University of Sydney Algebra Seminar

Serina Hu

Friday 21 March, 12-1pm, in Carslaw 175

Lie theory in \(\DeclareMathOperator{\Ver}{Ver} \Ver_4^+\) and Lie superalgebras in characteristic \(2\)

The simplest nontrivial higher Verlinde category, \(\Ver_4^+\), is a reduction of the category of supervector spaces to characteristic \(2\) (Venkatesh), so studying Lie theory in this category provides a theory of supergroups and superalgebras in characteristic \(2.\) In this talk, we first discuss representations of general linear groups in \(\Ver_4^+\), which can be viewed as a notion of general linear supergroups in characteristic \(2\). We classify their irreducible representations in terms of highest weights and conjecture a Steinberg tensor product theorem. We then define a Lie algebra in \(\Ver_4^+\) and prove a PBW theorem, which provides a notion of Lie superalgebra in characteristic \(2\), and discuss how to classify such Lie algebras. Finally, we define the notion of Lie superalgebra in \(\Ver_4^+\), which will unify both a pre-existing notion of Lie superalgebra in characteristic \(2\) as a \(Z/2\)-graded Lie algebra with squaring map (Bouarroudj et. al) and the notion of a Lie algebra in \(\Ver_4^+\). Time permitting, we will also discuss a natural lift of this notion to characteristic \(0\) (for perfect \(k\)), which we call a mixed Lie superalgebra over a ramified quadratic extension of the ring of Witt vectors \(W(k)\). This is joint work with Pavel Etingof.