Twisted Hopf algebras, Ringel-Hall algebras, and Green's categories

[摘要] The concept and some basic properties of a twisted Hopf algebra are introduced and investigated. Its unique difference from a Hopf algebra is that the comultiplication delta: A --> A circle times A is an algebra homomorphism, not for the componentwise multiplication on A circle times A, but for the twisted multiplication on A circle times A given by Lusztig's rule. Also, it is proved that any object A in Green's category has a twisted Hopf algebra structure, any morphism between objects is a twisted Hopf algebra homomorphism, the antipode s of A is self-adjoint under the Lusztig form (-, -) on A, and the Green polynomials M-a,M- b(t) share a so-called cyclic-symmetry. As examples, the twisted Ringel-Hall algebras, Ringel's twisted composition algebras, Lusztig's free algebras 'F and non-degenerate algebras F, the positive part U+ of the Drinfeld-Jimbo quantized enveloping algebras U, and Rosso's quantum shuffle algebra T(V) all are twisted Hopf algebras. The antipode and its inverse for a twisted Ringel-Hall are explicitly given, and all delta-primitive elements are determined, (C) 2000 Academic Press.

[发布日期] 2000-09-15