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A Note on Free Quantum Groups

Teodor Banica (2008)

Annales mathématiques Blaise Pascal

We study the free complexification operation for compact quantum groups, G G c . We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying G = G c .

Bar-invariant bases of the quantum cluster algebra of type A 2 ( 2 )

Xueqing Chen, Ming Ding, Jie Sheng (2011)

Czechoslovak Mathematical Journal

We construct bar-invariant [ q ± 1 / 2 ] -bases of the quantum cluster algebra of the valued quiver A 2 ( 2 ) , one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

Gateva-Ivanova, Tatiana (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...

Classifying bicrossed products of two Sweedler's Hopf algebras

Costel-Gabriel Bontea (2014)

Czechoslovak Mathematical Journal

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras E that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products H 4 H 4 . There are three steps in our approach. First, we explicitly describe the set of all matched pairs ( H 4 , H 4 , , ) by proving that, with the exception of the trivial pair, this set is parameterized by the ground field k . Then, for...

Graphs having no quantum symmetry

Teodor Banica, Julien Bichon, Gaëtan Chenevier (2007)

Annales de l’institut Fourier

We consider circulant graphs having p vertices, with p prime. To any such graph we associate a certain number k , that we call type of the graph. We prove that for p k the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

Invariants of the half-liberated orthogonal group

Teodor Banica, Roland Vergnioux (2010)

Annales de l’institut Fourier

The half-liberated orthogonal group O n * appears as intermediate quantum group between the orthogonal group O n , and its free version O n + . We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between O n * and U n , a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...

On the quantum groups and semigroups of maps between noncommutative spaces

Maysam Maysami Sadr (2017)

Czechoslovak Mathematical Journal

We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are...

Right coideal subalgebras of U q + ( 𝔰𝔬 2 n + 1 )

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group U q + ( 𝔰𝔬 2 n + 1 ) provided that q is not a root of 1. If q has a finite multiplicative order t > 4 ; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel u q + ( 𝔰𝔬 2 n + 1 ) . In particular, the total number of right coideal subalgebras that contain the coradical equals ( 2 n ) ! ! ; the order of the Weyl group defined by the root system of type B n .

The geometric reductivity of the quantum group S L q ( 2 )

Michał Kępa, Andrzej Tyc (2011)

Colloquium Mathematicae

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants A G is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...

Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples

Alfons Van Daele, Shuanhong Wang (2012)

Banach Center Publications

Let G be a finite group. Consider the algebra A of all complex functions on G (with pointwise product). Define a coproduct Δ on A by Δ(f)(p,q) = f(pq) where f ∈ A and p,q ∈ G. Then (A,Δ) is a Hopf algebra. If G is only a groupoid, so that the product of two elements is not always defined, one still can consider A and define Δ(f)(p,q) as above when pq is defined. If we let Δ(f)(p,q) = 0 otherwise, we still get a coproduct on A, but Δ(1) will no longer be the identity in A ⊗ A. The pair (A,Δ)...

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