Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2013
    • (edited Apr 8th 2013)

    I have added to groupoid convolution algebra the beginning of an Examples-section titled Higher groupoid convolution algebras and n-vector spaces/n-modules.

    Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a “higher C*-algebra” version of convolution algebras of higher Lie groupoids.

    I have labelled it as “under construction” to reflect the fact that this latter aspect is a bit experimental for the moment.

    The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor

    C:GrpdAlgopb2Mod

    (as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends double groupoids to convolution sesquialgebras, hence to 3-modules with basis (3-vector spaces).

    As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the “horizontally constant” double groupoid incarnation of BG, while the convolution algebra of G is the algebra of the “vertically discrete” double groupoid incarnation of BG.

    But next, if we simply replace the bare Algopb2Mod with the 2-category C*Algb of C*-algebras and Hilbert bimodules between them and assume (as seems to be the case) that C*-algebraic groupoid convolution is a 2-functor

    LieGrpdC*Algopn

    then the same argument goes through as before and yields convolution “C*-2-algebras” that look like Hopf-C*-algebras. Etc. Seems to go in the right direction…