Scalar Extension Hopf Algebroids

• Scalar extension Hopf algebroids are structures that extend Hopf algebra symmetries via base change and smash-product methods, preserving the key comodule, torsor, and groupoid properties.
• The construction involves changing scalars along an algebra map or forming a smash product with a braided commutative Yetter–Drinfeld module algebra, encapsulating geometric transitivity and weak equivalence.
• Recent advances integrate noncommutative balancing techniques, Lie-theoretic realizations, and covariant differential calculi to analyze homogeneous spaces and push-forward scenarios.

Scalar extension Hopf algebroids form a class of constructions in which Hopf-algebraic data are transported across a change of scalars or assembled into a Hopf algebroid over a nontrivial base algebra. In the literature represented here, the term has two established uses. In one use, scalar extension means base change of a Hopf algebroid along an algebra map $\phi:A\to B$, producing $(B,H_B)$ with $H_B=B\otimes_A H\otimes_A B$; in another, it means the Brzeziński–Militaru smash-product construction $A\sharp H$ attached to a braided commutative Yetter–Drinfeld $H$-module algebra $A$ (Kaoutit, 2015, Stojić, 2022). These two usages are related by the common theme that extension of scalars should preserve, or reconstruct, the essential comodule, torsor, and groupoid structure of the theory.

1. Terminological scope and basic patterns

A central point of terminology is that “scalar extension Hopf algebroid” does not denote a single universal construction across the recent literature. One strand studies base change of an existing Hopf algebroid, while another studies the smash product of a Hopf algebra with a braided commutative Yetter–Drinfeld module algebra. A later paper on calculi makes the distinction explicit: there, “scalar extension Hopf algebroid” means the Brzeziński–Militaru smash product $A\#H$, and not the base-change construction $S\otimes_R H\otimes_R S$ along an algebra map $\phi:R\to S$ (Kowalzig et al., 22 Jul 2025).

Usage	Input	Output
Base change	$(A,H)$ and $(B,H_B)$0	$(B,H_B)$1
Smash-product scalar extension	Hopf algebra $(B,H_B)$2 and braided commutative Yetter–Drinfeld $(B,H_B)$3-module algebra $(B,H_B)$4	$(B,H_B)$5 over base $(B,H_B)$6

In the base-change setting, scalar extension is the algebraic counterpart of pulling back affine groupoid schemes along maps of base rings. In the smash-product setting, the construction starts from Hopf-algebraic symmetry already acting and coacting on $(B,H_B)$7, and packages that symmetry into a bialgebroid or Hopf algebroid over $(B,H_B)$8. The two viewpoints meet in their emphasis on source and target maps, Takeuchi balancing, comodule categories, principal bundles, and categorical invariance under change of scalars.

2. Base change, geometric transitivity, and weak equivalence

For a commutative Hopf algebroid $(B,H_B)$9 over a field $H_B=B\otimes_A H\otimes_A B$0, the structure maps are

$H_B=B\otimes_A H\otimes_A B$1

with source and target

$H_B=B\otimes_A H\otimes_A B$2

Flatness means that $H_B=B\otimes_A H\otimes_A B$3 is a flat $H_B=B\otimes_A H\otimes_A B$4-module. The associated presheaf of groupoids has, on each algebra $H_B=B\otimes_A H\otimes_A B$5, objects $H_B=B\otimes_A H\otimes_A B$6 and arrows $H_B=B\otimes_A H\otimes_A B$7, with composition

$H_B=B\otimes_A H\otimes_A B$8

inverse $H_B=B\otimes_A H\otimes_A B$9, and identity $A\sharp H$0 (Kaoutit, 2015).

Base change along $A\sharp H$1 gives

$A\sharp H$2

with

$A\sharp H$3

$A\sharp H$4

$A\sharp H$5

The canonical morphism is $A\sharp H$6 with $A\sharp H$7.

The decisive structural result is the characterization of geometric transitivity. Under the standing hypotheses that $A\sharp H$8 is a commutative flat Hopf algebroid over $A\sharp H$9, with $H$0 and $H$1, the following are equivalent: $H$2 is faithfully flat; any two objects are fpqc locally isomorphic; for any extension $H$3, the extension $H$4 into $H$5 is faithfully flat; $H$6 is geometrically transitive; for any extension $H$7, the canonical morphism $H$8 is a weak equivalence; and for any extension $H$9, the trivial principal left $A$0-bundle $A$1 is a principal bi-bundle (Kaoutit, 2015).

Here weak equivalence means that the induced functor

$A$2

is an equivalence of symmetric monoidal $A$3-linear categories. In the geometrically transitive case, every extension of the base ring is Landweber exact, and the extension-of-scalars functor on comodules is exact. The same package of results yields transitivity of the character groupoid $A$4 and, more generally, transitivity of $A$5 for any field extension $A$6 with $A$7. It also yields weak equivalence of isotropy Hopf algebras

$A$8

at different $A$9-points, and their conjugacy in the sense formulated in the paper.

3. Smash-product scalar extension and symmetric Hopf algebroids

In the Brzeziński–Militaru construction, one fixes a Hopf algebra $A\#H$0 over $A\#H$1 and a braided commutative Yetter–Drinfeld $A\#H$2-module algebra $A\#H$3. In the left-right convention, $A\#H$4 carries a left $A\#H$5-action and a right $A\#H$6-coaction

$A\#H$7

satisfying the Yetter–Drinfeld compatibility

$A\#H$8

and braided commutativity

$A\#H$9

The smash product $S\otimes_R H\otimes_R S$0 has multiplication

$S\otimes_R H\otimes_R S$1

unit $S\otimes_R H\otimes_R S$2, source and target

$S\otimes_R H\otimes_R S$3

and coproduct and counit

$S\otimes_R H\otimes_R S$4

These formulas give the scalar extension left $S\otimes_R H\otimes_R S$5-bialgebroid in the Lu/Takeuchi sense (Stojić, 2022).

A significant correction to the classical literature concerns the antipode. The map

$S\otimes_R H\otimes_R S$6

had been written down earlier, but the published proof that $S\otimes_R H\otimes_R S$7 is an antihomomorphism covered only a special case. A complete proof is given in the later paper, which establishes

$S\otimes_R H\otimes_R S$8

for all $S\otimes_R H\otimes_R S$9, without assuming that the antipode of $\phi:R\to S$0 is invertible (Stojić, 2022). This resolves a genuine gap rather than a matter of presentation.

The same paper generalizes the construction. If $\phi:R\to S$1 is a braided commutative left-right Yetter–Drinfeld $\phi:R\to S$2-module algebra, $\phi:R\to S$3 is a braided commutative right-left Yetter–Drinfeld $\phi:R\to S$4-module algebra, and $\phi:R\to S$5 is an algebra antiisomorphism satisfying

$\phi:R\to S$6

then there is an algebra isomorphism

$\phi:R\to S$7

with inverse

$\phi:R\to S$8

Transporting the left and right bialgebroid structures across $\phi:R\to S$9 yields a symmetric Hopf algebroid over $(A,H)$0 and $(A,H)$1, again without requiring invertibility of $(A,H)$2.

4. Balancing subalgebras and noncommutative-base formulations

For scalar extension Hopf algebroids over a noncommutative base, a basic difficulty is that the kernel

$(A,H)$3

need not be a two-sided ideal in $(A,H)$4. Lu’s definition circumvents this by choosing a section $(A,H)$5. An alternative framework replaces the choice of section by a balancing subalgebra $(A,H)$6 such that $(A,H)$7 is a two-sided ideal in $(A,H)$8, $(A,H)$9, and $(B,H_B)$00 is multiplicative (Škoda et al., 2016).

For a scalar extension $(B,H_B)$01, with $(B,H_B)$02 a braided-commutative left-right Yetter–Drinfeld $(B,H_B)$03-module algebra and $(B,H_B)$04 bijective, the structure maps are

$(B,H_B)$05

$(B,H_B)$06

$(B,H_B)$07

The balancing-subalgebra construction starts from

$(B,H_B)$08

If $(B,H_B)$09 is the smallest unital subalgebra containing $(B,H_B)$10 and $(B,H_B)$11 is the subalgebra generated by $(B,H_B)$12 and $(B,H_B)$13, then the ideal $(B,H_B)$14 generated by the $(B,H_B)$15 satisfies

$(B,H_B)$16

The resulting theorem states that every scalar extension Hopf algebroid can be cast into the balancing-subalgebra axioms, with $(B,H_B)$17, $(B,H_B)$18, $(B,H_B)$19, and $(B,H_B)$20 (Škoda et al., 2016).

This reformulation is structurally important because it makes the balancing relations explicit inside a subalgebra of $(B,H_B)$21, rather than hiding them in a choice of section. The paper also proves that weak Hopf algebras fit the same framework, although those examples are not scalar extensions in the Brzeziński–Militaru sense.

5. Lie-theoretic realizations over universal enveloping algebras

A substantial Lie-theoretic class of scalar extension Hopf algebroids arises from universal enveloping algebras. Let $(B,H_B)$22 be an affine algebraic group over $(B,H_B)$23, let $(B,H_B)$24, let $(B,H_B)$25 be the Lie algebra of left-invariant derivations on $(B,H_B)$26, and let $(B,H_B)$27 and $(B,H_B)$28 be the corresponding bases of left- and right-invariant derivations. The adjoint representation gives representative functions $(B,H_B)$29 and $(B,H_B)$30 satisfying

$(B,H_B)$31

and similarly for $(B,H_B)$32. The coaction on $(B,H_B)$33 is

$(B,H_B)$34

extended antimultiplicatively, while the right $(B,H_B)$35-action is induced by the Hopf pairing

$(B,H_B)$36

These data make $(B,H_B)$37 a braided commutative right-left Yetter–Drinfeld module algebra over $(B,H_B)$38 (Stojić et al., 3 Jun 2025).

The smash product $(B,H_B)$39 is identified with the algebra of regular differential operators $(B,H_B)$40. In the resulting Hopf algebroid, the explicit left and right base-algebra structures are

$(B,H_B)$41

$(B,H_B)$42

where $(B,H_B)$43. The coproducts satisfy

$(B,H_B)$44

and similarly on the right side. The antipode is given on generators by

$(B,H_B)$45

The same paper gives a finite-dual variant with $(B,H_B)$46, using matrix coefficients $(B,H_B)$47 and $(B,H_B)$48 derived from the adjoint representation. A notable feature of this work is that it avoids completion issues by restricting to $(B,H_B)$49, $(B,H_B)$50, and $(B,H_B)$51, while still connecting the constructions to completed Heisenberg doubles and to differential-operator Hopf algebroids.

6. Covariant calculi and homogeneous-space variants

For right Hopf algebroids, the translation map replaces the antipode in many structural arguments. In the scalar extension Hopf algebroid $(B,H_B)$52 with invertible antipode, the translation map is

$(B,H_B)$53

This is the key input in the Hopf-algebroid analogue of the Woronowicz classification of left covariant first-order differential calculi (Kowalzig et al., 22 Jul 2025).

The classification theorem states that for a right Hopf algebroid $(B,H_B)$54 with suitable flatness, left $(B,H_B)$55-covariant first-order calculi are in bijection with left ideals $(B,H_B)$56. For the scalar extension Hopf algebroid $(B,H_B)$57 one has

$(B,H_B)$58

and therefore left $(B,H_B)$59-covariant calculi correspond to left $(B,H_B)$60-ideals in $(B,H_B)$61. An explicit family is obtained by taking $(B,H_B)$62 to be both a left $(B,H_B)$63-ideal and a left $(B,H_B)$64-ideal, and $(B,H_B)$65 a left $(B,H_B)$66-ideal; then

$(B,H_B)$67

is a left $(B,H_B)$68-ideal, and the corresponding calculus is

$(B,H_B)$69

with

$(B,H_B)$70

The same paper develops homogeneous-space variants. If $(B,H_B)$71 is a Hopf algebra surjection and $(B,H_B)$72 is a braided commutative algebra in $(B,H_B)$73, then there is a surjection of right Hopf algebroids

$(B,H_B)$74

whose left Hopf kernel is $(B,H_B)$75 with $(B,H_B)$76. If $(B,H_B)$77 is a Hopf–Galois extension, then $(B,H_B)$78 is a Hopf–Galois extension in the Hopf-algebroid sense and hence a principal homogeneous space. Covariant calculi on such spaces are classified by subobjects $(B,H_B)$79 in $(B,H_B)$80, and one again obtains explicit families from pairs $(B,H_B)$81.

7. Push-forward, noncentral base change, and Ehresmann–Schauenburg algebroids

A further base-change direction appears in the push-forward of Hopf–Galois extensions. If $(B,H_B)$82 is an $(B,H_B)$83-Galois extension and $(B,H_B)$84 is an algebra map, the noncentral push-forward is not the ordinary tensor-product algebra $(B,H_B)$85; instead, it is a twisted tensor product algebra

$(B,H_B)$86

built from a twisting map $(B,H_B)$87 (Landi et al., 23 Dec 2025).

The multiplication on $(B,H_B)$88 is

$(B,H_B)$89

where $(B,H_B)$90. The push-forward remains $(B,H_B)$91-Galois provided $(B,H_B)$92 satisfies the compatibility conditions used in the main theorem: $(B,H_B)$93-bilinearity through $(B,H_B)$94, normality, the one-sided distributive law, and $(B,H_B)$95-colinearity. Under these hypotheses, and assuming faithful flatness of $(B,H_B)$96 over $(B,H_B)$97, $(B,H_B)$98 is a faithfully flat right $(B,H_B)$99-comodule algebra with coinvariants $H_B=B\otimes_A H\otimes_A B$00, and the pushed-forward translation map is

$H_B=B\otimes_A H\otimes_A B$01

At the bialgebroid level, the paper compares the Ehresmann–Schauenburg bialgebroid

$H_B=B\otimes_A H\otimes_A B$02

of the original extension with

$H_B=B\otimes_A H\otimes_A B$03

for the pushed-forward extension. There is a morphism of bialgebroids

$H_B=B\otimes_A H\otimes_A B$04

and, more strongly, a comparison with the base-changed coring

$H_B=B\otimes_A H\otimes_A B$05

Assuming inner $H_B=B\otimes_A H\otimes_A B$06-linearity and invertibility of $H_B=B\otimes_A H\otimes_A B$07, the map

$H_B=B\otimes_A H\otimes_A B$08

is a coring isomorphism. In this sense, push-forward realizes scalar extension of Ehresmann–Schauenburg bialgebroids in the noncentral case.

This synthesis places scalar extension Hopf algebroids at the intersection of groupoid transitivity, Yetter–Drinfeld module algebras, Hopf–Galois theory, and differential geometry over noncommutative bases. Base change can characterize geometric transitivity through weak equivalence of comodule categories; smash-product scalar extension can produce left, right, and symmetric Hopf algebroids from braided commutative Yetter–Drinfeld data; balancing subalgebras provide a noncommutative-base replacement for Lu’s section; Lie-theoretic models over $H_B=B\otimes_A H\otimes_A B$09 identify the total algebra with differential operators; and recent work on calculi and push-forward shows that scalar extension remains a natural organizing principle for homogeneous spaces and noncentral base change.