nLab
Hopf action

In a strict symmetric monoidal category CC with symmetry τ\tau, a map :BAA\triangleright : B\otimes A\to A, where (B,Δ B)(B,\Delta_B) is a comonoid and (A,μ A)(A,\mu_A) a monoid, is a measuring if

Bμ A=μ A()(BτA)(Δ BAA):BAAA B\triangleright \mu_A = \mu_A\circ(\triangleright \otimes \triangleright)\circ (B\otimes \tau\otimes A)\circ (\Delta_B\otimes A\otimes A) : B\otimes A\otimes A\to A

where we wrote B=id BB=\mathrm{id}_B etc. If BB is in fact a bimonoid and if the measuring :BAA\triangleright:B\otimes A\to A is an action, then \triangleright is said to be a Hopf action. In the kk-linear case, a kk-algebra (A,μ A)(A,\mu_A) equipped with a Hopf action is called also a left BB-module algebra; it is the same as a monoid (=algebra) in the monoidal category of left BB-modules, where the monoidal structure is induced by the coaction Δ B\Delta_B. It is straightforward to modify the condition above to the case of non-strict symmetric monoidal categories. A dual concept is a Hopf coaction or equivalently, a notion of a BB-comodule algebra.

Created on April 12, 2009 22:38:13 by Zoran Škoda (195.37.209.180)