Zoran Skoda
Hopf smash product bimodule

Preliminaries: Hopf smash product algebras

If HH is a kk-bialgebra (or, in particular, kk-Hopf algebra) an action :HUU\triangleright: H\otimes U\to U of HH (as a kk-algebra) on a kk-algebra UU is a (left) Hopf action if it is also a measuring (of UU by an underlying coalgebra of HH); recall that an algebra measuring? of UU by a coalgebra CC is a bilinear map :CUU\triangleright: C\otimes U\to U such that h1 U=ϵ(h)1 Uh\triangleright 1_U = \epsilon(h) 1_U and h(uu)=(h (1)u)(h (2)u)h\triangleright (u \cdot u') = \sum (h_{(1)}\triangleright u)\cdot (h_{(2)}\triangleright u') for all hCh\in C, u,uUu,u'\in U. If HH acts on UU by a Hopf action, one also says that UU is an HH-module algebra (equivalently, a monoid in the monoidal category of HH-modules).

Bimodule measurings

If UU and VV are kk-algebras measured by a kk-coalgebra CC and MM is a UU-VV-bimodule, then we say that MM is measured by CC, if there is a kk-bilinear map M:CMM\triangleright_M:C\otimes M\to M such that for all uUu\in U, vVv\in V, mMm\in M, cCc\in C,

c(umv)=(c (1) Uu)(c (2) Mm)(c (3)v) c\triangleright (u m v) = \sum (c_{(1)}\triangleright_U u) (c_{(2)} \triangleright_M m)(c_{(3)}\triangleright v)

If CC is in addition a kk-bialgebra and U, V, M\triangleright_U,\triangleright_V,\triangleright_M actions of HH, we say that a measuring is Hopf action of CC on the UU-VV-bimodule MM.

Hopf smash product bimodule

Last revised on September 8, 2013 at 15:59:02. See the history of this page for a list of all contributions to it.