Algebras of Functions on Quantum Groups: Part I by Leonid I. Korogodski

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By Leonid I. Korogodski

The e-book is dedicated to the examine of algebras of services on quantum teams. The authors' method of the topic relies at the parallels with symplectic geometry, permitting the reader to exploit geometric instinct within the conception of quantum teams. The publication contains the idea of Poisson Lie teams (quasi-classical model of algebras of services on quantum groups), an outline of representations of algebras of capabilities, and the speculation of quantum Weyl teams. This booklet can function a textual content for an advent to the idea of quantum teams.

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Given a natural transformation ν : GF → 1C , we define θ = α(ν) : HomD (F, F ) → HomC (•, •) as follows: take g : F (C) → F (C ) in D, and put θC,C (g) = νC ◦ G(g) ◦ ηC Straightforward arguments show that θ is natural. Conversely, given θ : HomD (F, F ) → HomC (•, •) we define α−1 (θ) = ν : GF → 1C by 28 1 Generalities νC = θGF (C),C (εF (C) ) : GF (C) → C We leave it as an exercise to show that ν is natural, as needed, and that α and α−1 are inverses. The proof of the second statement is similar.

7). 8). Doi-Koppinen structures Let H be a bialgebra, A a right H-comodule algebra, and C a right H-module coalgebra. We call (H, A, C) a right-right Doi-Koppinen structure or DK structure over k. 11) for all a ∈ A, c ∈ C, and h ∈ H. The category of right-right Doi-Hopf structures over k is denoted by DK•• (k). DK•• (k) is a monoidal category, if we define (H, A, C) ⊗ (H , A , C ) = (H ⊗ H , A ⊗ A , C ⊗ C ) with the obvious structure maps. The unit element is (k, k, k). We will also consider the full subcategories H•• (k), HA•• (k), and HC•• (k) of DK•• (k), consisting of objects respectively of the form (H, H, H), (H, A, H), (H, H, C) The subcategory of DK•• (k) consisting of objects (H, A, C) and morphisms ( , α, γ) where H has a twisted antipode S, and where preserves the twisted antipode, is denoted by DKs•• (k).

For example, if M ∈ M(ψ)C A , then the corresponding object in Aop M(ψ ◦ τ )C is equal to M as a right C-comodule, but with left Aop -action given by aop · m = ma All the other isomorphisms are defined in a similar way, and we leave further details to the reader. 3 Entwined modules and the smash product Let A and B be k-algebras, and consider a map R : B ⊗ A → A ⊗ B. 25) or If this new multiplication makes A#R B into an associative algebra with unit 1#1, then we call A#R B a smash product, and (A, B, R) a smash product structure or a factorization structure.

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