I am wondering if there exist a universal construction of (co)-chain complex associated to a given algebra to study deformations as Hochschild homology for associative algebras, or Chevalley-Eilenberg cohomology for Lie algebras. I am interested in Lie bialgebras and their deformations. In particular, is it possible to use operads to construct a chain complex (as in simplicial homology)?
$\begingroup$
$\endgroup$
4
-
$\begingroup$ You should go to properads if you want to model bialgebras. A general theory that suffices for bialgebras is worked out in "Deformation theory of representations of prop(erad)s I and II" by Merkulov and Vallette. $\endgroup$– Gabriel C. Drummond-ColeCommented Nov 28, 2014 at 11:43
-
$\begingroup$ I believe that the case of Lie bialgebras was worked out earlier by hand in Y. Kosmann-Schwarzbach, Grand crochet, crochets de Schouten et cohomologies d’algèbres de Lie, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 1, 123–126., but I haven't actually read it. $\endgroup$– Gabriel C. Drummond-ColeCommented Nov 28, 2014 at 11:51
-
$\begingroup$ I agree with Gabriel's comments, but want to add one remark. There are different ways to model many-to-many operations: properads and props (which are essentially equivalent, by a deep result of Vallette's) allow compositions of arbitrary "genus" (or "loop order"); an alternate version, called "dioperads", uses only "tree-level" compositions. This in general makes a huge difference: the answers to homological-type questions can be very difference. I think that for Lie bialgebras, you happen to get the same answers in the different settings, but for other types of bialgebras you often do not. $\endgroup$– Theo Johnson-FreydCommented Nov 28, 2014 at 21:22
-
1$\begingroup$ A very down to earth approach connecting to deformation theory is in my joint paper with L. Guerra "The variety of Lie bialgebras" emis.de/journals/JLT/vol.13_no.2/16.html $\endgroup$– Nicola CiccoliCommented Dec 18, 2014 at 9:44
Add a comment
|