Modular finite W-algebras

Let ${\mathbb{k}}$ be an algebraically closed field of characteristic p > 0 and let G be a connected reductive algebraic group over ${\mathbb{k}}$. Under some standard hypothesis on G, we give a direct approach to the finite W-algebra $U(\mathfrak{g},e)$ associated to a nilpotent element $e \in \mathfrak{g} = \textrm{Lie}\ G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the p-centre of $U(\mathfrak{g},e)$, which allows us to define reduced finite W-algebras $U_{\eta}(\mathfrak{g},e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin’s equivalence of categories, generalizing recent work of the second author.

Two-parameter Quantum Superalgebras and PBW Theorem

A class of two-parameter quantum algebras [Formula: see text] is constructed. It is shown that [Formula: see text] is a Hopf superalgebra. Then the PBW basis of [Formula: see text] is described. For this purpose, some commutative relations of root vectors of [Formula: see text] are given.