Formal deformations of algebras over not necessarily commutative rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:36:49Z http://mathoverflow.net/feeds/question/21892 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21892/formal-deformations-of-algebras-over-not-necessarily-commutative-rings Formal deformations of algebras over not necessarily commutative rings S1 2010-04-19T21:04:55Z 2010-04-19T21:26:58Z <p>In Iain Gordon's survery article <a href="http://www.maths.ed.ac.uk/~igordon/pubs/ICRAsurvey_dec9.pdf" rel="nofollow">"Symplectic reflection algebras"</a> the concept of formal deformations of algebras over semisimple artinian (not necessarily commutative) rings is summarized (chapter 2). Unfortunately, deformations are needed in this generality and there are a few general things I don't understand: </p> <ol> <li><p>An algebra over a semisimple artinian $\mathbb{C}$-algebra $k$ is "defined" as a $k$-bimodule $A$ with a $k$-bimodule morphism $A \otimes_k A \rightarrow A$. Is this a standard definition and is it correct that there is no associativity or unity assumption? I could not find a single book defining an algebra over a not necessarily commutative ring.</p></li> <li><p>In the definition of a formal deformation given in the survey there also seems to be no associativity or unity assumption. Is this a standard definition and does Hochschild cohomology also work in this setting with the same interpretation? My problem is that when I don't restrict my deformations to be associative or have a unit, I might get a lot more deformations.</p></li> </ol> <p>Is there some literature discussing this in more detail?</p> http://mathoverflow.net/questions/21892/formal-deformations-of-algebras-over-not-necessarily-commutative-rings/21897#21897 Answer by Vladimir Dotsenko for Formal deformations of algebras over not necessarily commutative rings Vladimir Dotsenko 2010-04-19T21:26:58Z 2010-04-19T21:26:58Z <p>I recall your <a href="http://mathoverflow.net/questions/15107/algebra-unital-associative-algebra-better-terminology" rel="nofollow">question</a> on a related topic... </p> <p>I am sure that associativity is assumed here (and just omitted because the audience is unlikely to think of any other algebras); as for unitality, won't it be preserved automatically exactly because of that yoga "when we deform A we don't want to deform k"?</p>