The Representation Category of the Quantum Group of a Non-degenerate Bilinear Form

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C * -algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C * -algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.

show abstract

“…A combination of Proposition 6.2.6 in [6] and the results in [7] yields an alternative proof for the fact that A o (F 1 , F 2 ) = 0.…”

Section: Unitary Fiber Functors Preserving the Dimensionmentioning

confidence: 99%

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

Bichon

Rijdt

Vaes

2005

Commun. Math. Phys.

Self Cite

120

303

View full text Add to dashboard Cite

show abstract

“…Propositions 3.2, 3.3 and 3.4 in [Bi1] ensure that we have indeed this map ψ. Moreover, ψ is surjective by Theorem 4 and injective by Theorem 7.…”

Section: Corollarymentioning

confidence: 99%

“…By Theorem 4, there exist m ≥ 2 and F ∈ GL m (k) such that Z is isomorphic to B(E, F ) as a B(E)-Galois object. Bichon [Bi1,Propositions 3.3,3.4…”

Section: Corollarymentioning

confidence: 99%

“…Let n, m ≥ 1 be integers and let E ∈ GL n (k), F ∈ GL m (k) be invertible scalar matrices. Following Bichon [Bi1], we define the algebra B(E, F ) as the k-algebra generated by n × m variables z ij , i = 1, . .…”

Section: Hopf-galois Extensions and Bi-galois Objectsmentioning

confidence: 99%

“…Bichon [Bi1] has proved that the representation category of each Hopf algebra B(E) is monoidally equivalent to the one of O q (SL(2)), where q is a solution of the equation q 2 + Tr(E −1 E t )q + 1 = 0.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the classification of Galois objects over the quantum group of a nondegenerate bilinear form

Aubriot

2006

manuscripta math.

View full text Add to dashboard Cite

show abstract

Autoequivalences of Tensor Categories Attached to Quantum Groups at Roots of 1

Davydov

Etingof

Nikshych

2018

Progress in Mathematics

View full text Add to dashboard Cite

We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group u q (g), where q is a root of unity.To the memory of Bertram Kostant IntroductionLet k be an algebraically closed field of characteristic zero. Let G be a simple algebraic group over k, and let g = Lie(G) be its Lie algebra. Let q be a root of unity of odd order coprime to 3 if G is of type G 2 , and coprime to the determinant of the Cartan matrix of G. Let u q (g) be Lusztig's small quantum group attached to g [Lu1]. Then u q (g) is a quasitriangular Hopf algebra, so the category of its finite dimensional representationsRep u q (g) is a finite braided tensor category [EGNO]. One of the main goals of this paper is to compute the Picard group Pic(Rep u q (g)) of this category, i.e., the group of equivalence classes of invertible Rep u q (g)-module categories. Picard groups of braided tensor categories and, in particular, Brauer-Picard groups of tensor categories play a crucial role in classification of graded extensions [ENO] and also appear as symmetry groups of three-dimensional topological field theories [FS]. It is known that Pic(Rep u q (g)) is isomorphic to the group Aut br (Rep u q (g)) of braided autoequivalences of Rep u q (g) [DN, ENO]. We show under some restrictions on q that Aut br (Rep u q (g)) is isomorphic to the group Aut(g) of automorphisms of g, i.e., Aut br (Rep u q (g)) = Γ ⋉ G ad , where G ad is the adjoint group of G and Γ = Γ g is the automorphism group of the Dynkin diagram of g. Namely, we prove this when the order ℓ of q is sufficiently large and also for classicalMoreover, we show that Rep u q (g) has only two braidings (the standard one and its reverse) and deduce that any tensor autoequivalence of Rep u q (g) is necessarily braided.Thus, the group of tensor autoequivalences (also known as the group of biGalois objects) of Rep u q (g) is isomorphic to Γ ⋉ G ad . This generalizes the result of Bichon [Bi2], who proved this fact for g = sl 2 .Date: March 21, 2017. 1 2 ALEXEI DAVYDOV, PAVEL ETINGOF, AND DMITRI NIKSHYCHWe also consider the braided tensor category O q (G) − comod of finite dimensional comodules over the function algebra O q (G), which is the G-equivariantization of Rep u q (g).We show that every braided autoequivalence of O q (G) − comod comes from a Dynkin diagram automorphism if ℓ is sufficiently large, and prove a similar result in the non-braided case. This generalizes a result of Neshveyev and Tuset [NT1, NT2], who proved this when q is not a root of unity. We also show this for the classical groups SL N , Sp N , SO N if ℓ > N.As a tool, we introduce the notion of a finitely dominated tensor category. We show that the category of comodules over a finitely presented Hopf algebra is finitely dominated and prove that tensor autoequivalences of a finitely dominated category that preserve a tensor generator form an algebraic group. While this theory plays an auxiliary role in our paper, it may be of independent interest. We expect t...

show abstract

The Representation Category of the Quantum Group of a Non-degenerate Bilinear Form

Cited by 56 publications

References 17 publications

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

On the classification of Galois objects over the quantum group of a nondegenerate bilinear form

Autoequivalences of Tensor Categories Attached to Quantum Groups at Roots of 1

Contact Info

Product

Resources

About