Question

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...

Answer 1

These algebras were considered by Ramaiengar Sridharan a long time ago. See [Sridharan, R. Filtered algebras and representations of Lie algebras. Trans. Amer. Math. Soc. 100 1961 530--550. MR0130900 (24 #A754)]

If the map you are using to twist is not a Chevalley-Eilenberg cocycle, then things are ugly. In particular, you do not get a PBW-basis of the algebra (the cocycle condition is equivalent to the BPW property, in fact; this was in the general context of quadratic Koszul algebras a few years ago)

By the way: I call them Sridharan enveloping algebras, and I have heard others do the same.