University of Sydney Algebra Seminar

Serina Hu

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

Lie theory in ${\mathrm{Ver}}_4^{+}$ and Lie superalgebras in characteristic $2$

The simplest nontrivial higher Verlinde category, ${\mathrm{Ver}}_4^{+}$, 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 ${\mathrm{Ver}}_4^{+}$, 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 ${\mathrm{Ver}}_4^{+}$ 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 ${\mathrm{Ver}}_4^{+}$, 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 ${\mathrm{Ver}}_4^{+}$. 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.