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).