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.

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    $\begingroup$ For the commutative case, the Hochschild cohomology ring should be the polynomial ring in $n$ variables over $A$ for $m=2$ and for $m>2$ it should be the tenor product over $A$ of an exterior algebra in $n$ variables over $A$ and a polynomial ring in $n$ variables over $A$. For $n=\operatorname{char} K$ this is stated in "Hopf algebra structures and tensor products of group algebras" by Carlson and Iyengar. $\endgroup$ – Julian Kuelshammer Jul 7 '17 at 8:34
  • $\begingroup$ Possibly related: Claude Cibils, Tensor Hochschild homology and cohomology. $\endgroup$ – darij grinberg Jul 7 '17 at 10:10
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    $\begingroup$ I think the new title does not fit to the problem. $\endgroup$ – Mare Jul 7 '17 at 15:13
  • $\begingroup$ You probably mean that $J$ is the ideal generated by $x_1,\dots,x_n$ (the span of these elemens is not an ideal). $\endgroup$ – Victor Protsak Jul 7 '17 at 16:36
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    $\begingroup$ Actually, maybe this helps: Guillermo Ames, Leandro Cagliero, Paulo Tirao, 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... $\endgroup$ – darij grinberg Jul 7 '17 at 17:59

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.

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