Let $K$ be a field and $A$ the algebra $K\langle x_1,...,x_n\rangle/J^m$ for $n \geq 1$ and $m \geq 2$, where $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring in $n$ variables over $K$ and $J$ the ideal generated by $x_1,\dotsc,x_n$.

- What is the Lie algebra of the first Hochschild cohomology of $A$ (or at least its dimension)?
What is the Hochschild cohomology ring of that algebra?

The same questions, with the non-commutative polynomial ring replaced by a commutative one.

Tensor Hochschild homology and cohomology. $\endgroup$generatedby $x_1,\dots,x_n$ (the span of these elemens is not an ideal). $\endgroup$The GL-module structure of the Hochschild homology of truncated tensor algebras. The algebra they study is the same; but they study the homology, not the cohomology. But the projective resolution can't hurt... $\endgroup$1more comment