What does “control of a deformation problem” mean?

Is the expression “control of a deformation problem’ ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ever defined? Answer AttributionSource : Link , Question Author : Jim Stasheff , Answer Author : Community

Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms. A tensor functor F:C→D is called dominant (sometimes called “surjective”) if for any Y:D, there is an X:C such that Y is a subobject of FX. It is known (“On fusion categories” by Pavel Etingof, Dmitri Nikshych, and Viktor … Read more

Quantization of $S^2$ as $C^*$-algebra?

The general context for the question – is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695). The particular question is about the most simple example – sphere $S^2$ with the standard symplectic form. I think that corresponding $C^*$-algebra can be defined explicitly by generators and relations. … Read more

Is the Nichols-Richmond theorem true for integral fusion rings?

The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper. It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here): Theorem: Let $x \in {\small \rm Irr}(\mathcal{C})$ with ${\small \rm FPdim} \ x = 2$. Then at least one of the following holds: $G[x] \neq 1.$ $\mathcal{C}$ has a … Read more

Lusztig’s definition of quantum groups

In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form UQ(q)q that is rather different from the usual approach (see [1,Ch.9.1] for expample). As far as I understood, the translation goes as follows: Let g be a simple Lie algebra and I a set of simple roots in … Read more

Nichols Algebras as Braided Hopf Algebras

Given a Hopf algebra H and a Yetter–Drinfeld module V over H, it is well-known that V has an induced braided vector space structure, and so, one can consider it’s Nichols algebra which is a braided Hopf algebra in the braided monoidal category of Yetter–Drinfeld modules over H. The notion, however, of a Nichols algebra … Read more

Is the associated grouplike γ=uS(u)−1\gamma=uS(u)^{-1} of a quasi-triangular Hopf algebra always the square of another grouplike?

Let (H,R) be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for H fin. dim. pointed with G(H) abelian). Let u∈H be the Drinfeld’t element and γ=uS(u)−1∈G(H) the associated group-like element. Does there always exist another grouplike g∈G(H), s.t. g2=γ? EDIT: I just saw that if … Read more

Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, because the general question is hard to answer, Is it known a complete classification of indecomposable modules for … Read more

Do global bases exist for quantum enveloping algebras at qq nonroot of unity?

Take k to be a field, q∈k a nonroot of unity, and U=Uq(g) the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write U− for its negative part. SHORT VERSION: can you get a Global basis of U− in the sense of Kashiwara and Lusztig in these conditions (i.e. not just … Read more