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Some remarks on quantized Lie superalgebras of classical type
 Preprint. LINK INVARIANTS FROM LIE SUPERALGEBRAS 23
"... Abstract. In this paper we use the EtingofKazhdan quantization of Lie bisuperalgebras to investigate some interesting questions related to DrinfeldJimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the DJ type superalgebra associated to a Lie supera ..."
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Cited by 4 (4 self)
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Abstract. In this paper we use the EtingofKazhdan quantization of Lie bisuperalgebras to investigate some interesting questions related to DrinfeldJimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the DJ type superalgebra associated to a Lie superalgebra of type AG, with the distinguished Cartan matrix, is isomorphic to the EK 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 AG can be deformed to modules over the corresponding DJ type superalgebra and 2) a super version of the DrinfeldKohno Theorem. 1.
The Kontsevich integral and quantized Lie superalgebras
 Algebr. Geom. Topol
"... 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 Rmatrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that ..."
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Cited by 4 (3 self)
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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 Rmatrix, 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 AG. We use this result to investigate the LinksGould invariant. We also give a positive answer to a conjecture of PatureauMirand’s concerning invariants arising from the Lie superalgebra D(2, 1; α).
MULTIVARIABLE LINK INVARIANTS ARISING FROM sl(21) AND THE ALEXANDER POLYNOMIAL
, 2006
"... Abstract. In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(21). The usual quantum group invariant of links associated to (generic) representations of sl(21) is trivial. However, we modify this construction ..."
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Cited by 3 (3 self)
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Abstract. In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(21). The usual quantum group invariant of links associated to (generic) representations of sl(21) 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 LinksGould 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.
Classification of two and three dimensional Lie superbialgebras, arXiv:0901.4471 [mathph
"... Using adjoint representation of Lie superalgebras, we write the matrix form of super Jacobi and mixed super Jacobi identities of Lie superbialgebras. Then through direct calculations of these identities and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain an ..."
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Cited by 2 (2 self)
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Using adjoint representation of Lie superalgebras, we write the matrix form of super Jacobi and mixed super Jacobi identities of Lie superbialgebras. 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 superbialgebras.
SUPER SOLUTIONS OF THE DYNAMICAL YANGBAXTER EQUATION
, 2005
"... Abstract. A super dynamical rmatrix 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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Cited by 1 (1 self)
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Abstract. A super dynamical rmatrix 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.
ATG The Kontsevich integral and quantized Lie superalgebras
, 2005
"... 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 Rmatrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that t ..."
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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 Rmatrix, 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 AG. We use this result to investigate the LinksGould invariant. We also give a positive answer to a conjecture of PatureauMirand’s concerning invariants arising from the Lie superalgebra D(2, 1; α). AMS Classification 57M27; 17B65, 17B37
Dynamical Quantum Groups The Super Story
, 2007
"... 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 YangBaxter equations in the super setting and develop the super analog of the theory of dynamical quantum groups. 1. ..."
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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 YangBaxter equations in the super setting and develop the super analog of the theory of dynamical quantum groups. 1.
MONODROMY OF TRIGONOMETRIC KZ EQUATIONS
, 2006
"... Abstract. The famous DrinfeldKohno theorem for simple Lie algebras states that the monodromy representation of the KnizhnikZamolodchikov equations for these Lie algebras expresses explicitly via Rmatrices of the corresponding DrinfeldJimbo quantum groups. This result was generalized by the secon ..."
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Abstract. The famous DrinfeldKohno theorem for simple Lie algebras states that the monodromy representation of the KnizhnikZamolodchikov equations for these Lie algebras expresses explicitly via Rmatrices of the corresponding DrinfeldJimbo quantum groups. This result was generalized by the second author to simple Lie superalgebras of type AG. In this paper, we generalize the DrinfeldKohno theorem to the case of the trigonometric KnizhnikZamolodchikov equations for simple Lie superalgebras of type AG. The equations contain a classical rmatrix on the Lie superalgebra, and the answer expresses through the quantum Rmatrix 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.