By Leonid I. Korogodski

ISBN-10: 0821803360

ISBN-13: 9780821803363

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

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**Extra info for Algebras of functions on quantum groups. Part I**

**Example text**

See [Ge2], p. 13- 14). Define p(t) = -I { 1 if t if t > 0, ~ O. ~),p(t)) , ¢H(t) = (( ~ e~t )'1). From this it is very easy to calculate d7r~(F) and d7r~(H) , but there is a small difficulty in determining d7r~(G) . The best thing is to use the Fourier transformation F . One computes F 0 d7r~( G) 0 F- 1 = -27rimx 2 , and derives from this the formula in the following lemma. 2 Lemma. 3 resp. 4). 2. Models for infinitesimal representations and unitarizability 41 We want to describe the infinitesimal Schrodinger-Weil representation in ~pure ly algebraic way.

GJ with multiplier>.. -1 resp. >... After tensorizing the co cycles cancel and the result is an ordinary representation of G J . Summarizing we obtain the following result. 1 Theorem. The above construction gives a 1-1 correspondence between the irreducible unitary projective representations ofSL(2, R) with multiplier>.. and the irreducible unitary representations of G J (R) with non-trivial central character 'ljJm. The corresponding non-archimedean result is as follows . 2 Theorem. Let F be a p-adic field.

If this condition is fulfilled then H' is called regularly embedded, and G' = G I>( H' is called a regular semi direct product. The result of Mackey is now as follows. 2. 1 Theorem. 2) splits. Assume that H' is of type I and regularly embedded. For every a E Hi let G~ the stabilizer of a under the above action of G' on Hi, and G~ = {T E ~: TI H' is a multiple of a}. Then the induced representation e' Inde'T " is irreducible for every T E G~, and (ji is a disjoint union (ji = U {Indg~ T: T E G~} .

### Algebras of functions on quantum groups. Part I by Leonid I. Korogodski

by Donald

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