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

By Leonid I. Korogodski

The ebook is dedicated to the examine of algebras of capabilities on quantum teams. The authors' method of the topic relies at the parallels with symplectic geometry, permitting the reader to take advantage of geometric instinct within the idea of quantum teams. The e-book comprises 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 idea of quantum Weyl teams. This booklet can function a textual content for an advent to the speculation of quantum teams.

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M n ) = s ⋅ (s′ , m2 , . . , m n ) = (m2 , . . , m n ) = m′ d’où p(m) ⋅ ε = p(s)−1 ⋅ m′ = m. Soit maintenant x ∈ F(S) et soit m ∈ M′ (S) un mot de longueur minimale tel que p(m) = x ; nécessairement m est réduit. D’autre part, si m ∈ M′ (S)0 est un mot réduit tel que p(m) = x, on a x ⋅ ε = p(m) ⋅ ε = m. Ainsi, x ⋅ ε est l’unique mot réduit dont l’image dans F(S) est égale à x. 7). — a) L’application j ∶ S → F(S) est injective. 46 CHAPITRE 2. GROUPES b) Le groupe F(S) est sans torsion. c) Si Card(S) ⩾ 2, le centre de F(S) est réduit à l’élément neutre.

GROUPES ABÉLIENS 53 On dit que r est le rang de A et que les entiers d1 , . . , ds sont ses facteurs invariants. Démonstration. — Soit (x1 , . . , x n ) une quasi-base de A ; quitte à réordonner cette suite, on suppose que x1 , . . , xr sont d’ordre infini et que xr+1 , . . , x n sont d’ordre fini ; notons alors m i l’ordre de x i si i >. D’après la définition d’une quasi-base, le groupe A est isomorphe à Zr × ∏ni=r+1 (Z/m i Z). La fin de la preuve du théorème consiste à récrire le groupe T = ∏ni=r+1 (Z/m i Z) sous la forme requise.

12. — Soit A un groupe et soit B un sous-groupe de A. Les ensembles A/B et B/A des classes à droite et à gauche sont distincts en général. Cependant, l’application aB ↦ Ba−1 est une bijection de A/B sur B/A. Lorsque A/B (ou B/A) est fini, son cardinal est appelé indice de B dans A et noté (A ∶ B). 13). — Soit A un groupe fini opérant dans un ensemble fini X. Pour toute orbite O ∈ X/A, choisissons un élément xO de cette orbite et soit AO son fixateur. On a la relation : Card(X) = ∑ Card(O) = ∑ (A ∶ AO ).

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