R. B. Zhang

Cohomology of Lie superalgebras and their generalizations

The cohomology groups of Lie superalgebras and, more generally, of e Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L=sl(1|2), the cohomology groups H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of e Lie algebras.

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.

From Representations of the Braid Group to Solutions of the Yang-Baxter Equation

Abstract A systematic method is developed for constructing solutions of the Yang-Baxter equation from given braid group representations, arising from such finite dimensional irreps of quantum groups that any irrep can be affinized and the tensor product of the irrep with itself is multiplicity-free. The main tool used in the construction is a tensor product graph, whose circuits give rise to consistency conditions. A maximal tree of this graph leads to an explicit formula for the quantum R-matrix when the consistency conditions are satisfied. As examples, new solutions of the Yang-Baxter equation are found, corresponding to braid group generators associated with the symmetric and antisymmetric tensor irreps of Uq[gl(m)], a spinor irrep of Uq[so(2n)]. and the minimal irreps of Uq[E6] and Uq[E7].

Quantum group invariants and link polynomials

A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universalR-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groupsUq(E8),Uq(so(2m+1) andUq(gl(m)) are considered as examples, and corresponding link polynomials are obtained.

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).

A q-analogue of Bargmann space and its scalar product

A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmanns scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.

Super duality and Kazhdan-Lusztig polynomials

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type A.

Generalized Gel’fand invariants and characteristic identities for quantum groups

The generalized Gel’fand invariants for an arbitrary quantum group are explicitly constructed, and their eigenvalues in any irreducible representation are computed. These invariants enable one to develop characteristic identities for the quantum group, and as a natural application, these identities are used to construct projection operators for tensor products of representations. To illustrate the general theory, the quantum group Uq(gl(m)) is studied in detail.

Universal L operator and invariants of the quantum supergroup Uq(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.

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

It is shown that the quantum supergroup Uq(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 Uq(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.