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Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

If the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped, although the notion itself seems fairly natural.

Indeed, any associative algebra is readily endowed with a Lie algebra structure with the commutator $[a,b]=a\circ b-b\circ a$ which obeys the Leibniz rule $(*)$ but there could exist other Lie algebra structures which still obey $(*)$.

Any relevant references are greatly appreciated. Thanks in advance.

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    $\begingroup$ Is it not the same as a "non-commutative Poisson algebra" (ref1, ref2)? $\endgroup$
    – Igor Khavkine
    Commented Mar 11, 2016 at 15:46
  • $\begingroup$ Many thanks. In particular, ref.1 is obviously spot on. $\endgroup$
    – another-guest
    Commented Mar 11, 2016 at 16:27
  • $\begingroup$ Another possible direction to go: if one thinks of the Poisson algebra as incarnated equivalently in its Poisson Lie algebroid, then one may ask for the generalization to Lie algebroids over non-commutative base manifolds. This is well studied in the guise of "Lie-Rhinehart pairs" ncatlab.org/nlab/show/Lie-Rinehart+pair $\endgroup$
    – Urs Schreiber
    Commented Mar 11, 2016 at 21:21

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This seems to first have been considered by Dirac under the name "quantum Poisson bracket" - an easy accessible reference is Fock's "Fundamentals of Quantum Mechanics", discussion around formula (2.10) in the translation into English of 1978.

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