3
votes
1answer
153 views

example of a compact quantum group at a root of unity?

In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some ...
4
votes
2answers
445 views

$q$-Deforming Woronowicz's Leibniz Rule

The Woronowicz definition of a differential calculus over an algebra consists of a pair $(\Omega,$d$)$, where $\Omega$ is an $A-A$-bimodule, and $$ \text{d}:A \to \Omega, $$ is a bimodule map, ...
6
votes
2answers
297 views

Are the Drinfeld compact quantum groups simply connected ?

To fix notations : let G be simply connected simple compact group, and $U_q(\mathcal{G})$ the Drinfeld-Jimbo universal algebra quantization of its complexified algebra defined as usual, with q not ...
9
votes
3answers
556 views

Is the nc torus a quantum group?

The non-commutative n-torus appears in many applications of non-commutative geometry. To stay in the setting $n=2$: it is a C$^\ast$-algebra generated by unitaries $u$ and $v$, satisfying $u v = e^{i ...
2
votes
0answers
99 views

Haar Functionals and Coquasi-triangular Structures

In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
6
votes
4answers
932 views

Compact Quantum Groups from Hopf Algebras

For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way ...
10
votes
2answers
469 views

Relationship between “different” quantum deformations

This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too. Let $\mathfrak{g}$ be a (simple) Lie algebra and ...
10
votes
1answer
444 views

Generators of the Odd Dimensional Quantum Spheres

As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
10
votes
1answer
1k views

Grothendieck and Non-commutative Geometry?

When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the ...
5
votes
1answer
542 views

Weyl Character Formula for Quantum Groups

How much is known about the Weyl character formula for quantum groups? More specifically, has the formula been generalized to the general setting of deformed coordinate algebras $\mathbb{C}[G_q]$ of ...
3
votes
0answers
131 views

The Tangent Spaces of the Bicovariant Calculi over Quantum SU(3)

In Woronowicz's approach to differential calculi on Hopf algebras, a calculus over an algebra $A$ can be specified by a finite dimensional subspace of the dual of $A$. The best known example is his ...
3
votes
0answers
122 views

Restricting Differential Calculi to Quotients

Let $A$ be an algebra, $B \subset A$ a subalgebra, and $(\Omega^1(A),d)$ a first order differential calculus over $A$. Now, as is well known, the subalgebra of $\Omega^1(A)$ consisting of elements of ...
1
vote
0answers
184 views

Bicovariant Calculi on the Quantum Unitary Groups

The bicovariant differential calculi on quantum-$SU(n)$ have been classified (by Schmudgen I think) and have been shown to have non-classical dimension. My question is whether or not the bicovariant ...
8
votes
6answers
869 views

Hopf algebras arising as Group Algebras

Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
7
votes
1answer
573 views

Is there a good differential calculus for quantum SU(3)?

For quantum $SU(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $SU(2)$ by $a,b,c,d$, then the ideal of ker($\epsilon)$ corresponding to this calculus is $$ ...
3
votes
3answers
512 views

Basis of quantum SU(n)

As is well known, the set $\{a^ib^jc^k | i,j,k \in \mathbb{Z}\_{\geq 0},k>0\} \cup \{b^lc^md^n | l,m,n \in \mathbb{Z}\_{\geq 0}\}$ forms a basis for quantum $SU(2)$. Does anyone know of a basis ...
2
votes
2answers
545 views

Connes v Woronowicz - Cyclic Cohomology v Diff Calculi

Following on from my last two questions link text and link text: Is it correct (and useful) to say that the relationship between Connes' cyclic cohomology approach to de Rham cohomology and ...
2
votes
1answer
122 views

Classical Calculi as Universal Quotients

As is well known, every differential calculus $(\Omega,d)$ over an algebra $A$ is a quotient of the universal calculus $(\Omega_A,d)$, by some ideal $I$. In the classical case, when $A$ is the ...
5
votes
2answers
358 views

Quantum Frobenius II

In a previous question, I asked how Lusztig's quantum Frobenius generalizes the classical Frobenius map on a variety over a finite field. I was directed to a very interesting paper by Kumar and ...
2
votes
1answer
261 views

Basis for Universal Calculus

Can anyone give an explicit basis of the universal (noncommutative) differential calculus over an algebra $A$ with basis ${e_i}$. (The universal calculus over $A$ is the kernel of the multiplication ...
21
votes
6answers
2k views

Why Drinfel'd-Jimbo-type Quantum Groups?

Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
3
votes
3answers
728 views

Quantum Frobenius

In what sense does Lusztig's quantum Frobenius, defined on a quantum enveloping algebra, generalise the classical Frobenius mapping on a variety over a finite field?