Given a finite-dimensional Hopf algebra and a -module algebra (where is the Drinfeld double of ), Lu1996 has introduced the structure of a left -bialgebroid on the smash product algebra 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 -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 (where is a bialgebra and a left-right Yetter-Drinfeld -module algebra with Hopf action and a coaction , then the smash product inherits the left -bialgebroid structure where source , target is followed by the identification of vector spaces and the coproduct is obtained by extending along inclusion , that is,
which is an extension of along the canonical embedding . The counit is
If is a Hopf algebra the Brzeziński-Militaru construction gives a Lu-Hopf bialgebroid and if the antipode of is invertible then it gives also a Böhm-Szlachányi symmetric Hopf algebroid. If the antipode 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.
Heisenberg double 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)
Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Int. J. Math. 7, 1 (1996) pp. 47-70, q-alg/9505024, MR95e:16037, doi; On the Drinfeld double and the Heisenberg double of a Hopf algebra, Duke Math. J. 7:3 (1994) 763-776, MR1277953, doi
Tomasz Brzeziński, Gigel Militaru, Bialgebroids, -bialgebras and duality, J. Algebra 251 (2002) 279-294 [math.QA/0012164, doi:10.1006/jabr.2001.9101]
Standard reference is now
Remaining issues about the antipode are settled in
Every invertible counital 2-cocycle (Drinfeld twist) for a bialgebra , with inverse , induces a Drinfeld-Xu 2-cocycle for the scalar extension bialgebroid . The scalar extension for the -twisted data is isomorphic as a -bialgebroid to the -twist of .
Some corrections and a slight generalization are given in
An infinite-dimensional case of a Heisenberg double of a universal enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is described as a version of a completed Hopf algebroid in
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 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
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
Last revised on August 12, 2023 at 13:06:05. See the history of this page for a list of all contributions to it.