The Miyashita action in Hopf-Galois Theory
Meaning hidden behind cohomology
Dedicated to Prof. Szeto and Prof. Ikehata with gratitude and respect
Akira Masuoka (U. Tsukuba)
Cohomologiesare a useful tool, which is automatic asautomatic cars. But they sometimes hide meanings behind them, just as automatic cars prevent us from learning mechanism of cars. Let’s spend this time to find out such ahidden meaning.
It would be interesting to see that such an attempt is more natural and more successful in thenon-commutative worldrather than in the commutative world.
This talk is to show you results byRalf G¨unther, who was a PhD student 30 years ago in M¨unchen. He worked on Hopf-Galois extensions. But I will work mostly withstrongly graded algebras, in particular. TheMiyashita actionplays a role.
Basic definitions
Work over a field k, for simplicity.
Let G be a group. The identity element is denoted by e. Let A =⊕g ∈GAg be a G -graded algebra. Note 1 ∈ Ae. Definition A is said to be
◮ over an algebra C, if Ae = C ;
◮ central, if Ae⊂ Z (A), the center of A;
◮ strongly graded, if AgAh= Agh, g , h ∈ G ;
◮ crossed product, if each Ag contains a unit ug in A. Remark A G -crossed product is thus A =⊕g ∈GCug given by
weak action G y C , non-abel 2-cocycle σ : G × G → C× so that
ugc =gc ug, uguh= σ(g , h) ugh, where c ∈ C , g , h ∈ G .
Abelian situation
Let C be acommutative algebra.
Fact Thecentral G-crossed products over C are classified (up to iso.) by H2(G , C×), where C× is regarded as a trivial G -module:
{central G -crossed products over C }/ ≃ 1−1↔ H2(G , C×)
Fact Given N ▹ G , we have an exact sequence
0 →H1(G /N, C×) → H1(G , C×) → H1(N, C×)G
→ H2(G /N, C×)→ Hinf 2(G , C×)
Rem Given a G /N-crossed prod. A =⊕g ∈G/NAg over C , inf(A) is the G -graded subalgebra of A ⊗ kG given by
inf(A)g = Ag ⊗ kg , g ∈ G .
Q. What’s the meaning of the exactness at H2(G /N, C×)? H1(N, C×)G = HomG(N, C×)
= {hom. f : N → C×| f (ng) = f (n), n ∈ N, g ∈ G } Ans. Every f gives rise to a cent. G /N-crossed prod. over C ,
CG/(n − f (n) | n ∈ N) ( = ⊕
g ∈G/N
C g),
which isinflatedto the trivial G -crossed prod. CG (group alg.). Conversely, such a G /N-crossed prod. arises (uniquely up to the image of H1(G , C×) → H1(N, C×)G) in this way.
Rem. In ⊕g ∈G/NC g, {g } is supposed to be a set of
arbitrarily chosen and fixedrepresentatives in G for G /N, and f(ng)f (nh) g h = f (ngh) gh,
where ng = g g−1 (∈ N).
Non-abelian situation
Question. Can the result above be generalized fornon-central strongly graded algebras?
Answer by Ralf G¨unther, Comm. Algebra 27 (1999). Yes! Let A =⊕g ∈GAg be a strongly G -graded algebra over C = Ae. The Miyashita action AC xG,bg (b ∈ AC, g ∈ G ), is det. by
ba= abg, a∈ Ag.
Example If A =⊕g ∈GCug is a G -crossed product, then bg = u−g1bug, b ∈ AC, g ∈ G .
Indeed, if a= cug, then
ba= b(cug) = cbug = (cug)(ug−1bug) = a(ug−1bug).
AC is a G -graded subalgebra of A, and a G -algebra w.r.t. the Miyashita action.
AlgGG denotes the category of all G -graded G -algebras B=⊕g ∈GBg such that
Bgh⊂ Bh−1gh, g, h ∈ G . Example 1) AC ∈ AlgGG.
2) Let N ▹ G . Then kN ∈ AlgGG w.r.t. the conjugation N x G . The set AlgGG(kN, AC) is identified with the set of systems (bn)n∈N of elements bn∈ AC such that
bn∈ An, b1 = 1, bnbn′ = bnn′, bng = bg−1ng, where n, n′ ∈ N, g ∈ G .
Thm (G¨unther) C a fixed algebra, N ▹ G .
1) Given a strongly G -graded algebra A over C and a system (bn)n∈N as above, the quotient G /N-graded algebra of A
A/(bn− 1 | n ∈ N)
is a strongly G /N-graded algebra over C , which is inflated to the original A.
2) Conversely, every strongly G /N-graded algebra over C arises in this way.
Rem This refers toall strongly G /N-graded algebras.
Compare with the result in the abelian situation which refers to G/N-crossed products which are trivially inflated.
A categorical re-formulation is given by thepull-back of the categories:
Str.grC(G /N) Str.grC(G )
AlgGG/kN AlgGG
inf //
forget
//
( )C
Here
Str.grC(Γ) = the cat. of strongly Γ-graded algebras over C AlgGG/kN = the cat. of all pairs (B, f ) of B ∈ AlgGG
with a morphism f : kN → B from kN
The original G¨unther’s Theorem was formulated for Hopf-Galois extensions, more generally than for strongly graded algebras, for which Hopf algebra quotients H → H act for G → G /N. I will not formulate the theorem in the Hopf-algebra language, but will only show how naturally the notion of Hopf-Galois extensions arises, from the viewpoint of (non-commutative) geometry.
Translation from geometry to algebra
Easy! 1) Replace × with ⊗.
2) Replace a one-pt. space with k, the base field. 3) Reverse the direction of arrows.
space X ↔ algebra A
group 1unit→ G prod← G × G ↔ Hopf algebra k ← Hε → H ⊗ H∆ right G -space X act← X × G
↔ right H-comodule algebra A→ A ⊗ Hρ
G-orbits X /G ↔ H-coinvariants C = {a ∈ A | ρ(a) = a ⊗ 1} G-torsor over X /G , if X ×X/GX ← X × G≃ via (x, xg) ←p (x, g )
↔ H-Galois over C , if A ⊗CA→ A ⊗ H via a ⊗ a≃ ′ 7→ a ρ(a′).
Def Let H be a Hopf algebra. A right H-comodule algebra A= (A, ρ : A → A ⊗ H) is an H-Galois extension over the subalgebra C of H-coinvariants, if
β : A ⊗C A→ A ⊗ H, β(a ⊗ a′) = a ρ(a′)
is a bijection.
Example 1) If H = Map(G , k) with |G | < ∞, H-Galois = G -Galois, since β is then identified with
A⊗AG A→ Map(G , A), a⊗ a′ 7→ [g 7→ a(ga′)].
2) If H = kG , then H-Galois = strongly G -graded. Indeed, Ag ⊗Ae Ah→ Agh⊗ kh, a⊗ a′ 7→ aa′⊗ h
are summed up to β.
The Miyashita action for strongly graded algebras is generalized by theMiyashita-Ulbrich action for H-Galois extensions A/C .
β−1: A ⊗ H → A ⊗C A restricts to H = k ⊗ H → (A ⊗C A)C. TheMiyashita-Ulbrich action on AC is defined by
H→ (A ⊗C A)C “crip”→ EndZ(A)(AC).
AC is an H-comodule subalgebra of A. Given the MU action above it turns into an algebra of the braided category YDHH of
Yetter-Drinfeld modules. Those algebras generalize the AlgGG before, and play a role in quantum group theory.
