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I guess not. Majid has cocycles for every $n$, but from 3 on one can not always go to cohomology classes in a consistent way. His approach generalizes the group cohomology with abelian coefficients, Lie algebra cohomology etc. but also some nonabelian cocycles like Drinfeld twist and Drinfeld associators. The combinatorics is beautiful but ill understood. GS is abelian in nature.
Hi Jim,
yes, the links on the $n$Lab are Gerstenhaber-Schack cohomology and bialgebra cocycle.
(moved this from another discussion)
Urs wrote in another thread:
For instance here Zoran points to bialgebra cocycle. But the first thing that this entry mentions is that, a) there is a standard notion of bialgebra cohomology which is precisely in line with the above slogan, and b) there is a differnt notion promoted by a single person so far.
Come on, Urs, your understanding of the subject is much higher than the surprising level of argument you descended here to (“promoted by a single person”, “another bialgebra cohomology”, “standard” for GS).
Majid’s bialgebra cocycle framework is, to my knowledge, the only known candidate which puts Drinfeld twist (Majid 2-cocycle) and Drinfeld associator (Majid 3-cocycle) in a context defined for all $n$, and for all Hopf algebras (and generalizes well to very many bialgebra-like structures like quasibialgebras, weak Hopf algebras, von Neumann-Hopf algebras etc.), and Gerstenhaber-Schack cohomology can say nothing about those two Drinfeld’s cocycle. The importance of the Drinfeld’s twist and Drinfeld’s associator (in deformation quantization, in knot theory, quantum groups, CFT…) can not be underestimated. GS cohomology is simply abelian cohomology of certain category of modules (tetramodules in Boris Shoikhet’s insight). Majid’s cohomology simultaneously can describe also (for some special coefficients) group cohomology etc. all in the same framework. To one algebraic structure one can associate many categories of modules. I know several other (and good) cohomologies related to bialgebras, on some of them I have some unpublished work (including some little piece of research with V. Lunts); namely to a bialgebra you can associate abelian categories of modules, Hopf modules, bimodules, comodules, bicomodules, tetramodules, Yetter-Drinfeld modules and so on, and clearly each of them will have its own derived category and hence the derived cohomology (in nonflat case over a ring, additionally one will have Hochschild-like complexes which do not agree with the derived functor cohomology). Majid’s example is however to go into a noncommutative generalization; it has also a dual version which works as well and has some known and some new special cases of interest. GS cohomology is on the other side, just an algebra devised to control the deformation theory of bialgebras – it is the analogue of the Hochschild cohomology, in a way, not a Hopf generalization of nonabelian group cohomology.
In my experience, in Hopf algebra community, Drinfeld twist is solely (not mentioning other important examples) used more often than GS cohomology, so your claims of GS being standard and Drinfeld’s and Majid’s cocycles being nonstandard is just a wishful talking.
Well, Majid’s cocycle framework is not generalizing the group cohomology with arbitrary coefficients, but rather special. It is noncommutative in the sense that the values may be noncommutative and the cocycle conditions noncommutative. In its abelian case it is defined for all n, in nonabelian case only cocycles are defined for all n but not the cohomology classes, the latter in noncommutative special cases works only up to n=2. Some of its cases can be treated via abelian categories, but not the general case. I do not know why would one need to look at it from deformation point of view, I find the extension, Hopf-algebraic torsor etc. more important motivations here.
We consider e.g. the extensions of bialgebras
$1\to K \stackrel{i}\hookrightarrow G \stackrel{p}\to B \to 1$in the sense that $i$ and $p$ are bialgebra maps and $G\cong K\otimes B$ as a left $B$-comodule and right $A$-module. One wants to clasify such extensions; it is like nonabelian extension of groups, just put $K = k[K_0]$ where $K_0$ is a group and $k$ is a ground field. Similar problems have their extension theory expressed in terms of variants of Majid’s cocycles, look at chapter 6.3 of Majid’s book. This is a hard case, the easier case is where one wants just to extend a comodule algebra, not a bialgebra; comodule algebras are like $G$-spaces; there is a special case of such extensions which we call Hopf-Galois extensions and are considered as noncommutative principal bundles/torsors. Some of such extension problems can be expressed in terms of variants of Majid’s theory, some are still open problem.
Thank you Jim for interesting comments. For some special cases of Majid’s cohomology, like for the universal enveloping of a Lie algebra some known cohomologies like Lie algebra cohomology appear. Thus it is justified to say cocycles and cohomology, but indeed in full generality, that is for any Hopf algebra, there is some problem with cohomology for higher $n$, I mean not with the cocycles or coboundaries but with making a “quotient” for $n$ after 3, but it is OK in a number of special choices for $H$. I will tell you more tomorrow or the day after (I was busy the whole day today), and will send you the relevant chapter.
is carefully written as an equality NOT of the form = 0
Surely, as it is not abelian. One can put all at one side of equation only in the abelian case.
To bring everything on one side is about having inverses. The difference between abelian and nonabelian is rather in whether what remains on the right is called “0” or “1”.
Urs, of course you are right, but this is not what I meant. Jim is talking about having a coboundary operator and relation, if I understood right. If the relation for cocycle twisting (see bialgebra cocycle) is
$\chi^\gamma = (\partial_+ \gamma) \chi (\partial_- \gamma)$and $\partial \gamma = (\partial_+ \gamma)( \partial_- \gamma)$ then the nonabelianess means that we can not assemble $\partial \gamma$ at one side, from consecutive parts. In the abelian case, we can and the cohomology relation is $\chi^\gamma = \chi + (\partial \gamma)$. Thus the abelian relation is by assembling the cocycle at one side.We can not do that in nonabelian case. The coboundary is kind of distributed around in pieces in nonabelian case. That is what I meant. I think that for $n=3$, while we still have the coboundary operator, we do not know how to distribute well to have a good relation of cohomologuness.
Right.
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