TITLE: Hochschild cohomology of certain local algebras
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: The right term to look for is "truncated quiver algebras". 
There are two relevant references which I believe lead to a complete answer to your question in the non-commutative case. 
First, Section 8.2 of "Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras" by Guillermo Ames, Leandro Cagliero, and Paulo Tirao (http://www.sciencedirect.com/science/article/pii/S0021869309002907) suggests that the product in cohomology is zero in positive degrees (for $n>1$).
Second, Theorem 1.3 of the article "The Lie algebra structure of the first Hochschild cohomology group for monomial algebras" by Claudia Strametz (http://www.sciencedirect.com/science/article/pii/S1631073X02023464) has formulas for the Lie bracket on the first Hochschild cohomology in the general case of a monomial algebra (using combinatorics of Anick chains /  Bardzell resolution). It should be possible to unravel these formulas completely in your case.