Abstract. In this paper we use the Etingof-Kazhdan quantization of Lie bisuperalgebras to investigate some interesting questions related to Drinfeld-Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and 2) a super version of the Drinfeld-Kohno Theorem. 1.

Abstract. Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are the same. These constructions can be generalized to some classes of Lie superalgebras. In this paper we show that constructions 1) and 2) give the same invariants for the Lie superalgebras of type A-G. We use this result to investigate the Links-Gould invariant. We also give a positive answer to a conjecture of Patureau-Mirand’s concerning invariants arising from the Lie superalgebra D(2, 1; α).

Abstract. In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link invariant. This new invariant can be thought of as a multivariable version of the Links-Gould invariant. We also show that after a variable reduction our invariant specializes to the Conway potential function, which is a refinement of the multivariable Alexander polynomial.

Using adjoint representation of Lie superalgebras, we write the matrix form of super Jacobi and mixed super Jacobi identities of Lie super-bialgebras. Then through direct calculations of these identities and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie super-bialgebras.

Abstract. A super dynamical r-matrix r satisfies the zero weight condition if: [h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ h, λ ∈ h ∗. In this paper we classify super dynamical r−matrices with zero weight, thus extending the results of [6] to the graded case. 1.

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Abstract Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are the same. These constructions can be generalized to some classes of Lie superalgebras. In this paper we show that constructions 1) and 2) give the same invariants for the Lie superalgebras of type A-G. We use this result to investigate the Links-Gould invariant. We also give a positive answer to a conjecture of Patureau-Mirand’s concerning invariants arising from the Lie superalgebra D(2, 1; α). AMS Classification 57M27; 17B65, 17B37

Abstract. We review recent results in the study of quantum groups in the super setting. In particular, we provide an overview of results about solutions of the Yang-Baxter equations in the super setting and develop the super analog of the theory of dynamical quantum groups. 1.

Abstract. The famous Drinfeld-Kohno theorem for simple Lie algebras states that the monodromy representation of the Knizhnik-Zamolodchikov equations for these Lie algebras expresses explicitly via R-matrices of the corresponding Drinfeld-Jimbo quantum groups. This result was generalized by the second author to simple Lie superalgebras of type A-G. In this paper, we generalize the Drinfeld-Kohno theorem to the case of the trigonometric Knizhnik-Zamolodchikov equations for simple Lie superalgebras of type A-G. The equations contain a classical r-matrix on the Lie superalgebra, and the answer expresses through the quantum R-matrix of the corresponding quantum group. The proof is based on the quantization theory for Lie bialgebras developed by the first author and D. Kazhdan. 1.

Classical r-matrices of two and three