Non-smooth algebra with smooth representation variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:48:09Z http://mathoverflow.net/feeds/question/9738 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9738/non-smooth-algebra-with-smooth-representation-variety Non-smooth algebra with smooth representation variety Peter Samuelson 2009-12-25T08:37:16Z 2009-12-25T11:00:45Z <p>A not necessarily commutative algebra A (over C, say) is called formally smooth (or quasi-free) if, given any map $f:A \to B/I$, where $I \subset B$ is a nilpotent ideal, there is a lifting $F:A \to B$ that commutes with the projection. (The reason for the terminology is that if we restrict to the category of finitely generated commutative algebras, this condition is equivalent to Spec(A) being smooth. For more info see the paper "algebra extensions and nonsingularity" by Cuntz and Quillen, 1995.) It isn't hard to see that if A is formally smooth, then the representation varieties $Rep_\mathbb{C}(A,V)$ are smooth (V is finite dimensional). Does anyone know of an example of an algebra that is not formally smooth, but whose representation varieties are smooth? </p> <p>One almost-answer is the Weyl algebra $A = \mathbb{C}\langle x,y\rangle/(xy - yx = 1)$. This isn't formally smooth, but its representation varieties are all empty. (To see this, take the trace of $xy - yx = 1$ to get $0 = n$.) This doesn't seem like it should count as answer, does anyone know a better one?</p> http://mathoverflow.net/questions/9738/non-smooth-algebra-with-smooth-representation-variety/9744#9744 Answer by lieven lebruyn for Non-smooth algebra with smooth representation variety lieven lebruyn 2009-12-25T11:00:45Z 2009-12-25T11:00:45Z <p>Take any semi-simple lie algebra g and consider its enveloping algebra U(g). As all finite dimensional representations are semi-simple, every representation variety rep_n U(g) is a finite union of orbits, whence smooth. No such U(g) is formally smooth.</p>