Discrete Quantum Groups

Quasi-Discrete Locally Compact Quantum Groups

Let A be a C *-algebra. Let A ⊗ A be the minimal C *-tensor product of A with itself and let M (A ⊗ A) be the multiplier algebra of A ⊗ A. A comultiplication on A is a non-degenerate *-homomorphism ∆ : A → M (A ⊗ A) satisfying the coassociativity law (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆ where ι is the identity map and where ∆ ⊗ ι and ι ⊗ ∆ are the unique extensions to M (A ⊗ A) of the obvious maps on A ⊗ A. We ...

Paragrassmann Integral, Discrete Systems and Quantum Groups

This report is based on review paper [1]. Some aspects of differential and integral calculi on generalized grassmann (paragrassmann) algebras are considered. The integration over paragrassmann variables is applied to evaluate the partition function for the Z p+1 Potts model on a chain. Finite dimensional paragrassmann representations for GL q (2) are constructed. Generalizations of grassmann al...

Structural Properties of Inner Amenable Discrete Groups

Rieffel Type Discrete Deformation of Finite Quantum Groups

We introduce a discrete deformation of Rieffel type for finite (quantum) groups. Using this, we give an example of a finite quantum group A of order 18 such that neither A nor its dual can be expressed as a crossed product of the form C(G1) ⋊τ G2 with G1 and G2 ordinary finite groups. We also give a deformation of finite groups of Lie type by using their maximal abelian subgroups.

Characterizations of compact and discrete quantum groups through second duals

A locally compact group G is compact if and only if L(G) is an ideal in L(G), and the Fourier algebra A(G) of G is an ideal in A(G)∗∗ if and only if G is discrete. On the other hand, G is discrete if and only if C0(G) is an ideal in C0(G)∗∗. We show that these assertions are special cases of results on locally compact quantum groups in the sense of J. Kustermans and S. Vaes. In particular, a vo...