Quantum Supergroup Research Articles

The supersymmetric t− J model with quantum group invariance

An integrable quantum group deformation of the supersymmetric t− J model is introduced. Open-boundary conditions lead to an spl q (2, 1) invariant hamiltonian. A general procedure to obtain such invariant models is proposed. To solve the model a generalized nested algebraic Bethe ansatz is constructed and the Bethe ansatz equations are obtained. The quantum supergroup structure of the model is investigated.

QUANTUM SUPERGROUPS AND LINK POLYNOMIALS

A general method for constructing invariants for quantum supergroups is applied to obtain a closed formula for link polynomials. For type I quantum supergroups, a realization of the braid group and corresponding link polynomial is determined, for each irreducible representation of the quantum supergroup in a certain class. Although these realizations are not matrix representations in the usual sense, nevertheless link polynomials are defined which are generalizations of those previously obtained from quantum groups. To illustrate the theory, link polynomials corresponding to the defining representations of the quantum supergroups Uq [gl(m|n)], Uq [C (m + 1)] are determined explicitly.

QUANTUM SUPERGROUPS, LINK POLYNOMIALS AND REPRESENTATION OF THE BRAID GENERATOR

Unlike the quantum group case, it is shown that the braid generator σ is not always diagonalizable on V ⊗ V, V an irreducible module for a quantum supergroup. Nevertheless a generalization of the Reshetikhin form of the braid generator, obtained previously for quantum groups, is determined corresponding to every finite dimensional standard cyclic module V of a quantum supergroup. This result is applied to obtain a general closed formula for link polynomials arising from standard cyclic modules of a quantum supergroup belonging to a certain class. As explicit examples we determine link polynomials corresponding to the rank 2 symmetric tensor representation of Uq [gl(m|m)] and the defining representation of Uq [osp(2n|2n)].

All quantum group structures on the supergroup GL(1 mod 1)

All quantum group structures are found on the supergroup GL(1 mod 1). These structures are described by two 2-parameter families, GLq, Q(1 mod 1) and GLh1,h2(1 mod 1). Each family possesses a central multiplicative quantum superdeterminant, which allows one to define the quantum supergroups SLq,Q(1 mod 1) and SLh1,h2(11).

Finite dimensional irreducible representations of the quantum supergroup Uq (gl(m/n))

Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite-dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.

Baxterization of ther-matrix for the adjoint representation of U q [D(2, 1;?)

The Baxterization procedure is applied to the Braid group representation arising from the adjoint representation of the quantum supergroup U(q)[D(2, 1: alpha)]. This yields a two-parameter family of solutions to the (parameter dependent) Yang-Baxter equation of interest in supersymmetric lattice models.

Braid group representations arising from quantum supergroups with arbitrary q and link polynomials

The theory of quantum supergroups is applied to the construction of braid group representations and link invariants. A method developed in an earlier publication is reviewed, which produces link polynomials from braid group representations generated by quantum supergroup invariant Ř matrices. When q is a generic complex parameter, this method, applied to the universal R matrices, yields hierarchies of link polynomials associated with different irreps of quantum supergroups. When q is a root of unity, suitable constraints are imposed on quantum supergroups to turn them into finite-dimensional quasitriangular Z2-graded Hopf algebras. The corresponding universal R matrices are obtained, which, in turn, lead to series of link polynomials upon using the above mentioned method. As concrete examples, the link polynomials related to the vector irreps of the quantum supergroups Uq(osp(M/2n)), all the irreps of Uq(osp(1/2)) with generic q, and a class of doubly atypical irreps of Uq(gl(2/1)) with q a root of unity are studied in detail.

Two variable link polynomials from quantum supergroups

New two variable link polynomials are constructed corresponding to a one-parameter family of representations of the quantum supergroup U q [gl(2 | 1)]. Their connection with the Kauffman polynomials is also investigated.

The two-parameter deformation of the supergroup GL(1|1), its differential calculus and its Lie algebra

We construct a right-invariant differential calculus on the quantum supergroup GL q,s (1|1) and we show that its quantum Lie algebra with comultiplication is isomorphic to that which we obtain using the Reshetikhin-Takhtajan-Faddeev approach.

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U q (osp(1/2n)) is essentially isomorphic to the quantum group U -q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.

Canonical differential calculus on quantum general linear groups and supergroups

We specify a set of relations between non-commuting matrix elements and their differentials, defined in terms of an R-matrix satisfying the braid relation, which are uniquely determined by the requirements of consistency with the relations between non-commuting coordinates and their differentials. We also give a necessary condition for the existence of a matrix inverse (antipode) in the form of an additional equation to be satisfied by the R-matrix.

Universal L operator and invariants of the quantum supergroup U q (gl(m/n))

A spectral parameter-dependent solution of the graded Yang–Baxter equation is obtained, which is universal in the sense that it lives in Uq(gl(m/n))⊗End(V) with V the vector module of Uq(gl(m/n)). The invariants of this quantum supergroup are constructed using this solution.

Invariants of the quantum supergroup Uq(gl(m/1))

A spectral parameter dependent solution of the graded Yang-Baxter equation is obtained, which in Uq(gl(m/1))(X)End(V) with V being the vector module of Uq(gl(m/1)). Invariants of this quantum supergroup are constructed using this solution and a general method developed in an earlier publication.

Universal R matrices and invariants of quantum supergroups

A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.

Graded representations of the Temperley–Lieb algebra, quantum supergroups, and the Jones polynomial

It is shown that from each self-dual representation of a quantum supergroup with nonvanishing q-superdimension, a graded representation of the Temperley–Lieb algebra can be constructed, which, in turn, gives rise to a solution of the graded Yang–Baxter equation. As examples, the vector irrep of Uq (osp(M/2n)) and all the finite-dimensional irreps of Uq (osp(1/2)) are studied, and the corresponding graded Ř-matrices are obtained explicitly. A general method is also given to obtain link polynomials from graded Ř-matrices. Applied to the ones associated with graded representations of the Temperley–Lieb algebra, the method yields the Jones polynomial.

Supercovariant systems of q-oscillators and q-supersymmetric hamiltonians

We construct the q-superoscillator algebra of n bosonic and m fermionic q-oscillators covariant under the coactions of the quantum supergroup SU q ( n\\)-invariant form bilinear in q-superoscillator variables describes the q-deformation of the hamiltonian for the supersymmetric oscillator in quantum mechanics. It is shown that such a hamiltonian is also invariant under the action of the U q( U(n\\m)) superalgebra, generated by the bilinears constructed from the q-superoscillators. The simplest case n= m=1 is considered in detail. The formulae relating the supercovariant q-oscillators with the independent ones are given.

Solution of the graded Yang-Baxter equation associated with the vector representation of

Abstract A general method of constructing spectral parameter-dependent solutions of the graded Yang-Baxter equation using quantum supergroups is briefly reviewed. Using this method, a quantum R-matrix associated with the vector representation of U q ( osp ( M 2n )) is obtained for all M and n.

QUANTUM SUPERGROUPS AND SOLUTIONS OF THE YANG-BAXTER EQUATION

A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp (1/2) and gl (m/n).

Some aspects of quantum groups and supergroups

Some features of Manin’s construction of quantum groups are developed and extended to supergroups.

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).