nLab scalar extension bialgebroid

Motivation and historical appearance

Given a finite-dimensional Hopf algebra $H$ and a $D(H)$-module algebra $A$ (where $D(H)$ is the Drinfeld double of $H$), Lu1996 has introduced the structure of a left $A$-bialgebroid on the smash product algebra $A\sharp H$ 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 $H$-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.

Definition

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

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

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

$$\epsilon^{A\sharp H}:A\sharp H\to A,\,\,\,\, \epsilon^{A\sharp H}(a\sharp h) = a\epsilon(h).$$

Scalar extension Hopf algebroid

If $H$ is a Hopf algebra the Brzeziński-Militaru construction gives a Lu-Hopf bialgebroid and if the antipode $S$ of $H$ is invertible then it gives also a Böhm-Szlachányi symmetric Hopf algebroid. If the antipode $S$ 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 $H^*\sharp H$ 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

Literature

• 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, $\times_A$-bialgebras and duality, J. Algebra 251 (2002) 279-294 [math.QA/0012164, doi:10.1006/jabr.2001.9101]

Standard reference is now

• G. Böhm, Hopf algebroids, in Handbook of algebra, vol. 6, ed. M. Hazewinkel, arXiv:math.RA/0805.3806

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 $F = \sum F^1\otimes F^2$ (Drinfeld twist) for a bialgebra $H$, with inverse $F^{-1} = G^1\otimes G^2$, induces a Drinfeld-Xu 2-cocycle $\mathcal{G}=\sum 1\sharp G^1\otimes_A 1\sharp G^2$ for the scalar extension bialgebroid $A\sharp H$. The scalar extension for the $F$-twisted data $A_F\sharp H^F$ is isomorphic as a $A_F$-bialgebroid to the $\mathcal{G}$-twist of $A\sharp H$.

• 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

• Zoran Škoda, Martina Stojić, Comment on “Twisted bialgebroids versus bialgebroids from Drinfeld twists”, arXiv:2308.05083

An infinite-dimensional case of a Heisenberg double of a universal enveloping algebra $U(g)$ 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 $U(g)$ 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

• X. Han, Shahn Majid, Bisections and cocycles on action Hopf algebroids, arXiv:2305.12465

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

• Zoran Škoda, Martina Stojić, Hopf algebroids with balancing subalgebra, J. Alg. 598 (2022) 445–469 arXiv:1610.03837 doi

category: algebra