nLab
measuring

In a strict symmetric monoidal category DD with unit 1\mathbf{1}, symmetry τ\tau, left and right unit coherences ll and rr, a map :CAA\triangleright : C\otimes A\to A of a (counital) comonoid (C,Δ C,ϵ)(C,\Delta_C,\epsilon) on a (unital) monoid (A,μ A,η A)(A,\mu_A,\eta_A) is a measuring, or we say that CC measures AA if

Cμ A=μ A()(CτA)(Δ CAA):CAAA, C\triangleright \mu_A = \mu_A\circ(\triangleright \otimes \triangleright)\circ (C\otimes \tau\otimes A)\circ (\Delta_C\otimes A\otimes A) : C\otimes A\otimes A\to A,

and

(Cη)=ηr C(ϵ1):C1A. \triangleright \circ (C\otimes \eta) = \eta\circ r_C\circ (\epsilon\otimes \mathbf{1}): C\otimes \mathbf{1}\to A.

In Sweedler notation, we also write c(ab)=(c (1)a)(c (2)b)c\triangleright (ab) = \sum (c_{(1)}\triangleright a)(c_{(2)}\triangleright b) and c1 A=ϵ(c)1 Ac\triangleright 1_A = \epsilon(c) 1_A. A Hopf action is a special case of measuring which is also an action of a bimonoid where B=(C,μ C)B=(C,\mu_C). Measurings are used to define the (cocycled) crossed product algebras, see also cleft extension (of an algebra by a bialgebra). For measurings and module algebras see

  • S. Montgomery, Hopf algebras and their actions on rings, CBMS 82, AMS 1993.

  • A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer, 1997;

and for (co)module (co)algebras and generalizations see also

  • Shahn Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

There are also more elaborate versions of measurings, which play role in a Galois theory, see

  • Tomasz Brzeziński, On modules associated to coalgebra Galois extensions, J. Algebra, 215, no. 1, 1999, 290-317.

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. For Hopf actions on bimodules, one can define a bimodule version of a (Hopf) smash product algebra, see Hopf smash product bimodule.

Revised on September 8, 2013 16:01:17 by Zoran Škoda (31.45.169.1)