Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim\_K$ of its hochschild homology $HH\_k(A;A)$?


Is there any software to compute Hochschild homology?



Hochschild homology of upper triangular matrix algebra?
Sasha's argument is pretty technological. You can really do this almost by hand, though.


Let $A$ be the algebra in question, let $r$ be its Jacobson radical (that is, the subspace of strictly upper triangular matrices), and let $E$ be the subalgebra of the diagonal matrices in $A$ (which is a complement to $r$) Notice that both $A$ and $r$ are $E$-bimodules.


The algebra $A$ has a projective resolution as a bimodule of the form $A\otimes\_E r^{\otimes\_E \*}\otimes\_E A$ which looks exactly like the Hochschild resolution but the inner copies of $A$ have been replaced by $r$, and all tensor products involved are over $E$ and not over the base field; the differentials in the complex have exactly the same formula as the usual Hochschild differential. This can be checked easily —it is a nice exercise— but you can find the details in a nice paper by Claude Cibils on square-zero algebras, if I recall correctly. (This is like the reduced Hochschild resolution, but instead of killing the copy of $k$ inside $A$, we kill the whole of $E$; almost anything useful that one wants to do equires that we be aware of this complex, so it is important to keep it at hand)


Now, $HH\_\*(A)$ is $Tor^{A^e}\_\*(A,A)$, so it can be computed as the homology of the complex obtained from $A\otimes\_E r^{\otimes\_E \*}\otimes\_E A$ by applying the functor $A\otimes\_{A^e}(\mathord-)$. You should explicitly describe this complex: its only non-zero term is the $0$th one, so its homology is very, very easy to compute!


(The same thing can be done for every triangular algebra, that is, every algebra whose ordinary quiver is acyclic)




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There are various programs people have written to compute Hochschild homology and cohomology; for example, I understand that Ed Green and his students have written code to do non-comm. Groebner bases on quotients of paths algebras and, probably, to compute (co)homology, and there are others (I have written code to handle very special cases, for example) I am not aware of any other approach apart from «try to be smart about n.c. Groebner bases, work hard for a minimal resolution, and then just do linear algebra.» The first step is pretty well understood for quotients of path algebras, say; the second one can be don algorithmically, I think; the third one is of course very well understood. In practice, interesting examples tend to result in huuuuge computations, and the result is somewhat unenlightening.


