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1. • CommentRowNumber1.
   • CommentAuthorfpaugam
   • CommentTimeJan 18th 2012
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   Author: fpaugam
   Format: TextHow is Majid's bialgebra cohomology related to GS cohomology for bialgebras? I saw the recent paper by Shoikhet where GS cohomology is k-monoidally understood. Is there a similar understanding of Majid's cohomology?

   How is Majid's bialgebra cohomology related to GS cohomology for bialgebras?
   
   I saw the recent paper by Shoikhet where GS cohomology is k-monoidally understood.
   
   Is there a similar understanding of Majid's cohomology?
2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 18th 2012
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   Author: zskoda
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeJan 19th 2012
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   Author: jim_stasheff
   Format: TextWhat is GS? Gerstenhaber-Schack?

   What is GS? Gerstenhaber-Schack?
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 19th 2012
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   Author: Urs
   Format: MarkdownItexHi Jim, yes, the links on the $n$Lab are _[[Gerstenhaber-Schack cohomology]]_ and _[[bialgebra cocycle]]_.

   Hi Jim,

   yes, the links on the $n$Lab are Gerstenhaber-Schack cohomology and bialgebra cocycle.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJan 19th 2012
   • (edited Jan 19th 2012)
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   Author: zskoda
   Format: MarkdownItex(moved this from another discussion) Urs wrote in [another thread](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3446&Focus=28605#Comment_28605): > 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 module]]s, 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.

   (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.

6. • CommentRowNumber6.
   • CommentAuthorfpaugam
   • CommentTimeJan 22nd 2012
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   Author: fpaugam
   Format: TextOups! I didn't want to make a war on this... I like the idea that quantum group deformations correspond to deforming k-monoidal infinity-categories of modules (or dg, if you prefer), where k=1 or 2 starting from infinity-monoidal (i.e. monoidal symmetric) categories of modules on a commutative thing. May one understand the Drinfeld twist in a similar fashion? I thought it was more something like keeping the k-monoidal category of modules fixed (gauge transformation). I want to use this to deform quantize homogeneous spaces, but i would like to have a clear understanding of the deformation theory (a la DDT in the Lurie's ICM sense) that underlies Majid's cohomology. In the algebra deformation, his cocycles are just convolution invertible elements of the dual that are also counital. What kind of categorical structures are we deforming when we look at these things? k-monoidal infinity-categories of modules/comodules? I don't think that Majid's cohomology is really more noncommutative than the GS cohomology, because he is able to define cocycles in every degrees, and this is not really possible when one works with group non abelian cohomology, because one can't define BBG when G is noncommutative, for example GL_n. What do you think?

   Oups! I didn't want to make a war on this...
   
   I like the idea that quantum group deformations correspond to deforming k-monoidal infinity-categories of modules (or dg, if you prefer),
   where k=1 or 2 starting from infinity-monoidal (i.e. monoidal symmetric) categories of modules on a commutative thing.
   
   May one understand the Drinfeld twist in a similar fashion? I thought it was more something like keeping the k-monoidal category of modules fixed (gauge transformation). I want to use this to deform quantize homogeneous spaces, but i would like to have a clear understanding of the deformation theory (a la DDT in the Lurie's ICM sense) that underlies Majid's cohomology. In the algebra deformation, his cocycles are just convolution invertible elements of the dual that are also counital. What kind of categorical structures are we deforming when we look at these things? k-monoidal infinity-categories of modules/comodules?
   
   I don't think that Majid's cohomology is really more noncommutative than the GS cohomology, because he is able to define cocycles in every degrees, and this is not really possible when one works with group non abelian cohomology, because one can't define BBG when G
   is noncommutative, for example GL_n.
   
   What do you think?
7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJan 22nd 2012
   • (edited Jan 22nd 2012)
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   Author: zskoda
   Format: MarkdownItexWell, 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorfpaugam
   • CommentTimeJan 23rd 2012
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   Author: fpaugam
   Format: TextWhat do you mean by extension and hopf-algebraic torsor?

   What do you mean by extension and hopf-algebraic torsor?
9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeJan 23rd 2012
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   Author: zskoda
   Format: MarkdownItexWe 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 extension]]s 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.

   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.

10. • CommentRowNumber10.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 25th 2012
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    Author: jim_stasheff
    Format: Text@5 and other 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 as I read these comments, there seems to be a disconnect between cocycles and cohomology what is the benefit of *calling* a solution of the equation a cocycle? labelling Drinfeld's twist and associator as cocycles - OK but a rose by any other name... what does Majid's pov give beyond GS? also: ``variants of Majid's cocycles, look at chapter 6.3 of Majid's book'' are we far from having a general theory? could some one send the relevant pages - wading through 640 pages is not for me and not at that price also notice the appearance of the associahedron in relation to his associator

    @5 and other
    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
    
    as I read these comments, there seems to be a disconnect between cocycles and cohomology
    what is the benefit of *calling* a solution of the equation a cocycle?
    
    labelling Drinfeld's twist and associator as cocycles - OK but a rose by any other name...
    what does Majid's pov give beyond GS?
    
    also: ``variants of Majid's cocycles, look at chapter 6.3 of Majid's book''
    
    are we far from having a general theory?
    could some one send the relevant pages - wading through 640 pages is not for me and not at that price
    
    also notice the appearance of the associahedron in relation to his associator
11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJan 25th 2012
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    Author: zskoda
    Format: MarkdownItexThank 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.

    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.

12. • CommentRowNumber12.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 26th 2012
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    Author: jim_stasheff
    Format: Textin re: http://ncatlab.org/nlab/show/bialgebra%20cocycle notice that the Drinfeld associator condition is carefully written as an equality NOT of the form = 0 and moreover corresponds to comparing the usual two ways to go from a(b(cd)) to ((ab)c)d around the associahedron so Majid's condition might better be written that way which is fine for cocycles though not for defining a coboundary operator

    in re: http://ncatlab.org/nlab/show/bialgebra%20cocycle
    notice that the Drinfeld associator condition is carefully written
    as an equality NOT of the form = 0
    and moreover corresponds to comparing the usual two ways to go
    from a(b(cd)) to ((ab)c)d around the associahedron
    so Majid's condition might better be written that way
    which is fine for cocycles
    though not for defining a coboundary operator
13. • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJan 26th 2012
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    Author: zskoda
    Format: MarkdownItex> 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2012
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    Author: Urs
    Format: MarkdownItexTo 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".

    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”.

15. • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeJan 27th 2012
    • (edited Jan 27th 2012)
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    Author: zskoda
    Format: MarkdownItexUrs, 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.

    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.

16. • CommentRowNumber16.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 27th 2012
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    Author: jim_stasheff
    Format: Text@15 Zoran: thanks coboundary is kind of distributed around in pieces in nonabelian case again Drinfeld's associator conditions are a very good example also brings to mind parity complexes

    @15 Zoran: thanks
    coboundary is kind of distributed around in pieces in nonabelian case
    again Drinfeld's associator conditions are a very good example
    also brings to mind parity complexes
17. • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeJan 27th 2012
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    Author: zskoda
    Format: MarkdownItexRight.

    Right.