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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...

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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

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up vote 10 down vote accepted

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.

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+1: Well spotted! –  José Figueroa-O'Farrill Feb 14 '11 at 20:53
    
That was one of the very first papers I ever read :) –  Mariano Suárez-Alvarez Feb 14 '11 at 20:55
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Rediscovery of neglected older mathematics will (I confidently predict) be a major pastime of future generations, unless we somehow persuade fewer people to take up the subject -;) –  Jim Humphreys Feb 14 '11 at 23:11
    
@Mariano: It's Ramaiyengar and not Ramiengar Sridharan –  S.C. Jun 14 '13 at 10:12

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