Lie Super-bialgebraand Quantization Of The Super Virasoro Algebra

Quantizing Lie (super)algebras is an important method for constructing quantum groups. In this process, the Drinfeld twist element F plays a key role, and it is closely related to the Lie (super-)bialgebra structures. Therefore, quantization and Lie (super-) bialgebra structures of Lie (super)algebras are inseparable.In this paper, we study Lie super-bialgebra structures and quantization of the super Virasoro algebra C, which is defined as an infinite-dimensional Lie superalgebra over C with a basis{Ln,Fn,Gn|n∈Z}, and satisfying the following nontrivial relations: The Z2-grading is defined by requiring that deg(Li)=deg(Fi)= 0, deg(Gi)= 1, for i∈Z.Firstly, we shall generalize the Michaelis theorem to the super case. That is, for any Lie superalgebra L containing linearly independent elements a and b satisfying [a, b]= kb with |a|=|b|= 0, 0≠k∈C, then the cobracket δ with r= a(?)b-b(?)a equips L with the structure of a triangular coboundary Lie super-bialgebra.Secondly, we shall prove that all derivations from the super Virasoro algebra C to its adjoint module V=L(?)L are inner. In other words, H1 (L, V)= 0. By using this, we shall prove that all Lie super-bialgebra structures on C are triangular coboundary.Finally, let satisfying [X, Y]= Y. Thus X and Y generate a two-dimensional nonabelian Lie sub-algebra of L. Set r= X(?)Y-Y(?)X. Then δr equips C with the structure of a triangular coboundary Lie super-bialgebra. Based on this, we construct the Drinfeld twist element F. By using F, we deform the standard Hopf superalgebra (U/(L)[t]], μ, l, Δ0,S0,ε) on the universal enveloping algebra U(L) of L and obtain a class of noncommutative and noncocommutative Hopf superalgebras (U/(?)[t]], μ, i, Δ, S, ε).