# “Twisted” universal enveloping algebra?

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...

-
I take it that $Q$ is a cocycle? If so, then what you are defining is a quotient of the universal enveloping algebra of the central extension of $\mathfrak{g}$. –  José Figueroa-O'Farrill Feb 14 '11 at 20:45
Yes, it should be a cocycle, forgot to tell it. Where can I find about central extensions? –  Pietro Tortella Feb 14 '11 at 21:00
Any treatment of Lie algebra cohomology should do it. I think it even goes back to the original paper of Chevalley--Eilenberg, but I'n not sure. –  José Figueroa-O'Farrill Feb 14 '11 at 21:03
Indeed, it's §26 in the Chevalley--Eilenberg paper: ams.org/mathscinet-getitem?mr=24908 –  José Figueroa-O'Farrill Feb 14 '11 at 22:32
By the way, older issues of AMS journals are now freely available. For the Chevalley-Eilenberg paper see: e-math.ams.org/journals/tran/1948-063-01/… –  Jim Humphreys Feb 16 '11 at 13:56