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Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well defined class in $HH^0(A)\simeq \left( A/[A,A] \right)^*$, the zero-th Hochschild cohomology of $A$. My question is

What are the higher analogs of this construction? namely which representation theoretic datum of $A$ produces classes in $HH^*(A)$?

By $HH^*(A)$ I mean $HH^*(A, A^*) \simeq (HH_*(A, A) )^*$, the Hochschild cohomology with coefficients on the bimodule $A^*$.

Here are things that I know and that may help clarify the question.

  • The Dennis Trace/Hattori-Stallings Trace/Connes-Chern Character defines a map from the $K$-theory of $A$ to Hochschild homology $HH_*(A)$. The fact that it factors through cyclic homology is irrelevant here, the important point is that it is a map between covariant functors and I'm asking for a contravariant target. In particular, in degree $0$ this is the map from $K^0(A)$ that to a finitely generated projective $A$-module $P$ defined by an idempotent $e \in \mathrm{Mat}_{n}(A)$ it assigns $\mathrm{tr} (e) \in A/[A,A] \simeq HH_0(A)$, instead of the above mentioned trace.

  • There is $K$-homology of Baum, Douglas, Kasparov, Gelfand and others in several flavours: analytic, geometric, operator algebras (see [1] for definitions and relations between them ). Roughly speaking, these theories are built out of (homotopy classes of) pairs $(H,F)$, a representation of $A$ on a Hilbert superspace $H$ endowed with an odd Fredholm operator $F$ such that $F^2 = \mathrm{id}_H$ plus some extra properties (essentially that some power of $[F,a]$ be a trace class operator for every $a \in A$). These theories indeed produce maps to Hochschild cohomology so I guess I could ask instead of the above:

    Is there a purely algebraic or representation theoretic version of $K$-homology of an associative algebra $A$?

  • I could try to mimic the construction of $K$-homology restricting to finite dimensional representations. For example given a finite dimensional representation $M$ of $A$ and an endomorphism $P$ with $P^2=1$ we always have the class $$ a \otimes b \mapsto \mathrm{tr}_M P[P,a][P,b] $$ which is well defined in $HH^1(A)$ (the higher analogs are well defined as well). This situation arises when you have for example a self extension of the form $$ 0 \rightarrow M \rightarrow E \rightarrow M \rightarrow 0$$ of $A$-modules, and consider $P = \begin{pmatrix} 0 & \mathrm{id}_M \\ \mathrm{id}_M & 0 \end{pmatrix}$. So in particular there is a way to associate to self-extensions of modules a class in $HH^{odd}(A)$.

  • Finally typically considering compactly supported objects or some form of finite objects is helpful to change functoriality. So:

    Is there a finite $K$-theory of associative algebras such that the corresponding chern-character has Hochschild cohomology as target instead of homology?

I'd appreciate any reference to the literature in these subjects, I'm looking to construct concrete explicit classes in $HH^*(A)$ or $HC^*(A)$ and not interested in any derived, up to homotopy or $A_\infty$ versions of these maps.

[1] A. Connes Noncommutative differential geometry Pub. Mat. IHES 62, 2 (1985) pp 41-144

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  • $\begingroup$ $F$ is $P$ I guess? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 14, 2018 at 4:55
  • $\begingroup$ Thanks, corrected, its important that (M,P) is not the (H,F) from K-homology (although they are related) as M does not need to be Z/2 graded and P does not need to be odd. $\endgroup$
    – Reimundo Heluani
    Commented Jul 14, 2018 at 5:13
  • $\begingroup$ Concerning the question - I don't know enough to really judge but I guess if you want $HH^*$ to be the target then you either have to restrict to smooth algebras or else have the source some form of $K$-homology, since in examples coming from spaces $HH^*$ relates to de Rham currents, i. e. to a form of de Rham homology, as opposed to $HH_*$ which relates to de Rham cohomology $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 14, 2018 at 5:18
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    $\begingroup$ Small comment. The construction you mention is obtained from functoriality of Hochschild homology. Namely, $A\rightarrow \mathrm{End}(M)$ gives rise to $HH_\bullet(A)\rightarrow HH_\bullet(\mathrm{End}(M))\cong k$, where I use that $\mathrm{End}(M)$ is Morita equivalent to $k$ since $M$ is finite-dimensional. Finally, you identify $HH^\bullet(A, A^*)\cong HH_\bullet(A)^*$. $\endgroup$
    – Pavel Safronov
    Commented Jul 14, 2018 at 8:27
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    $\begingroup$ If $V$ is a Tate vector space, there is a natural trace map $HH_\bullet(\mathrm{End}^{cts}(V))\rightarrow k[1]$, where $\mathrm{End}^{cts}$ is the algebra of continuous endomorphisms. So, if $V$ carries a continuous $A$-action, you get a class in $HH_1(A)^*$. $\endgroup$
    – Pavel Safronov
    Commented Jul 14, 2018 at 11:56

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