If $A$ is a ($\mathbb Z$-)graded algebra, we can define its Hochschild cohomology in the usual way, via the standard complex of $A$-bimodules:

$$C_*(A)=\cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A.$$

Since $A$ is graded, then so is the envelopìng algebra $A^{op}\otimes A$, therefore there is a ($\mathbb Z$-)graded $Hom$ in the category of $A$-bimodules, that we can use to define the Hochschild cohomology complex,

$$C^{*,\bullet}(A)=Hom_{A^{op}\otimes A}^{\bullet}(C_*,A).$$

Therefore Hochschild cohomology is bigraded $HH^{*,\bullet}(A)$ (I'm just interested in $A$ as a bimodule of coefficients).

Following some computations of mine, the $(1,0)$-cochain $\delta$ which multiplies each homogenoeus element in $A$ by its degree

$$\delta(x)=|x|\cdot x$$

seems to be a cocycle, and its cohomology class

$$\{\delta\}\in HH^{1,0}(A)$$

is surprisingly (in my opinion) non-trivial since it can somehow detect the degree of any other cohomology class $y\in HH^{p,q}(A)$, $|y|=q$, via the Lie bracket $$[\{\delta\},y]=|y|\cdot y.$$ That formula actually holds at the cochain level. It's turned out to play an important role in a current project and I wonder whether anybody has seen this class, or anything like that before. Is there maybe any mistake in what I say?

Notice that it is a very particular phenomenon of the graded setting. In the ungraded case $\delta$ is trivial already as a cochain.

EDIT: The degree of $y$ is $q$, not $p$ (which is what was written before).

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    $\begingroup$ Avramov and Herzog in "Jacobian criteria ..." use the Euler morphism (in the commutative setting) $\delta \in Hom(\Omega_A,A) = HH¹(A)$ in a way that reminds me your question (compare e.g. their lemma 1.15 with your formula). Sorry if too unrelated to your question. $\endgroup$ – Vinteuil Feb 19 '14 at 20:00
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    $\begingroup$ Your cocycle is in fact an homogeneous derivation of the algebra (tangent to the obvious action of the multiplicative group), the so called Eulerian derivation. It shows up in many contexts, usually more or less implicitly. For example, you can see it in Goodwillie's theorem on cyclic homology of graded algebras. $\endgroup$ – Mariano Suárez-Álvarez Feb 19 '14 at 22:24
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    $\begingroup$ Notice that sometimes the class of $\delta$ in cohomology is zero. For example, if $A$ is the Weyl algebra $\mathbb C\langle x,y|xy-yx=1\rangle$, graded with $x$ in degree $1$ and $y$ in degree $-1$, then the Eulerian derivation coincides with $[xy,\mathord-]$, so its class in $HH^1$ vanishes. The class of $\delta$ cannot be zero if the algebra is, say, has component of degree zero contained in the center, of course. (I don't understand you last sentence: how do you define $\delta$ for an ungraded algebra?) $\endgroup$ – Mariano Suárez-Álvarez Feb 19 '14 at 22:27
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    $\begingroup$ This cochain appears in a beautiful symplectic geometry paper of Abouzaid and Smith. See theorem 2.6 of arxiv.org/pdf/1311.5535v1.pdf. $\endgroup$ – user36931 Feb 20 '14 at 14:45
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    $\begingroup$ I should say that the upshot of theorem 2.6 is a characterization of formality for an $A_{\infty}$ algebra via this cochain. $\endgroup$ – user36931 Feb 20 '14 at 17:58

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