Is there any reference treating explicitly Lie-Rinehart pairs over non-commutative base algebras?
Wondering around in the literature, any reference to Lie-Rinehart algebras define them as pairs $(A,L)$ where $A$ is a commutative algebra over some field $\Bbbk$ (or even commutative ring) and $L$ is a Lie algebra satisfying certain additional properties.
However, in the nlab page about Lie-Rinehart pairs seems to be more or less explicitly written that commutativity of $A$ is not necessary. Namely, therein a Lie-Rinehart pair is defined as a couple $(A,L)$ where
- $A$ is a non-commutative associative and unital algebra
- $L$ is a Lie algebra
- $L$ acts on $A$ by $\Bbbk$-linear derivations (i.e. we have a Lie algebra map $\omega:L \to \mathsf{Der}_{\Bbbk}(A)$)
- $A$ acts on $L$ by $\Bbbk$-linear endomorphisms (i.e. we have an algebra map $A \to \mathsf{End}_{\Bbbk}(L)$) such that $$[X,aY] = \omega(X)(a)Y + a [X,Y]$$ for all $X,Y \in L$ and $a \in A$.
In light of this, I am a bit surprised of not finding papers in non-commutative geometry treating this slightly more general "non-commutative analogue" of Lie-Rinehart algebras. Is there anybody aware of some reference doing it? Maybe under a different name than "Lie-Rinehart algebras" or "Lie-Rinehart pairs"?
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$\begingroup$A non-commutative version of a Lie-Rinehart algebra was introduced by Michel Van den Bergh in Non-commutative quasi-Hamiltonian spaces in Poisson geometry in mathematics and physics, Volume 450 of Contemp. Math., pages 273–299. Amer. Math. Soc., Providence, RI, 2008.
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