Twisted Hopf Algebras, Ringel–Hall Algebras, and Greenʹs Categories, : With an appendix by the referee

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 δ: A → A A is an algebra homomorphism, not for the componentwise multiplication on A A, but for the twisted multiplication on A A given by Lusztigʹs rule. Also, it is proved that any object A in Greenʹs 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 Ma, b(t) share a so-called cyclic-symmetry. As examples, the twisted Ringel–Hall algebras, Ringelʹs twisted composition algebras, Lusztigʹs free algebras ′F and non-degenerate algebras F, the positive part U+ of the Drinfeld–Jimbo quantized enveloping algebras U, and Rossoʹs 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 δ-primitive elements are determined.