nLab scalar extension bialgebroid



Motivation and historical appearance

Given a finite-dimensional Hopf algebra HH and a D(H)D(H)-module algebra AA (where D(H)D(H) is the Drinfeld double of HH), Lu1996 has introduced the structure of a left AA-bialgebroid on the smash product algebra AHA\sharp H as a noncommutative generalization of the algebra of functions on an action groupoid. In this finite-dimensional case, the category of left-right Yetter-Drinfeld modules is equivalent to the category of Yetter-Drinfeld HH-modules. In this new context of Yetter-Drinfeld module algebras, the construction has been generalized by the so called Brzeziński-Militaru construction or scalar extension bialgebroid in BrzezinskiMilitaru2002. They also proved a converse, roughly that the formulas given for the scalar extension bialgebroid for an arbitrary smash product satisfy the axioms of a bialgebroid only if the module algebra which is also a comodule is in fact a braided commutative Yetter-Drinfeld algebra.


If (H,Δ,ϵ)(H,\Delta,\epsilon) (where Δ(h)=h (1)h (2)\Delta(h) = \sum h_{(1)}\otimes h_{(2)} is a bialgebra and AA a left-right Yetter-Drinfeld HH-module algebra with Hopf action :HAA\triangleright:H\otimes A\to A and a coaction ρ:AAH\rho: A\to A\otimes H, then the smash product AHA\sharp H inherits the left AA-bialgebroid structure (AH,α,β,Δ AH,ϵ AH)(A\sharp H,\alpha,\beta,\Delta^{A\sharp H},\epsilon^{A\sharp H}) where source α(a)=a1\alpha(a) = a\sharp 1, target β\beta is ρ\rho followed by the identification of vector spaces AHAHA\sharp H\cong A\otimes H and the coproduct Δ AH\Delta^{A\sharp H} is obtained by extending Δ\Delta along inclusion HAHH\to A\sharp H, that is,

Δ AH:AH(AH) A(AH),ahah (1)1h (2), \Delta^{A\sharp H} : A\sharp H\to (A\sharp H)\otimes_A (A\sharp H), \,\,\,\,a\sharp h\mapsto a\sharp h_{(1)}\otimes 1\sharp h_{(2)},

which is an extension of Δ\Delta along the canonical embedding HAHH\to A\sharp H. The counit is

ϵ AH:AHA,ϵ AH(ah)=aϵ(h). \epsilon^{A\sharp H}:A\sharp H\to A,\,\,\,\, \epsilon^{A\sharp H}(a\sharp h) = a\epsilon(h).

Scalar extension Hopf algebroid

If HH is a Hopf algebra the Brzeziński-Militaru construction gives a Lu-Hopf bialgebroid and if the antipode SS of HH is invertible then it gives also a Böhm-Szlachányi symmetric Hopf algebroid. If the antipode SS is not invertible then the data of a Yetter-Drinfeld module algebra has to be replaced by a compatible pair of a left-right Yetter-Drinfeld module algebra and a right-left Yetter-Drinfeld module algebra yielding again a symmetric Hopf algebroid Stojic2023.

Special case: Heisenberg double

Heisenberg double H *HH^*\sharp H of a finite-dimensional Hopf algebra is a specific example of a Hopf algebroid. There are many infinite-dimensional cases of Heisenberg double which can be cast into bialgebroids, but most often complicated issues with completions are involved (e.g. in MSSncphasespace2017)

Twists of scalar extension bialgebroids


Standard reference is now

Remaining issues about the antipode are settled in

  • M. Stojić, Scalar extension Hopf algebroids, Journal of algebra and applications 2023 doi arXiv:2208.11696

Every invertible counital 2-cocycle F=F 1F 2F = \sum F^1\otimes F^2 (Drinfeld twist) for a bialgebra HH, with inverse F 1=G 1G 2F^{-1} = G^1\otimes G^2, induces a Drinfeld-Xu 2-cocycle 𝒢=1G 1 A1G 2\mathcal{G}=\sum 1\sharp G^1\otimes_A 1\sharp G^2 for the scalar extension bialgebroid AHA\sharp H. The scalar extension for the FF-twisted data A FH FA_F\sharp H^F is isomorphic as a A FA_F-bialgebroid to the 𝒢\mathcal{G}-twist of AHA\sharp H.

  • Andrzej Borowiec, Anna Pachoł, Twisted bialgebroids versus bialgebroids from Drinfeld twist, J. Phys. A50:5 (2017) arXiv:1603.09280

Some corrections and a slight generalization are given in

An infinite-dimensional case of a Heisenberg double of a universal enveloping algebra U(g)U(g) of a finite dimensional Lie algebra over a field of characteristic zero is described as a version of a completed Hopf algebroid in

  • S. Meljanac, Z. Škoda, M. Stojić, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Phys. 107:3, 475–503 (2017) doi arXiv:1409.8188

While all the statements in the above article are rigorous, the axioms are not the most natural. A natural version where the Heisenberg double of U(g)U(g) has been described as an internal Hopf algebroid (of an internal scalar extension type) in a symmetric monoidal category of (countably cofinite) filtered-cofiltered vector spaces in

  • Martina Stojić, Completed Hopf algebroids (in Croatian: Upotpunjeni Hopfovi algebroidi), Ph. D. thesis, University of Zagreb, 2017, 306 pp. pdf

Other articles include

Scalar extension Hopf algebroids can be recast also in the form fitting the axioms of the Hopf algebroid with a balancing subalgebra, see Sec. 4 in

category: algebra

Last revised on August 12, 2023 at 13:06:05. See the history of this page for a list of all contributions to it.