On the Hochschild homology of quantum $ \mathit{SL}(N)$

Tom Hadfield; Ulrich Krähmer

doi:10.1016/j.crma.2006.03.031

Algebra/Homological Algebra

On the Hochschild homology of quantum

SL (N)

[Sur l'homologie de Hochschild de quantum

SL (N)

]

Présenté par : Alain Connes

Tom Hadfield ¹ ; Ulrich Krähmer ²

¹ School of Mathematical Sciences, Queen Mary, University of London, 327 Mile End Road, London E1 4NS, UK
² Instytut Matematyczny Polskiej Akademii Nauk, Ul. Sniadeckich 8, 00956 Warszawa, Poland

Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 9-13.

Résumé
Abstract

Nous démontrons que l'anneau standard quantique des coordonnées $A : = k_{q} [SL (N)]$ satisfait l'analogue de van den Bergh de la dualité de Poincaré dans l'(co)homologie de Hochschild. Le bimodule de la dualité est $A_{σ}$ , le A-bimodule qui est A comme un espace vectoriel, avec la multiplication à droite tordue par l'automorphisme modulaire σ de la fonctionnelle de Haar. Ceci implique $H_{N^{2} - 1} (A, A_{σ}) ≅ k$ , et généralise notre résultat précédent pour $k_{q} [SL (2)]$ .

We show that the quantized coordinate ring $A : = k_{q} [SL (N)]$ satisfies van den Bergh's analogue of Poincaré duality for Hochschild (co)homology with dualizing bimodule being $A_{σ}$ , the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism σ of the Haar functional. This implies that $H_{N^{2} - 1} (A, A_{σ}) ≅ k$ , generalizing our previous result for $k_{q} [SL (2)]$ .

Reçu le : 2005-10-26
Accepté le : 2006-03-14
Publié le : 2006-06-21

DOI : 10.1016/j.crma.2006.03.031

Affiliations des auteurs :

Tom Hadfield ¹ ; Ulrich Krähmer ²

¹ School of Mathematical Sciences, Queen Mary, University of London, 327 Mile End Road, London E1 4NS, UK
² Instytut Matematyczny Polskiej Akademii Nauk, Ul. Sniadeckich 8, 00956 Warszawa, Poland

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     title = {On the {Hochschild} homology of quantum $ \mathit{SL}(N)$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {9--13},
     publisher = {Elsevier},
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     year = {2006},
     doi = {10.1016/j.crma.2006.03.031},
     language = {en},
}

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Tom Hadfield; Ulrich Krähmer. On the Hochschild homology of quantum $ \mathit{SL}(N)$. Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 9-13. doi : 10.1016/j.crma.2006.03.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.031/

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