Crossed modules of Hopf algebras: an approach via monoids Gabriella - - PowerPoint PPT Presentation

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Crossed modules of Hopf algebras: an approach via monoids Gabriella - - PowerPoint PPT Presentation

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Crossed modules of Hopf algebras: an approach via monoids Gabriella B ohm Wigner Research Centre for Physics, Budapest Quantum groups and their analysis Summer school and workshop at University of Oslo 6 th of August 2019 Crossed module of

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Crossed modules of Hopf algebras: an approach via monoids Gabriella B¨

  • hm
Wigner Research Centre for Physics, Budapest

Quantum groups and their analysis Summer school and workshop at University of Oslo

6th of August 2019
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Crossed module

  • f groups [Whitehead 1941]
a•a generalization of a normal subgroup N ⊳ G to non-injective N → G a•a diverse applications a•a equivalent to: ◮ strict 2-group (= category object in the category of groups) a• equivalent to:a ◮ simplicial group whose Moore complex has length 1 a•a concise categorical proof [G Janelidze 2003]
  • f groupoids [Brown, ˙
I¸ cen 2003]
  • f Hopf algebras [Fern´
andez Vilaboa et al. 2006] [Aguiar 1997] [Majid 2012] [Faria Martins 2016] [Gran, Sterck, Vercruysse 2019] [Emir 2019] a•a working definitions a•a bits of the equivalent forms a•a no abstract categorical treatment Unified treatment ? a simplified review of [GB arXiv:1803.03418 1803.04124 1803.04622]
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Idea

view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects
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Idea

view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects
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Monoids in monoidal categories

  • Definition. A monoidal category consists of
a•a a category C a•a functors C × C juxtaposition C ✶ I
  • a•a coherent natural isomorphisms (− =) ≡
− (= ≡) a•a coherent natural isomorphisms I − − − I
  • (omitted throughout).
Examples: (set, ×), (span, ), (vec, ⊗), (clg, ⊗).
  • Definition. A monoid in a monoidal category consists of
a•a an object A a•a morphisms AA m A I u
  • s.t.
AAA m1 1m AA m
  • A
m A A u1 1u AA m
  • A
m A commute. Examples: ordinary monoids, small categories, algebras, bialgebras.
  • Definition. A monoid morphism is A
f A′ in C s.t. f .m = m′.ff and f .u = u′.
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Factorization of monoids

For monoid morphisms A f C B g
  • s.t. q := AB
fg CC m C is invertible monoid morphism ⇔ monoid morphisms s.t. A f
  • a
  • C
c
  • B
g
  • b
  • D
A f
  • a
  • C
c
  • B
g
  • b
  • D
BA gf ba DD m
  • CC
m (∗) D C q−1 AB ab DD m
  • .
commutes. C c D → A f C c D C c
  • B
g
  • C
q−1 AB ab DD m D → A a D B b
  • multiplicative iff (∗) commutes
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Idea

view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects
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Category objects

For a category with pullbacks and a given object B, the category of spans B A s
  • t
  • is monoidal via the pullback
B .
  • Definition. A category object is a monoid in the category of spans:
B i A s
  • t
  • A
B A. c
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Relative category objects

In a categorical Hopf algebra B A
  • A
B A
  • in [Fern´
andez Vilaboa et al. 2006], the cotensor product B is not a pullback: for coalgebra maps A s B C t
  • a• A
B C := { i ai ⊗ ci ∈ A ⊗ C | i ai 1 ⊗ s(ai 2) ⊗ ci = i ai ⊗ t(ci 1) ⊗ ci 2} a• is a subcoalgebra iff a → a1 ⊗ s(a2) and c → t(c1) ⊗ c2 are coalgebra maps a• (then the counits ε induce coalgebra maps in the bottom row) a• a factorization D
  • a
  • c
  • A
A ⊗ C 1⊗ε
  • A
B C
  • j
  • j A ⊗ C ε⊗1 C
exists a• iff d →a(d1)⊗c(d2) is a coalgebra map

Idea: only a relative pullback wrt a suitable admissible class of spans.

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Admissible class of spans

  • Definition. A class S of spans in any category is admissible if
X A f
  • g Y ∈ S ⇒ X ′
X f ′
  • A
f
  • g Y
g ′ Y ′ ∈ S ∀f ′, g ′ and X A f
  • g Y ∈ S ⇒ X
A f
  • B
h
  • h A
g Y ∈ S ∀h. Examples: a•a The class of all spans in any category is admissible. a•a In coalg the class C := { X A f
  • g Y | a → f (a1)⊗g(a2) is a coalgebra map}
a•a is admissible. a•a In bialg the class B := { X A f
  • g Y | a → f (a1)⊗g(a2) is a coalgebra map}
a•a is admissible.
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Relative pullback

  • Definition. The S-relative pullback of any A
s B C t
  • is A
A B C pA
  • pC C ∈ S
. s.t. in D c
  • a
  • h
  • A
B C pC pA C t
  • A
s B
  • the blue square commutes
  • if A
D a
  • c C ∈ S and the exterior commutes then ∃! h
  • A
A B C pA
  • D
f
  • g E
& C A B C pC
  • D
f
  • g E ∈ S ⇒ A
B C D f
  • g E ∈ S
  • E
D g
  • f A
B C pA A & E D g
  • f A
B C pC C ∈ S ⇒ E D g
  • f A
B C ∈ S Examples.
  • If S = {all spans} then S-relative pullback=pullback.
  • If A s B
C t
  • are bialgebra maps s.t. a → a1 ⊗ s(a2), c → t(c1) ⊗ c2 are
coalgebra maps then A B C (ε⊗1).j (1⊗ε).j C t
  • A
s B is a B-relative pullback.
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Relative category

  • Theorem. Let S be an admissible class of spans in a category s.t. if A
A s B, B C t
  • C ∈ S, then there exists the S-relative pullback A
B C. Then for any B for which B B B ∈ S, there is a monoidal category: a•a objects are the spans B A t
  • s B s.t. A
A s B, B C t
  • C ∈ S
a•a morphisms are the span morphisms a•a monoidal product is the S-relative pullback with the unit B B B.
  • Example. For a cocommutative bialgebra B there is a monoidal category:
a•a objects are the spans B A t
  • s B of bialgebras s.t. a → a1 ⊗ s(a2) and
a•a a → t(a1) ⊗ a2 are coalgebra maps a•a morphisms are the maps of bialgebra spans a•a monoidal product is the cotensor product over B with the unit B B B.
  • Definition. For S and B as in the theorem, an S-relative category — with object
  • f objects B — is a monoid in the above monoidal category:
B i A s
  • t
  • A
B A. c
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Idea

view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects
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Relative category in the category of bialgebras

In bialg, a B-relative category is given by bialgebra maps B i A s
  • t
  • A
B A ={ i ai ⊗ci ∈ A ⊗ C| i ai 1⊗s(ai 2)⊗ci = i ai ⊗t(ci 1)⊗ci 2} c
  • s.t. a → a1⊗s(a2) & a → t(a1)⊗a2 are coalgebra maps (⇒ B is cocommutative)
s.t. i and c are maps of spans s.t. c is associative with the unit i . For A B I = {y ∈ A | y1 ⊗ s(y2) = y ⊗ 1} j A B, i
  • q = (A
B I) ⊗ B j⊗i A ⊗ A m A , y ⊗ b → yi(b) has the inverse a → a1i(z(s(a2))) ⊗ s(a3) whenever B is a Hopf algebra with antipode z.
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Split epimorphisms versus actions

[Radford 1985] split epimorphism monoid and comonoid in mod(B) B i A s
  • t
  • A
B A c
  • ⇔
(Y , B ⊗ Y ⊲ Y ) s.t. ← − Hopf algebra (bb′) ⊲ y = b ⊲ (b′ ⊲ y) 1 ⊲ y = 1 b ⊲ (yy ′) = (b1 ⊲ y)(b2 ⊲ y ′) b ⊲ 1 = ε(b)1 (b⊲y)1⊗(b⊲y)2 =b1⊲y1⊗b2⊲y2 ε(b⊲y) = ε(b)ε(y) B i A s
  • →
( A B I,B⊗(A B I) i⊗jA⊗A m A q−1 (A B I)⊗B 1⊗ε A B I) b ⊗ y → i(b1)yi(z(b2)) B 1⊗− Y ⊗ B =: A ε⊗1
  • →
(Y , B ⊗ Y ⊲ Y ) a → a1 ⊗ s(a2) is a coalgebra map ⇔ B is cocommutative.
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Reflexive graphs versus pre-crossed modules

. Hopf algebra →B i A s
  • t
  • A
B A c
  • bialgebra map
⇔ bialgebra maps s.t. A B I j
  • k
  • A
t
  • B
i
  • B
A B I j
  • k
  • A
t
  • B
i
  • B
B ⊗ (A B I) i⊗j 1⊗k B ⊗ B m
  • A ⊗ A
m B A q−1
  • (A
B I) ⊗ B k⊗1 B ⊗ B m
  • .
commutes; i.e. . k(b1 ⊲ y)b2 = bk(y) . 1st Peiffer condition a → t(a1) ⊗ a2 is a coalgebra map ⇔ y → k(y1) ⊗ y2 is a coalgebra map
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Category objects versus crossed modules

. Hopf algebra →B i A s
  • t
  • A
B A c
  • For A
B I j A 1i A B A A i1
  • q2 = (A
B I)⊗A j⊗1 A⊗A (1i)⊗(i1) (A B A)⊗(A B A) m A B A, y⊗a→yit(a1)⊗a2 has the inverse
  • n an⊗cn →
n ani(z(t(cn 1 )))⊗cn 2 . . A B I (1i).j j
  • A
B A c
  • A
i1
  • A
⇔ A B I (1i).j j
  • A
B A c
  • A
i1
  • A
s.t. A ⊗ (A B I) 1⊗j 1⊗j A ⊗ A m
  • A ⊗ A
(i1)⊗(1i) (A B A) ⊗ (A B A) m A A B A q−1 2
  • (A
B I) ⊗ A j×1 A ⊗ A m
  • .
i.e. y ′y = (k(y ′ 1) ⊲ y)y ′ 2 . 2nd Peiffer condition
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  • Theorem. For a cocommutative Hopf algebra B, there is an equivalence between
  • B-relative categories B
i A s
  • t
  • A
B A c
  • f bialgebras
  • crossed modules (B, Y
k B , B ⊗ Y ⊲ Y ) of bialgebras; i.e. i.e. - B-module algebras and B-module coalgebras (Y , ⊲) i.e. - bialgebra maps k s.t. k(b1 ⊲ y)b2 = bk(y) (k(y1) ⊲ y ′)y2 = yy ′ and i.e. - bialgebra maps k s.t. y → k(y1) ⊗ y2 is a coalgebra map. Proof. factorization of monoids
  • crossed module ⇐
⇒ reflexive graph with unital composition B i A s
  • t
  • A
B A c
  • c is a morphism of spans & associative ⇐ universality of relative pullback.
The equivalence restricts to B-relative categories and crossed modules of Hopf algebras cf. [Fern´ andez Vilaboa et al. 2006] and of cocommutative Hopf algebras.

¡For cocommutative Hopf algebras there is more!

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Idea

view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects
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The Moore complex of a simplicial bialgebra

. [Emir 2019]
  • Definition. The Moore complex of a simplicial bialgebra
H0 σ0 H1 ∂0
  • ∂1
  • σ0
  • σ1
H2 · · · Hn−1 ∂1
  • ∂0
  • ∂2
  • σ0
  • σn−1
  • Hn · · ·
∂0
  • ∂1
. . .
  • ∂n
  • is the normalized chain complex · · · Mn+1
Dn Mn Dn−1 · · · D0 M0 where . Mn Hn
  • ∂n
  • ∂n−1
. . .
  • ∂1
  • 1ε(−):x→x1⊗1⊗x2
  • Hn⊗Hn−1⊗Hn is a joint equalizer for
∂k:x →x1⊗∂k(x2)⊗x3 Dn is the restriction of Hn+1 ∂0 Hn . If all Hn are Hopf algebras then so are all Mn.
  • Definition. The Moore complex has length ℓ if Mn ∼
= I for all n > ℓ.
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  • Theorem. The following are equivalent.
  • B-relative categories of cocommutative Hopf algebras
  • crossed modules of cocommutative Hopf algebras
  • cocommutative simplicial Hopf algebras whose Moore complex has length 1.
Proof. simplicial Hopf algebra ← → B-relative category H truncation − → H0 σ0 H1 ∂0
  • ∂1
  • H2 ∼
= H1 H0 H1 ∂1
  • B
i A s
  • t
  • 1i
  • i1
A B A · · · c
  • p1
  • p2
  • nerve
← −
  • B
i A s
  • t
  • A
B A c
  • For a cocommutative simplicial Hopf algebra Hn ∼
= H
  • H0
n 1 .
  • The corresponding crossed module is (M0,M1
D0 M0,M0⊗M1 b⊗x→σ0(b1)xσ0(z(b2) ) M1 ).
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SLIDE 22 Analogous proofs work in the category of monoids in any monoidal category with a monoidal admissible class of spans S s.t. the S-relative pullbacks of those morphisms A f B C g
  • exist for which A
A f C & C B g
  • B ∈ S.
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Further questions

  • Crossed modules of associative/ Lie/ Jordan algebras do not fit.
¿A broader framework including them?
  • ¿Higher categories vs higher dimensional crossed cubes?
Crossed modules of monoids do not seem to be monoids themselves in any suitable monoidal category ⇒ naive iteration does not work.
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Thank you!