On principal finite W-algebras for the Lie superalgebra D(2, 1; α)

Abstract

We study finite W-algebras associated to even regular (principal) nilpotent elements for the family of simple exceptional Lie superalgebras D(2, 1; α) and for the universal central extension of 𝔭𝔰𝔩(2|2). We give a complete description of these finite W-algebras in terms of generators and relations.

On the finite W-algebra for the Lie superalgebra Q(N) in the non-regular case

“…Recently Y. Zeng and B. Shu have proved this conjecture for a basic Lie superalgebra g over C of any type except D(2, 1; α), where α ∈Q ( [20], Theorem 0.1). In [13] it is proven for D(2, 1; α) and a regular nilpotent χ. We proved this conjecture for g = Q(n) in the regular case in [12] (Corollary 4.9).…”

“…The general definition of a finite W -algebra was given by A. Premet in [15] (see also [7]). In the case of Lie superalgebras, finite W -algebras have been extensively studied by mathematicians and physicists in [1,2,[10][11][12][13][14][18][19][20].…”