Quantum Groups and Their Representations

A class of Hilbert space representations of the quantum plane and the quantum algebra Uq(sl2)

“…Thus, in the case where F 1 and F 2 are essentially bounded, the set {T 1 , T 2 } of operators yields a bounded operator representation of the quantum plane C 2 q with q = 0, ±1 ( [4,5]). …”

“…For a complex number q ∈ C \ {0, −1, 1}, the quantum algebra U q (sl 2 ) is defined to be the complex associative algebra with unit 1 generated by four elements E, F, K, K −1 subject to the following relations [4,5]: (cf. also [9]), there is a general scheme to construct a representation of U q (sl 2 ) from a representation of the quantum plane C 2 q .…”

Equivariant Cyclic Cohomology of H-Algebras

“…One can develop a similar theory for Hopf comodule algebras and prove the analogues of Theorem 4.2 and Corollary 4.2, for cosemisimple Hopf algebras. It is known that compact quantum groups in the sense of Woronowicz are cosemisimple [11].…”

“…In particular, let G be a compact Lie group acting smoothly on a complete locally convex algebra A, and let H = Rep(G) ⊂ C ∞ (G) be the Hopf algebra of representable functions on G [11]. Then, the dual of Corollary 4.2 reduces to Proposition 3.4 in [16].…”

“…By a Yetter-Drinfeld H-algebra [11] we mean an algebra A that satisfies the following conditions: 1)A is a left H-algebra, 2)A is a right H op -comodule algebra, i.e., the coaction ρ : A → A ⊗ H, satisfies…”