Fibered quadratic Hopf algebras related to Schubert calculus. (English) Zbl 0962.16028
The paper is concerned with the study of a family of quadratic associative algebras \({\mathcal E}_n\) introduced by S. Fomin and A. N. Kirillov [Advances in geometry, Prog. Math. 172, 147-182 (1999; Zbl 0940.05070)]. A Hopf algebra structure is defined on the twisted group algebra \({\mathcal E}_n\{{\mathcal S}_n\}\), where \({\mathcal S}_n\) is the symmetric group. This structure is used to prove that \({\mathcal E}_n\) can be decomposed as a graded module into a tensor product, one of the tensor factors being \({\mathcal E}_{n-1}\).
Reviewer: Sorin Dascalescu (Bucureşti)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 16S37 | Quadratic and Koszul algebras |
| 16S35 | Twisted and skew group rings, crossed products |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Keywords:
quadratic algebras; Hopf algebras; cohomology of flag manifolds; twisted group algebras; graded algebras; tensor productsCitations:
Zbl 0940.05070References:
| [1] | Abe, E., Hopf Algebras (1980), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0476.16008 |
| [2] | De Concini, C.; Procesi, C., Quantum Groups. Quantum Groups, Lecture Notes in Mathematics, 1565 (1993), Springer-Verlag: Springer-Verlag Berlin, p. 31-140 · Zbl 0795.17005 |
| [3] | Fomin, S.; Kirillov, A. N., Quadratic algebras, Dunkl elements, and Schubert calculus, Adv. in Geometry. Quadratic algebras, Dunkl elements, and Schubert calculus, Adv. in Geometry, Progress in Math., 172 (1999), Birkhäuser: Birkhäuser Boston, p. 147-182 · Zbl 0940.05070 |
| [5] | Roos, J.-E., Some non-Koszul algebras. Some non-Koszul algebras, Adv. in Geometry, Progress in Math., 172 (1999), Birkhäuser: Birkhäuser Boston, p. 385-389 · Zbl 0962.16022 |
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