Generalized root systems, convex orders and a relation with quantized enveloping algebras of semisimple Lie superalgebras

IVÁN ANGIONO

Córdoba

Congreso; XV Escuela Latinoamericana de Matemática; 2011

FaMAF (UNC), CIMPA

Let g be a semisimple Lie superalgebra. If we consider different Cartan subalgebras, the associated sets of roots can be different. A first generalization of the notion of a root system was given in [S], where the author defines odd symmetries relating different sets of roots. It was also introduced the notion of quantized enveloping algebras of semisimple Lie superalgebras, and the consideration of their positive part as in the classical (non super) case. It gives place to examples of Nichols algebras, related with the classification of pointed Hopf algebras [AAY]. In this context the definition of the Weyl groupoid and the associated root systems appears in [H], generalizing Serganova's ideas in the super Lie context. These root systems can be used to describe different properties of quantized enveloping algebras and the classical enveloping algebras. In this context, it is important to have a characterization of convex orders. These orders have been studied firstly related with semisimple Lie algebras, and last time because of the relation with Lusztig isomorphisms and canonical bases of quantum groups and small quantum groups. In this talk we introduce the notion of generalized root systems and analyze their relation with Lie superalgebras. We show how to characterize convex orders for root systems of Nichols algebras and obtain also the relation between these orders and some PBW bases. This can be applied to obtain a presentation by generators and relations of Nichols algebras of diagonal type, whose associated root system is finite. [AAY] N. Andruskiewitsch, I. Angiono and H. Yamane, On pointed Hopf superalgebras. Accepted in Contemp. Math. arxiv:1009.5148, 18 pp [A] I. Angiono, Presentation of Nichols algebras of diagonal type by generators and relations, submitted. arXiv:1008.4144. [HS] I. Heckenberger and H.-J. Schneider, Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid. arXiv:0909.0293. [H] I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164, 175-188 (2006). [S] V. Serganova, On generalizations of root systems, Comm. Algebra 24 (1996), 4281–4299.