Let $A$ be an associative unital $k$-algebra, and let $M$ be a bimodule of $A$. The Hochschild cohomology of $A$ with coefficients in $M$ can be defined as $$HH^{n}(A,\,M)=\mathrm{Ext}^{n}_{A^{e}}(A,\,M)$$ where $A^{e}$ is the enveloping algebra of $A$.
Since both $A$ and its $k$-linear dual $A^{*}$ are $A$-bimodules, we have two cohomology theories $HH^{\bullet}(A,\,A)$ and $HH^{\bullet}(A,\,A^{*})$. Most of papers I saw call $HH^{\bullet}(A,\,A)$ the Hochschild cohomology of $A$ and denote it by $HH^{\bullet}(A)$. However, in Loday's book, he used $HH^{\bullet}(A)$ to denote $HH^{\bullet}(A,\,A^{*})$.
So it seems like that we have two different definitions of $HH^{\bullet}(A)$, the Hochschild cohomology of $A$. And these two definitions have their own advantages.
For example, $HH^{\bullet}(A,\,A)$ is a Gerstenhaber algebra and even a Batalin-Vilkovisky algebra for some algebra $A$. Moreover, if $A$ is smooth, we can relate $HH^{\bullet}(A,\,A)$ to the polyvector fields as the dual of differential forms.
On the other hand, $HH^{\bullet}(A,\,A^{*})$ is a contravariant functor from the category of $k$-algebras to the category of $k$-vector spaces. This definition of $HH^{\bullet}(A)$ is also related to the cyclic cohomology $HC^{\bullet}(A)$ of $A$, and we have the long exact sequence $$\longrightarrow HC^{n-2}(A)\longrightarrow HC^{n}(A)\longrightarrow HH^{n}(A)\longrightarrow $$ of Hochschild cohomology and cyclic cohomology.
I want to know that which definition of $HH^{\bullet}(A)$ is more popular and has more significant results in research. Are they defined from different points of view? Thanks a lot.
