Twisted internal coHom for conic algebras

S. GRILLO; H. MONTANI

COMMUNICATIONS IN ALGEBRA

Marcel Dekker, Inc.

Lugar: Monticello, New York; Año: 2004 vol. 32 p. 839 - 853

0092-7872

Adapting the idea of twisted tensor products to the category of conic algebras (CA), i.e. finitely generated graded algebras, we define a family of objects hom^{Υ}[B,A] there, one for each pair A,B∈CA, with analogous properties to its internal coHom objects hom[B,A], but representing spaces of transformations whose coordinate rings and the ones of their respective domains do not commute among themselves. They give rise to a CA^{op}-based category different from that defined by the function (A,B)↦hom[B,A]. The mentioned non commutativity is controlled by a collection of twisting maps τ_{A,B}. We show, under certain circumstances, that (bi)algebras end^{Υ}[A]≐hom^{Υ}[A,A] are counital 2-cocycle twistings of the corresponding coEnd objects end[A]. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups GL_{q}(n) and their multiparametric versions.