Lie algebras in ${\text{Ver}}_4^{+}$

Abstract

We develop Lie theory in the category ${\text{Ver}}_4^{+}$ over a field of characteristic 2, the simplest tensor category which is not Frobenius exact, as a continuation of arXiv:2406.10201. We provide a conceptual proof that an operadic Lie algebra in ${\text{Ver}}_4^{+}$ is a Lie algebra, i.e. satisfies the PBW theorem, exactly when its invariants form a usual Lie algebra. We then classify low-dimensional Lie algebras in ${\text{Ver}}_4^{+}$, construct elements in the center of $U(\mathfrak{gl}(X))$ for $X\in {\text{Ver}}_4^{+}$, and study representations of $\mathfrak{gl}(P)$, where $P$ is the indecomposable projective of ${\text{Ver}}_4^{+}$.

37 pages, 9 tables