Homological invariants relating the super Jordan plane to the Virasoro algebra

TL;DR

This paper investigates homological invariants of the super Jordan plane Nichols algebra, revealing its Hochschild homology, cohomology, and Lie structures, and establishing properties like being $K_2$ and finitely generated Yoneda algebra.

Contribution

It provides the first detailed homological analysis of the super Jordan plane Nichols algebra, including its Hochschild invariants and Lie algebra structure related to the Virasoro algebra.

Findings

The algebra $A$ is $K_2$.

The Yoneda algebra of the bosonization of $A$ is finitely generated.

The algebra $A$ has a Lie subalgebra of the Virasoro algebra in its first cohomology.

Abstract

Nichols algebras are an important tool for the classification of Hopf algebras. Within those with finite GK dimension, we study homological invariants of the super Jordan plane, that is, the Nichols algebra $A=B(V(-1,2))$. These invariants are Hochschild homology, the Hochschild cohomology algebra, the Lie structure of the first cohomology space - which is a Lie subalgebra of the Virasoro algebra - and its representations $H^n(A,A)$ and also the Yoneda algebra. We prove that the algebra $A$ is $K_2$ . Moreover, we prove that the Yoneda algebra of the bosonization of $A$ is also finitely generated, but not $K_2$ .