nLab measuring

Measuring of coalgebras of algebra morphisms and algebras was introduced by Sweedler in

  • M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, Benjamin 1969.

Given a commutative ring kk, a kk-coalgebra (C,Δ C,ϵ)(C,\Delta_C,\epsilon) and kk-algebras (A,μ,η)(A',\mu',\eta') and (A,μ,η)(A,\mu,\eta), a CC-measuring from AA' to AA is a kk-linear map f:CMod k(A,A)f: C\to Mod_k(A',A) such that

ev(fμ A)=μ A(evev)(fAfA)(Cτ A,CA)(Δ CAA):CAAA, ev\circ (f\otimes\mu_{A'}) = \mu_A\circ (ev\otimes ev)\circ(f\otimes A'\otimes f\otimes A')\circ (C\otimes \tau_{A',C}\otimes A')\circ (\Delta_C\otimes A'\otimes A') : C\otimes A'\otimes A'\to A,

and fη=ηϵ C:CAf\circ \eta' = \eta\circ\epsilon_C:C\to A where τ\tau is the transposition of factors and evev the evaluation of a map. In Sweedler notation, the requirements on ff are f(c)(a Ab)=f(c (1))(a) Af(c (2))(b)f(c)(a\cdot_{A'} b) = f(c_{(1)})(a)\cdot_A f(c_{(2)})(b) and f(c)(1 A)=ϵ(c)1 Af(c)(1_{A'}) = \epsilon(c)1_{A}.

We can instead of f:CMod k(A,A)f:C\to Mod_k(A',A) consider its adjoint map f˜:CAA\tilde{f}:C\otimes A'\to A and the convolution algebra Mod k(C,A)Mod_k(C,A). Then, ff is a measuring (or some say taht f˜\tilde{f} is a measuring iff the other adjoint f˜ *:AMod k(C,A)\tilde{f}^*: A'\to Mod_k(C,A) is a map of algebras.

As a special case, we can consider the case of the identity map AAAA\overset{A}\to A and talk about a CC-measuring of a monoid AA, including in the general case of a 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) is a measuring of the comonoid CC on a (unital) monoid (A,μ A,η A)(A,\mu_A,\eta_A), 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,


(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. Note that CC is not a monoid so a measuring is not an action (although the notation may suggest this). However if CC is a bialgebra, we may consider when it is an action. A Hopf action is a measuring which is also an action of a bimonoid B=(C,Δ C,ϵ,μ C,η C)B=(C,\Delta_C,\epsilon,\mu_C,\eta_C). In other words, AA is a monoid in the monoidal category of BB-modules (also called BB-module monoid or BB-module algebra).

Measurings of algebras by bialgebras 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.

Other works include

  • M. Hyland, I. Lo’pez Franco, C. Vasilakopoulou, Hopf measuring comonoids and enrichment, Proc. Lond. Math. Soc. (3) 115:5 (2017) 1118–1148

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) Vv) c\triangleright (u m v) = \sum (c_{(1)}\triangleright_U u) (c_{(2)} \triangleright_M m)(c_{(3)}\triangleright_V 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.

There is a dual notion of a comeasuring, see

  • Mitsuhiro Takeuchi, Matched pairs of groups and bismash products of hopf algebras, Comm. Alg. 9:8 (1981) 841–882 doi

See also related nnLab page measuring coalgebra about the related enrichment (in the cocommutative case) and wikipedia:measuring coalgebra which has a bit of both.

Given an algebra AA and a coalgebra CC, a map ρ:CAC\rho:C\to A\otimes C is a left AA-comeasuring of CC if

c Ac (1) Cc (2) C=(c (1)) A(c (2)) Ac (1) Cc (2) C \sum c^A\otimes c^C_{(1)} \otimes c^C_{(2)} = \sum (c_{(1)})^A (c_{(2)})^A\otimes c_{(1)}^C\otimes c_{(2)}^C

and (Aϵ)ρ=ϵ(A\otimes\epsilon)\circ\rho = \epsilon, where we denote ρ(c)=c Ac C\rho(c) = \sum c^A\otimes c^C and Δ(c)=c (1)c (2)\Delta(c) = \sum c_{(1)}\otimes c_{(2)}. If AA is a bialgebra then AA-comeasurings of coalgebras (with additional data) appear in the definition of (cocycle) cross coproduct coalgebras.

Last revised on January 31, 2023 at 19:01:36. See the history of this page for a list of all contributions to it.