Etingof–Kazhdan quantization of Lie superbialgebras

Abstract

For every semi-simple Lie algebra g one can construct the Drinfeld-Jimbo algebra U DJ h (g). This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U DJ h (g), Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A g . He proved that particular categories of modules over the algebras U DJ h (g) and A g are tensor equivalent. Analogous constructions of the algebras U DJ h (g) and A g exist in the case when g is a Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the EtingofKazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

“…where (h, h ′ ) is the supertrace form on h. By the same computations as those in [16] (also see [50,51]), we can show that the radical Rad of the bilinear form is generated by the Serre relations and higher order Serre relations obeyed by the elements E a,a+1 . Thus U q (b; T ) :=Ũ q (b; T )/Rad coincides with the quantum Borel subalgebra of U q (g, φ; T ).…”

“…The notion of quasi Hopf algebras was introduced by Drinfeld in the seminal papers [9,10]. Generalisation to the super context was treated in [64, §II] (see also [16]). B.1.…”

“…It was shown in [64] (also see [16,17]) that the universal enveloping superalgebra U(g; T ) over the power series ring T = C [[t]] can be endowed with the structure of a braided quasi Hopf superalgebra, where the universal R-matrix was constructed from the quadratic Casimir operator and the associator was obtained by using solutions of a KZ equation following the strategy of Drinfeld [9]. The Drinfeld-Jimbo quantum supergroup U q (g; T ) [4,50,70] over T is the Etingof-Kazhdan quantisation of U(g; T ) [16,17]. It follows from a general property of the Etingof-Kazhdan quantisation that U q (g; T ) and U(g; T ) are equivalent (cf.…”

“…To consider quantum supergroups [4,50,70] from a deformation theoretical point of view [16,17,44,64], we will need the notion of quasi Hopf superalgebras [64,§II], which will be briefly discussed in Appendix B. Let g = g0 ⊕ g1 be a complex Lie superalgebra [22,39] with even subspace g0 and odd subspace g1.…”

“…Our study is based on the deformation theoretical treatment of quantum supergroups [16,17,44,64] and makes essential use of the Etingof-Kazhdan quantisation [13,14]. It was shown in [64] (also see [16,17]) that the universal enveloping superalgebra U(g; T ) over the power series ring T = C [[t]] can be endowed with the structure of a braided quasi Hopf superalgebra, where the universal R-matrix was constructed from the quadratic Casimir operator and the associator was obtained by using solutions of a KZ equation following the strategy of Drinfeld [9].…”