nLab crossed product algebra



Smash product algebra


Given a kk-bialgebra HH, and a left Hopf action \triangleright of HH on a kk-algebra AA, one defines the crossed product algebra AHA\sharp H (in Hopf algebra literature also called the smash product algebra or Hopf smash product; distinguish from the rather different smash product in topology) as the kk-algebra whose underlying vector space is AHA\otimes H and the product is given by

(ah)(ah)=a(h (1)a)h (2)h. (a\otimes h)(a'\otimes h') = \sum a (h_{(1)}\triangleright a')\otimes h_{(2)}h'.

The idea is that if the bialgebra HH is in fact a Hopf algebra embedded as 1HAH1\otimes H\subset A\sharp H – whatever the product in the latter is (but assumed to satisfy (a1)(1h)=ah(a\otimes 1)(1\otimes h) = a\otimes h) – and if the action is inner within AHA\sharp H, i.e. ha=h (1)aS(h (2))h\triangleright a = \sum h_{(1)} a S(h_{(2)}), then we have

(h (1)a)h (2)=h (1)aS(h (2))h (3)=h (1)aϵ(h (2))=ha, \sum (h_{(1)}\triangleright a) h_{(2)} = \sum h_{(1)} a S(h_{(2)}) h_{(3)} = \sum h_{(1)} a \epsilon(h_{(2)})= h a \,,

and hence the formula for the product above is a tautology: ahah=a(h (1)a)h (2)ha h a' h' = a(h_{(1)}\triangleright a') h_{(2)} h'.

Similarly, given a right Hopf action of HH on AA, one defines the crossed product algebra HAH\sharp A whose underlying space is HAH\otimes A. The left and right versions are isomorphic if HH has an invertible antipode; this extends the correspondence between the left and right actions obtained by composing with the antipode map.


Every smash product algebra of the form AHA\sharp H is naturally equipped with a monomorphism AA1AHA\mapsto A\sharp 1\hookrightarrow A\sharp H of algebras and with a right HH-coaction ahaΔ(h)(AH)Ha\otimes h\mapsto a\otimes \Delta(h)\in (A\sharp H)\otimes H making AHA\sharp H into a right HH-comodule algebra. Map γ:h1h\gamma: h\mapsto 1\otimes h, HAHH\hookrightarrow A\sharp H is then a map of right HH-comodule algebra (where the coaction on HH is Δ\Delta), and A1AHA\otimes 1\subset A\sharp H is the subalgebra of HH-coinvariants.

If HH is a Hopf algebra, then the homomorphism γ\gamma is a convolution invertible linear map with convolution inverse γ 1\gamma^{-1} defined by γ 1(h)=γ(Sh)\gamma^{-1}(h)=\gamma(Sh) for hHh\in H, where SS is the antipode of HH. Conversely,

Proposition Let HH be a Hopf algebra, EE a right HH-comodule algebra, and γ:HE\gamma:H\to E a map of right HH-comodule algebra. Clearly HH acts on E coHE^{co H} by ha=γ(h (1))aγ(Sh (2))h\triangleright a = \sum \gamma(h_{(1)}) a\gamma(Sh_{(2)}) for aE coHa\in E^{co H} and hHh\in H, where the product on the right-hand side is in EE. Conclusion: EE coHHE\cong E^{co H}\sharp H where the smash product is with respect to that action.

Cocycled crossed product

There is also a more general cocycled crossed product. For a bialgebra HH and an algebra UU, if we consider the category C(U,H)C(U,H) of extensions UEU\hookrightarrow E which are compatibly left UU-modules and right HH-comodules, and where U=E coHU=E^{\mathrm{co}H}, then the crossed product algebras are the canonical representatives of cleft Hopf-Galois extensions which are a more invariant concept.

Let UU be an algebra, HH a Hopf algebra, :HUU\triangleright : H\otimes U\to U a measuring, i.e. a kk-linear map satisfying h(uv)=(h (1)u)(h (2)v)h\triangleright(u v)=\sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright v) for all hHh\in H, u,vUu,v\in U, and which we assume unital, i.e. h1=ϵ(h)1h\triangleright 1 = \epsilon(h)1 for all hHh\in H. We do not assume that \triangleright is an action.

Let further a (convolution) invertible kk-linear map σHom k(HH,U)\sigma \in Hom_k(H\otimes H,U) be given. For readability in longer calculations, one usually writes σ(h,k)\sigma(h,k) rather than σ(hk)\sigma(h\otimes k).

We say that σ\sigma is a 2-cocycle (relative to the measuring \triangleright) if the following two cocycle identities hold

h(ku)=σ(h (1),k (1))((h (2)k (2))u)σ 1(h (3),k (3)) h\triangleright (k\triangleright u) = \sum \sigma(h_{(1)},k_{(1)}) ((h_{(2)}k_{(2)})\triangleright u) \sigma^{-1}(h_{(3)},k_{(3)})
[h (1)σ(k (1),m (1))]σ(h (2),k (2)m (2))=σ(h (1),k (1))σ(h (2)k (2),m). \sum [h_{(1)}\triangleright\sigma(k_{(1)},m_{(1)})]\sigma(h_{(2)},k_{(2)}m_{(2)})=\sum \sigma(h_{(1)},k_{(1)})\sigma(h_{(2)}k_{(2)},m) \,.

The 2-cocycle σ\sigma is normalized if σ(h,1)=ϵ(h)1=σ(1,h)\sigma(h,1) = \epsilon(h)1 = \sigma(1,h).

These identities clearly generalize the classical factor system?s in group theory (linearly extended to the case of group algebras, for the finite groups at least). Therefore it is an example of a nonabelian cocycle in Hopf algebra theory. However, its role in the general theory is less well understood than the group case.

Define the cocycled crossed product on UHU\otimes H by

(uh)(vk)=u(h (1)v)σ(h (2),k (1))h (3)k (2) (u \sharp h)(v\sharp k) = \sum u (h_{(1)}\triangleright v) \sigma(h_{(2)},k_{(1)})\sharp h_{(3)} k_{(2)}

for all h,kHh,k\in H, u,vUu,v\in U. The cocycled crossed product is an associative algebra iff σ\sigma is a cocycle.

If so, we call U σHU\sharp_\sigma H cocycled crossed product algebra. Map 1Δ H:U σH(U σH)H1\otimes\Delta_H:U\sharp_\sigma H\to (U\sharp_\sigma H)\otimes H is a right HH-coaction, making U σHU\sharp_\sigma H into a right HH-comodule algebra, which is cleft extensions]] are always isomorphic (as HH-extensions) to the cocycled crossed product algebras.

If σ(h,k)=ϵ(h)ϵ(k)1 U\sigma(h,k)=\epsilon(h)\epsilon(k)1_U then we say that σ\sigma is a trivial cocycle and then the compatibility conditions above reduce to demanding that the measuring \triangleright is an action. The cocycled crossed product then reduces to the usual smash product algebra.

Theorem. Suppose we are given two measurings ,:HUA\triangleright,\triangleright':H\otimes U\to A with normalized cocycles σ,τ\sigma, \tau respectively. Then there exists an isomorphism of HH-extensions of UU, i:U σHU τHi: U\sharp_\sigma H\cong U\sharp_\tau H (i.e. an isomorphism of kk-algebras, left UU-modules and right HH-comodules) iff there is an invertible element fHom k(H,U)f\in Hom_k(H,U) such that for all uUu\in U, h,kHh,k\in H

hu=f 1(h (1))(h (2)u)f(h (3)),h\triangleright' u = \sum f^{-1}(h_{(1)})(h_{(2)}\triangleright u) f(h_{(3)}),
τ(h,k)=f 1(h (1))[h (2)f 1(k (1))]σ(h (3),k (2))f(h (4)k (3)).\tau(h,k) = \sum f^{-1}(h_{(1)})[h_{(2)}\triangleright f^{-1}(k_{(1)})]\sigma(h_{(3)},k_{(2)})f(h_{(4)}k_{(3)}).

The isomorphism ii is then given by

i(u σh)=uf(h (1)) τh (2)i(u\sharp_\sigma h) = \sum u f(h_{(1)})\sharp_\tau h_{(2)}


Related nnLab entres include cleft extension, crossed product C*-algebra, noncommutative torsor, Hopf-Galois extension

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  • Y. Doi, Equivalent crossed products for a Hopf algebra, Comm. Alg. 17 (1989), 3053–3085, MR91k:16027, doi

  • S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, AMS 1993.

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

Last revised on April 25, 2024 at 13:13:15. See the history of this page for a list of all contributions to it.