By Olivier Schiffman, Pavel Etingof

Revised moment variation. The textual content covers the cloth provided for a graduate-level path on quantum teams at Harvard collage. The contents hide: Poisoon algebras and quantization, Poisson-Lie teams, coboundary Lie bialgebras, Drinfelds double development, Belavin-Drinfeld class, countless dimensional Lie bialgebras, Hopf algebras, Quantized common enveloping algebras, formal teams and h-formal teams, countless dimensional quantum teams, the quantum double, tensor different types and quasi Hopf-algebras, braided tensor different types, KZ equations and the Drinfeld classification, Quasi-Hpf enveloping algebras, Lie associators, Fiber functors and Tannaka-Driein duality, Quantization of finite Lie bialgebras, common buildings, common quantization, Dequantization and the equivalence theorem, KZ associator and a number of zeta capabilities, and Mondoromy of trigonometric KZ equations. Probems are given with each one topic and a solution secret's incorporated. desk of contents Poisson algebras and quantization Poisson-Lie teams Coboundary Lie bialgebras Drinfeld's double building Belavin-Drinfeld type (I) endless dimensional Lie bialgebras Belavin-Drinfeld category (II) Hopf algebras Quantized common enveloping algebras Formal teams and h-formal teams countless dimensional quantum teams The quantum double Tensor different types and quasi-Hopfalgebras Braided tensor different types KZ equations and the Drinfeld class Quasi-Hopf quantized enveloping algebras Lie associators Fiber functors and Tannaka-Krein duality Quantization of finite dimensional Lie bialgebras common structures common quantization Dequantization and the Equalivalence

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If N is a norm al s ubgro up o f G an d H is any s ubgro up of G, prove th at NHis a s ubgro up o f G. 4. Show th at the intersection o ftwo norm al s ubgro ups o f G is a norm al s ubgro up o f G. 5. If His a s ubgro up o f G an d Nis a norm al s ubgro up o f G, show th at H n Nis a norm als ub gro up o f H. 6. Show th at eve ry s ubgro up o f an abe li an gro up is norm al . * 7. Is the converse o f Problem 6 tr ue ? If yes , prove it, i f no, give an ex ample o f a non -abeli an gro up all o fwhose s ubgro ups ar enorm al .

Before we pro ceed with the proof it se lf it might be advi sa ble to see what it i s that we are going to prove. In part (a) we want to show that if two e el ment s e and fin G en joy the property that for every a E G, a = a e = e a = a · f = f · a, then e = f In part ( b) our aim i s to show that if x · a = a · x = e and y · a = a ·y = e, where a ll of a, x,y are in G, then X = y. • · 33 Sec. 3 35 Some Preliminary lemmas Problems I . In the following determine whether the systems described are groups.

Amongst mathematicians neither the beauty nor the significance of the first example which we have chosen to discuss-groups-is disputed. 1 Defi nition of a G ro u p At this juncture it is advisable to recall a situation discussed in the first chapter. For an arbitrary nonempty set S we defined A (S) to be the set of all one-to-one mappings of the set S onto itself. For any two elements a, T e A (S) we introduced a product, denoted by a o -r, and on further investi gation it turned out that the following facts were true for the elements of A (S) subject to this product : l .