Let $\mathfrak{g}$ be a $k$-Lie algebra, and $Q: \bigwedge^2 \mathfrak{g}^* \rightarrow k$; define $U_Q(\mathfrak{g})$ to be the quotient of the full tensor algebra over $\mathfrak{g}$ by the ideal generated by elements of the form $x\otimes y - y \otimes x -[x,y] - Q(x,y)$. This definition does not depends properly on $Q$ but only in its cohomology class in the Chevallay cohomology.

Has anyone seen this kind of algebra appear somewhere and/or has a name for them? They appeared to me from a deformation...