n) and sl(m/n) are obtained through the use of Young tableaux techniques. The starting point is the graded permutation action, first introduced by Dondi and Jarvis, on V⊗l. The isomorphism between this group of actions and the symmetric group Sl enables the graded generalization of the Young symmetrizers, and hence of the column relations and Garnir relations, to be made. Consequently, corresponding to each partition of l an irreducible GL(m/n) module may be obtained as a submodule of V⊗l. A basis for the module labeled by the partition λ is provided by GL(m/n)–standard tableaux of shape λ defined by Berele and Regev. The reduction of an arbitrary tableau to standard form is accomplished through the use of graded column relations and graded Garnir relations. The standardization procedure is algorithmic and allows matrix representations of the Lie superalgebras gl(m/n) and sl(m/n) to be constructed explicitly over the field of rational numbers. All the various steps of the standardization algorithm are exemplified, as well as the explicit construction of matrices representing particular elements of gl(m/n) and sl(m/n).

IRF MODELS ASSOCIATED WITH REPRESENTATIONS OF THE LIE SUPERALGEBRAS gl(m|n) AND sl(m|n)

We present two families of exactly solvable interaction round a face (IRF) models associated with representation of the Lie superalgebras gl (m|n) and sl (m|n). These IRF models are the generalizations of integrable spin chains with bosons and fermions. We also presents fusion models associated with higher representation of gl (m|n) and sl (m|n). We introduce restricted IRF models both for gl (m|n) and sl (m|n).

Branching rules for a class of typical and atypical representations of gl(m‖n)

The irreducible representations of the Lie superalgebra gl(m‖n) with highest weights of the form Λ=(λ1,λ2,...λm‖ω̇) are investigated using a recently introduced induced module construction for atypical modules. The gl(m‖n)↓gl(m‖n−1) branching rules are obtained and a suitable Gel’fand–Tsetlin basis is introduced. The class of representations considered includes some multiply atypical irreducible representations of gl(m‖n) and all irreducible representations of gl(m‖1).

Graded solutions of the Yang-Baxter relation and link polynomials

From a family of graded solvable models the authors derive representations of the braid group associated with the Lie superalgebra gl(M mod N) and give explicitly a general form of the Markov traces on the representations. The braid operators thus obtained satisfy the Hecke algebra. The authors construct composite solvable models and obtain link polynomials from the braid operators for the composite models.

Finite-dimensional representations of the Lie superalgebra gl(2/2) in a gl(2)⊕gl(2) basis. II. Nontypical representations

All finite-dimensional irreducible representations of the general linear Lie superalgebra gl(2/2) are written down in matrix form. The basis within each representation space is chosen in such a way that it makes evident the decomposition of gl(2/2) into irreducible representations of its even subalgebra gl(2)⊕gl(2). Special attention is devoted to the analysis of all nontypical representations and some indecomposible representations.

Irreducible finite-dimensional representations of the Lie superalgebra gl(n/1) in a Gel’fand–Zetlin basis

All finite-dimensional irreducible representations of the general linear Lie superalgebra gl(n/1) are studied. For each representation, a concept of a Gel’fand–Zetlin basis is defined. Expressions for the transformation of the basis under the action of the generators are written down.

Highest weight representations for gl(m/n) and gl(m+n)

It is shown how vector coherent state (VCS) theory enlightens the simultaneous discussion of representation theory for classical Lie algebras and superalgebras and provides an optimal framework for the explanation of the noted similarities and dissimilarities of their representations. Reducibility, atypicality, and the positive-definitiveness of inner products in VCS representation spaces are discussed. The discussion is exemplified through the parallel and explicit construction of highest weight ladder representations of the Lie algebra gl(m+n) and superalgebra gl(m/n) in gl(m)⊕gl(n) bases.

Finite-dimensional representations of the Lie superalgebra gl(2/2) in a gl(2)⊕gl(2) basis. I. Typical representations

In a series of two papers all finite-dimensional irreducible representations and some indecomposible representations of the general linear Lie superalgebra gl(2/2) are constructed in a basis suitable for the decomposition gl(2/2)⊇gl(2)⊕gl(2). In this paper each induced gl(2/2) module W is represented as a direct sum of its irreducible gl(2)⊕gl(2) submodules Vi, 1≤i≤16. The basis Γ in W is chosen to consist of the union of all Γi, where Γi is an appropriate basis in each Vi. Expressions for the transformation of Γ under the action of the generators are written down for all induced and hence, also, for all typical gl(2/2) modules.

Essentially typical representations of Lie superalgebrasgl (n|m) in the Gel'fand-Tsetlin basis

Essentially typical representations of Lie superalgebrasgl (n|m) in the Gel'fand-Tsetlin basis

Finest grading of the Lie superalgebra gl(n/n)

The author shows that the group P2n of generalised Pauli matrices in 2n dimensions provides a finest grading of the Lie superalgebra gl(n/n) and of its subalgebras sl(n/n) and A(n-1,n-1).

Extremal weights of finite-dimensional representations of the Lie superalgebragl n/m

We investigate a class of irreducible finite-dimensional representations of the Lie superalgebragln/m, which are constructed in terms of creation and annihilation operators acting in certain Fock superspaces. For these representations we calculate the highest weights relative to representatives of all (n+m)!/n!m! conjugacy classes of Borel subalgebras ofgln/m.

Representations of the Lie superalgebras gl (n, m) and Q(n) on the space of tensors

Representations of the Lie superalgebras gl (n, m) and Q(n) on the space of tensors