2-Crossed Modules of Hopf Algebras Kadir EMR Joint work with: Joo - - PowerPoint PPT Presentation

2-Crossed Modules of Hopf Algebras

2-Crossed Modules of Hopf Algebras Kadir EMİR

Joint work with: João Faria Martins

11.08.2016 Category Theory ’2016 Dalhousie and St Mary’s, Halifax Nova Scotia, Canada

2-Crossed Modules of Hopf Algebras Abstract

Abstract

In this talk, we ❖ define the Moore complex of any simplicial (cocommutative) Hopf algebra by using Hopf kernels which are defined quite different from the kernels of groups or various well-known algebraic structures, ❖ see that these Hopf kernels only make sense in the case of cocommutativity, ❖ give some applications of the Moore complex such as iterated Peiffer pairings and simplicial decomposition of simplicial cocommutative Hopf algebras, ❖ introduce the notion of 2-crossed modules of (cocommutative) Hopf algebras and continue to talk about its categorical properties such as its relations with simplicial objects, Lie algebras and groups.

2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions

Hopf Algebraic Conventions

Let H be a Hopf algebra. We will use Sweedler’s notation to denote the coproduct in H: ∆(x) =

(x)

x′ ⊗ x′′ where x ∈ H And therefore a Hopf algebra H is cocommutative if, for each x ∈ H we have:

• (x)

x′ ⊗ x′′ =

(x)

x′′ ⊗ x′ The identity of a Hopf algebra is 1H and the counit map is ǫ: H → . The identity map is µ: λ ∈ → λ1H ∈ H.

2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions

Hopf Algebraic Conventions

Let H be a Hopf algebra. We will use Sweedler’s notation to denote the coproduct in H: ∆(x) =

(x)

x′ ⊗ x′′ where x ∈ H And therefore a Hopf algebra H is cocommutative if, for each x ∈ H we have:

• (x)

x′ ⊗ x′′ =

(x)

x′′ ⊗ x′ The identity of a Hopf algebra is 1H and the counit map is ǫ: H → . The identity map is µ: λ ∈ → λ1H ∈ H. Definition 1 If I is a bialgebra, we say that I is an H-module algebra if there exist a left action ρ: H ⊗ I → I, denoted by x ⊗ v ∈ H ⊗ I → x ◮ρ v ∈ I, such that, for each x ∈ H and u, v ∈ I. 1H ◮ρ v = v and x ◮ρ 1I = ǫ(x)1I x ◮ρ (uv) =

(x)

(x′ ◮ρ u)(x′′ ◮ρ v)

2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions

Hopf Algebraic Conventions

Let H be a Hopf algebra. We will use Sweedler’s notation to denote the coproduct in H: ∆(x) =

(x)

x′ ⊗ x′′ where x ∈ H And therefore a Hopf algebra H is cocommutative if, for each x ∈ H we have:

• (x)

x′ ⊗ x′′ =

(x)

x′′ ⊗ x′ The identity of a Hopf algebra is 1H and the counit map is ǫ: H → . The identity map is µ: λ ∈ → λ1H ∈ H. Definition 1 If I is a bialgebra, we say that I is an H-module algebra if there exist a left action ρ: H ⊗ I → I, denoted by x ⊗ v ∈ H ⊗ I → x ◮ρ v ∈ I, such that, for each x ∈ H and u, v ∈ I. 1H ◮ρ v = v and x ◮ρ 1I = ǫ(x)1I x ◮ρ (uv) =

(x)

(x′ ◮ρ u)(x′′ ◮ρ v) A left action of H on I is said to make I an H-module coalgebra if, for all x ∈ H and v ∈ I, we have: ∆(x ◮ρ v) =

(x)

• (v)

(x′ ◮ρ v′) ⊗ (x′′ ◮ρ v′′) ǫ(x ◮ρ v) = ǫ(x) ǫ(v)

2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions

Hopf Algebraic Conventions

Definition 2 (Majid, [5]) Let I, H be cocommutative Hopf algebras, where I is an H module algebra and coalgebra, under the the action ρ: H ⊗ I → I. We have (!) a bialgebra , moreover a Hopf algebra I ⊗ρ H, called the “smash product of H and I, with underlying vector space I ⊗ H, identity 1I ⊗ 1H and (for all u, v ∈ I and x, y ∈ H): (u ⊗ρ x)(v ⊗ρ y) =

(x)

• u (x′ ◮ρ v)

⊗ρ x′′y ∆(u ⊗ρ x) =

(u)

• (x)

(u′ ⊗ρ x′) ⊗ρ (u′′ ⊗ρ x′′) S(u ⊗ρ x) = 1I ⊗ρ S(x) S(u) ⊗ρ 1H

2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions

Hopf Algebraic Conventions

Definition 2 (Majid, [5]) Let I, H be cocommutative Hopf algebras, where I is an H module algebra and coalgebra, under the the action ρ: H ⊗ I → I. We have (!) a bialgebra , moreover a Hopf algebra I ⊗ρ H, called the “smash product of H and I, with underlying vector space I ⊗ H, identity 1I ⊗ 1H and (for all u, v ∈ I and x, y ∈ H): (u ⊗ρ x)(v ⊗ρ y) =

(x)

• u (x′ ◮ρ v)

⊗ρ x′′y ∆(u ⊗ρ x) =

(u)

• (x)

(u′ ⊗ρ x′) ⊗ρ (u′′ ⊗ρ x′′) S(u ⊗ρ x) = 1I ⊗ρ S(x) S(u) ⊗ρ 1H

• Example 3

If H is a cocommutative Hopf algebra (normally only in that case (!)), then H itself has a natural H-module algebra and coalgebra structure, which is given by the “adjoint action" (where x, y ∈ H): ρ(x ⊗ y) =

• (x)

x′yS(x′′) . = x ad y

2-Crossed Modules of Hopf Algebras Preliminaries Crossed Module of Hopf Algebras

Crossed Modules of Cocommutative Hopf Algebras

Definition 4 (Majid) A crossed module of (cocommutative) Hopf algebras [4, 5] is given by a Hopf algebra map: ∂ : I → H where I is an H-module bialgebra such that satisfying (for all x ∈ H and u, v ∈ I): ∂(x ◮ρ v) = x ad ∂(v) ∂(u) ◮ρ v = u ad v.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Cocommutative Hopf Algebra

Simplicial Cocommutative Hopf Algebra

A simplicial (cocommutative) Hopf algebra is a collection of (cocommutative) Hopf algebras Hn (n ∈ ) together with Hopf algebra maps called faces and degeneracies: dn

i :

Hn → Hn−1 , 0 ≤ i ≤ n sn+1

j

: Hn → Hn+1 , 0 ≤ j ≤ n which are to satisfy the following simplicial identities: (i) didj = dj−1di if i < j (ii) sisj = sj+1si if i ≤ j (iii) disj = sj−1di if i < j djsj = dj+1sj = id disj = sjdi−1 if i > j + 1 Any simplicial cocommutative Hopf algebra can be pictured as:

• =

H3

H2

d2

• d1
• d0
• H1

d1

• d0
• s0
• s1
• H0

s0

• We will denote the category of simplicial cocommutative Hopf algebras by Simp.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Cocommutative Hopf Algebra

Simplicial Cocommutative Hopf Algebra

A simplicial (cocommutative) Hopf algebra is a collection of (cocommutative) Hopf algebras Hn (n ∈ ) together with Hopf algebra maps called faces and degeneracies: dn

i :

Hn → Hn−1 , 0 ≤ i ≤ n sn+1

j

: Hn → Hn+1 , 0 ≤ j ≤ n which are to satisfy the following simplicial identities: (i) didj = dj−1di if i < j (ii) sisj = sj+1si if i ≤ j (iii) disj = sj−1di if i < j djsj = dj+1sj = id disj = sjdi−1 if i > j + 1 Any simplicial cocommutative Hopf algebra can be pictured as:

• =

H3

H2

d2

• d1
• d0
• H1

d1

• d0
• s0
• s1
• H0

s0

• We will denote the category of simplicial cocommutative Hopf algebras by Simp.

An n-truncated simplicial (cocommutative) Hopf algebra T rn = {Hn, . . . , H1, H0} is a simplicial cocommutative Hopf algebra obtained by forgetting any dimensions of greater than n in . We denote category of n-truncated simplicial Hopf algebras by TrnSimp.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

Hopf kernels appear in [1]. Let f, g : A → B be two Hopf algebra maps. Define subalgebras: LEqual(f, g) = {x ∈ A | (f ⊗ id)∆(x) = (g ⊗ id)∆(x)} REqual(f, g) = {x ∈ A | (id ⊗ f)∆(x) = (id ⊗ g)∆(x)} which implies f(x) = g(x), the property of equalizer of group maps.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

Hopf kernels appear in [1]. Let f, g : A → B be two Hopf algebra maps. Define subalgebras: LEqual(f, g) = {x ∈ A | (f ⊗ id)∆(x) = (g ⊗ id)∆(x)} REqual(f, g) = {x ∈ A | (id ⊗ f)∆(x) = (id ⊗ g)∆(x)} which implies f(x) = g(x), the property of equalizer of group maps.

!

△The problem is:

∆ LEqual(f, g) ⊆ LEqual(f, g) ⊗ A and ∆ REqual(f, g) ⊆ A ⊗ REqual(f, g) and S LEqual(f, g) = REqual(f, g) and S REqual(f, g) = LEqual(f, g) So they don’t form a (sub) Hopf algebra structure.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

Hopf kernels appear in [1]. Let f, g : A → B be two Hopf algebra maps. Define subalgebras: LEqual(f, g) = {x ∈ A | (f ⊗ id)∆(x) = (g ⊗ id)∆(x)} REqual(f, g) = {x ∈ A | (id ⊗ f)∆(x) = (id ⊗ g)∆(x)} which implies f(x) = g(x), the property of equalizer of group maps.

!

△The problem is:

∆ LEqual(f, g) ⊆ LEqual(f, g) ⊗ A and ∆ REqual(f, g) ⊆ A ⊗ REqual(f, g) and S LEqual(f, g) = REqual(f, g) and S REqual(f, g) = LEqual(f, g) So they don’t form a (sub) Hopf algebra structure. At this point: HEqual(f, g) = {x ∈ A | (id ⊗ f ⊗ id)(∆ ⊗ id)∆(x) = (id ⊗ g ⊗ id)(∆ ⊗ id)∆(x)} defines a sub Hopf algebra of A, moreover the equalizer object in the category of Hopf algebras.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

In the cocommutative case: LEqual(f, g) = HEqual(f, g) = REqual(f, g) and we follow REqual(f, g) in the rest.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

In the cocommutative case: LEqual(f, g) = HEqual(f, g) = REqual(f, g) and we follow REqual(f, g) in the rest. Since the base field .1H (or shortly ) is the zero object of the category of Hopf algebras, zero morphism is µ ◦ ǫ: A → B. Therefore the kernel of any Hopf algebra map f : A → B is. HKer(f) = HEqual(f, µ ◦ ǫ)

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

In the cocommutative case: LEqual(f, g) = HEqual(f, g) = REqual(f, g) and we follow REqual(f, g) in the rest. Since the base field .1H (or shortly ) is the zero object of the category of Hopf algebras, zero morphism is µ ◦ ǫ: A → B. Therefore the kernel of any Hopf algebra map f : A → B is. HKer(f) = HEqual(f, µ ◦ ǫ) Example 5 Given the identical map id: H → H we have: RKer(id) = .1H.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Hopf Kernels

In the cocommutative case: LEqual(f, g) = HEqual(f, g) = REqual(f, g) and we follow REqual(f, g) in the rest. Since the base field .1H (or shortly ) is the zero object of the category of Hopf algebras, zero morphism is µ ◦ ǫ: A → B. Therefore the kernel of any Hopf algebra map f : A → B is. HKer(f) = HEqual(f, µ ◦ ǫ) Example 5 Given the identical map id: H → H we have: RKer(id) = .1H. Lemma 6 If a cocommutative Hopf algebra map: f : H → I is an injective linear map, then RKer(f) = .1H.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Now we can use Hopf kernels for our further constructions!

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

The Moore Complex

Definition 7 (Moore Complex) For a simplicial cocommutative Hopf algebra , the Moore complex (N , ∂) is the chain complex of cocommutative Hopf algebras defined by: NHn =

n−1

∩

i=0

RKer(di) with the morphisms ∂n : NHn → NHn−1 induced from dn by restriction.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

The Moore Complex

Definition 7 (Moore Complex) For a simplicial cocommutative Hopf algebra , the Moore complex (N , ∂) is the chain complex of cocommutative Hopf algebras defined by: NHn =

n−1

∩

i=0

RKer(di) with the morphisms ∂n : NHn → NHn−1 induced from dn by restriction. Definition 8 (Length of the Moore Complex) We call a Moore Complex with length n, iff NHi is equal to zero object, namely .1H, for each i > n. We denote the category of simplicial objects with Moore Complex length n by Simp≤n.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

Applications of the Moore complex?

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex

• 1. Simplicial Decomposition

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Decomposition

Simplicial Decomposition

Lemma 9 For any simplicial cocommutative Hopf algebra there exists a left action (for all i ≤ n − 1): si(Hn−1) ⊗ RKer di

• −

→ RKer di

• (a, x)

− → a ◮ρi x . = a ad x where di : Hn → Hn−1. Thus we have smash product Hopf algebra: RKer(di) ⊗ρi si(Hn−1).

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Decomposition

Simplicial Decomposition

Lemma 9 For any simplicial cocommutative Hopf algebra there exists a left action (for all i ≤ n − 1): si(Hn−1) ⊗ RKer di

• −

→ RKer di

• (a, x)

− → a ◮ρi x . = a ad x where di : Hn → Hn−1. Thus we have smash product Hopf algebra: RKer(di) ⊗ρi si(Hn−1). Theorem 10 In , we have (for all n ∈ and i ≤ n − 1): Hn∼ =RKer(di) ⊗ρi si(Hn−1)

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Decomposition

Simplicial Decomposition

Lemma 9 For any simplicial cocommutative Hopf algebra there exists a left action (for all i ≤ n − 1): si(Hn−1) ⊗ RKer di

• −

→ RKer di

• (a, x)

− → a ◮ρi x . = a ad x where di : Hn → Hn−1. Thus we have smash product Hopf algebra: RKer(di) ⊗ρi si(Hn−1). Theorem 10 In , we have (for all n ∈ and i ≤ n − 1): Hn∼ =RKer(di) ⊗ρi si(Hn−1) Proof. The map: φ : Hn − → RKer(di) ⊗ρi si(Hn−1) x − →

• (x)

x′sidi(S(x′′)) ⊗ sidi(x′′′) defines the required Hopf algebra isomorphism.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Decomposition

Simplicial Decomposition

Theorem 11 Let be a simplicial cocommutative Hopf algebra. We can decompose Hn for any n ≥ 0 as: Hn ∼ = · · · NHn⊗sn−1NHn−1

• ⊗ · · · sn−2 · · · s1NH1
• ⊗
• · · ·

s0NHn−1 ⊗ s1s0NHn−2

• ⊗ · · · ⊗ sn−1sn−2 · · · s0NH0
• For istance:

H1 ∼ = NH1 ⊗ρ0 s0(NH0) and H2 ∼ = NH2 ⊗ρ1 s1(NH1) ⊗ρ0

• s0(NH1) ⊗ρ0 s1s0(NH0)

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Decomposition

• 2. Iterated Peiffer Pairings

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Iterated Peiffer Pairings

Iterated Peiffer Pairings

In this section we give a method to obtain for any dimensional Peiffer pairings in a simplicial Hopf algebra by iteration. We use an analogous technique which introduced in [2] and examined in [6] for the case of groups. Let be a simplicial cocommutative Hopf algebra and N be the Moore complex of it. We take the elements (α, β) from S(n) (the poset of surjective maps, [6]) with α ∩ β = ∅ and β < α, with respect to lexicographic ordering in S(n) where α = (il, · · · , i1), β = (jm, · · · , j1) ∈ S(n). The pairings we need Fα,β are defined as composites in the diagram: NHn−#α ⊗ NHn−#β

Fα,β sα⊗sβ

• NHn

Hn ⊗ Hn

ad

Hn

pn

• where:

sα = sil . . . si1 : NHn−#α → Hn, sβ = sjm . . . sj1 : NHn−#β → Hn, and p: Hn → NHn is defined by the composition: p(x) = pn−1 . . . p0(x)

• f the kernel generator maps:

pi : x − →

• (x)

x′ sidiS(x′′) ∈ RKer(di)

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Iterated Peiffer Pairings

Calculating for n = 2

Suppose α = (0), β = (1), x, y ∈ NH1 and calculate F(0)(1)(x, y) as follows: F(0)(1)(x, y) = p1p0 [(s0(x) ad s1(y)] = p1

 

• (s0(x)ads1(x))

[s0(x) ad s1(y)]′ s0d0(S(s0(x) ad s1(y))′′)

 

= p1

(x)(y)

• s0(x′) ad s1(y′)

S s0(x′′) ad s0s0d0(y′′)

• = p1
• (x)
• s0(x′) ad s1(y)

S s0(x′′) ad 1

• ∵ (??)

= p1

• s0(x) ad s1(y)

=

• (x)(y)
• s0(x′) ad s1(y′)

s1d1

• S

s0(x′′) ad s1(y′′) =

• (x)(y)
• s0(x′) ad s1(y′)

S s1(x′′) ad s1(y′′) belongs to NH2.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Iterated Peiffer Pairings

n = 3 Case

The possible peiffer elements of N3 are: a) for all x ∈ NH1 and y ∈ NH2, F(1,0)(2)(x, y) =

(x)(y)

[s1s0(x′) ad s2(y′)] S [s2s0(x′′) ad s2(y′′)] F(2,0)(1)(x, y) =

• (x)(y)

(s2s0(x′) ad s1(y′)) S (s2s1(x′′) ad s1(y′′)) S [(s2s0(x′′′) ad s2(y′′′)) S (s2s1(x′′′′) ad s2(y′′′′))] b) for all x ∈ NH2 and y ∈ NH1, F(0)(2,1)(x, y) =

(x)(y)

(s0(x′) ad s2s1(y′)) S (s1(x′′) ad s2s1(y′′)) S [s2s1(y′′′)S (s2(x′′′) ad s2s1(y′′′′))] c) for all x, y ∈ NH2, F(0)(1)(x, y) =

(x)(y)

(s0(x′) ad s1(y′)) S (s1(x′′) ad s1(y′′)) S [s2(y′′′)S (s2(x′′′) ad s2(y′′′′))] F(0)(2)(x, y) =

(y)

(s0(x) ad s2(y′)) S(s2(y′′)) F(1)(2)(x, y) =

(x)(y)

[s1(x′) ad s2(y′)] S [s2(x′′) ad s2(y′′)] Remark that Fα,β ∈ NHn implies that S(Fα,β) ∈ NHn.

2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Iterated Peiffer Pairings

What to do, with these Peiffer pairings?

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: Simp≤1 → XHopf

Theorem 12 Let be a simplicial cocommutative Hopf algebra with Moore complex of length 1. We have a crossed module of cocommutative Hopf algebras: ∂1 : NH1 → H0

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: Simp≤1 → XHopf

Theorem 12 Let be a simplicial cocommutative Hopf algebra with Moore complex of length 1. We have a crossed module of cocommutative Hopf algebras: ∂1 : NH1 → H0 with the left action: x ◮ρ y = s0(x) ad y for all x ∈ H0, y ∈ NH1, where ∂1 is the restriction of d1.

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: Simp≤1 → XHopf

Theorem 12 Let be a simplicial cocommutative Hopf algebra with Moore complex of length 1. We have a crossed module of cocommutative Hopf algebras: ∂1 : NH1 → H0 with the left action: x ◮ρ y = s0(x) ad y for all x ∈ H0, y ∈ NH1, where ∂1 is the restriction of d1. Proof. The first crossed module condition is clear. Also since the Moore complex of is with length 1, we have: d2

• F(0)(1)(x, y)

= ǫ(x)ǫ(y)1H which proves the latter.

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

Now, other way round!?

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The coskeleton functor

Remark 13 We know that the category of Hopf algebras has both limits and colimits [7].

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The coskeleton functor

Remark 13 We know that the category of Hopf algebras has both limits and colimits [7]. It means that we can use the construction of the (co)skeleton functor introduced in [3]; to obtain a simplicial Hopf algebra from any k-truncated simplicial Hopf algebra!

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The coskeleton functor

Remark 13 We know that the category of Hopf algebras has both limits and colimits [7]. It means that we can use the construction of the (co)skeleton functor introduced in [3]; to obtain a simplicial Hopf algebra from any k-truncated simplicial Hopf algebra! By using this idea, we can give the following theorem: Theorem 14 Suppose any truncation of a simplicial cocommutative Hopf algebra . The Moore complex of its coskeleton cosk(Trk( )) has the following properties: N cosk(Trk( ))

r

= for r > k + 1, (namely, the Moore complex of cosk(Trk( )) is with length k + 1) N cosk(Trk( ))

k+1

∼ = RKer ∂k : N( )K → N( )k−1

• ,

∂k+1 is injective.

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: XHopf → Simp≤1

The Functor: XHopf → Simp≤1. Let ∂ : H → I be a crossed module of cocommutative Hopf algebras (with the action ◮). We can define a 1-truncated simplicial cocommutative Hopf algebra: H ⊗◮ I

d1

• d0

I

s0

• with Hopf algebra maps:

d0(h ⊗ x) = ǫ(h)x, d1(h ⊗ x) = ∂(h)x, s0(x) = (1 ⊗ x)

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: XHopf → Simp≤1

The Functor: XHopf → Simp≤1. Let ∂ : H → I be a crossed module of cocommutative Hopf algebras (with the action ◮). We can define a 1-truncated simplicial cocommutative Hopf algebra: H ⊗◮ I

d1

• d0

I

s0

• with Hopf algebra maps:

d0(h ⊗ x) = ǫ(h)x, d1(h ⊗ x) = ∂(h)x, s0(x) = (1 ⊗ x) There exists an action of H ⊗ I on H with (for h, k ∈ H and x ∈ I) with: (h ⊗ x) ◮⋆ k = (∂(h)x) ◮ρ k

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: XHopf → Simp≤1

The Functor: XHopf → Simp≤1. Let ∂ : H → I be a crossed module of cocommutative Hopf algebras (with the action ◮). We can define a 1-truncated simplicial cocommutative Hopf algebra: H ⊗◮ I

d1

• d0

I

s0

• with Hopf algebra maps:

d0(h ⊗ x) = ǫ(h)x, d1(h ⊗ x) = ∂(h)x, s0(x) = (1 ⊗ x) There exists an action of H ⊗ I on H with (for h, k ∈ H and x ∈ I) with: (h ⊗ x) ◮⋆ k = (∂(h)x) ◮ρ k which lead us to define 2-truncated simplicial cocommutative Hopf algebra: H ⊗◮⋆ (H ⊗◮ I)

d2

• d1
• d0

H ⊗◮ I

d1

• d0
• s0
• s1
• I

s0

• with the Hopf algebra maps defined by:

d0(h ⊗ k ⊗ x) = ǫ(h)k ⊗ x, d1(h ⊗ k ⊗ x) = hk ⊗ x, d2(h ⊗ k ⊗ x) = h ⊗ ∂(k)x and s0(h ⊗ x) = (1 ⊗ h ⊗ x), s1(h ⊗ x) = (h ⊗ 1 ⊗ x)

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: XHopf → Simp≤1

The Functor: XHopf → Simp≤1. Let ∂ : H → I be a crossed module of cocommutative Hopf algebras (with the action ◮). We can define a 1-truncated simplicial cocommutative Hopf algebra: H ⊗◮ I

d1

• d0

I

s0

• with Hopf algebra maps:

d0(h ⊗ x) = ǫ(h)x, d1(h ⊗ x) = ∂(h)x, s0(x) = (1 ⊗ x) There exists an action of H ⊗ I on H with (for h, k ∈ H and x ∈ I) with: (h ⊗ x) ◮⋆ k = (∂(h)x) ◮ρ k which lead us to define 2-truncated simplicial cocommutative Hopf algebra: H ⊗◮⋆ (H ⊗◮ I)

d2

• d1
• d0

H ⊗◮ I

d1

• d0
• s0
• s1
• I

s0

• with the Hopf algebra maps defined by:

d0(h ⊗ k ⊗ x) = ǫ(h)k ⊗ x, d1(h ⊗ k ⊗ x) = hk ⊗ x, d2(h ⊗ k ⊗ x) = h ⊗ ∂(k)x and s0(h ⊗ x) = (1 ⊗ h ⊗ x), s1(h ⊗ x) = (h ⊗ 1 ⊗ x) Consider the image under the cos2 functor. The Moore complex of this simplicial cocommutative Hopf algebra supposed to be 3 normally. But we already checked that NH2 will be trivial, which implies NH3 as well. Therefore the length of the Moore complex is actually length one.

2-Crossed Modules of Hopf Algebras More on Crossed Modules The Functor: Simp≤1 → XHopf

The Functor: XHopf → Simp≤1

The above constructions yield the following functors: Theorem 15 The category of crossed modules of cocommutative Hopf algebras is equivalent to the category of Simplicial cocommutative Hopf algebras with Moore complex length 1. This can be pictured as: Tr2Simp

cos2

• Simp≤1

X1

XHopf

( )2

2-Crossed Modules of Hopf Algebras 2-Crossed Modules

2-Crossed Module

Definition 16 A 2-crossed module of cocommutative Hopf algebras is given by a chain complex: K

∂2

− → H

∂1

− → I

• f Hopf algebras with left actions of I on H and K and also on itself by adjoint action together with an I-equivariant

bilinear function (called the Peiffer lifting): { ⊗ } : H ⊗I H − → K, satisfying the following axioms, for all x, y, z ∈ H and k, l ∈ K: 1) K

∂2

− → H

∂1

− → I is a complex of I-modules and comodules, 2) ∂2{x ⊗ y} =

(x)(y)

(x′ ad y′) ∂1(x′′) ρ S(y′′), 3) {∂2(k) ⊗ ∂2(k)} =

(l)

(k ad l′) S(l′′), 4) {x ⊗ yz} =

(x)(y)

{x′ ⊗ y′} (∂1(x′′) ad y′′) ′

ρ {x′′′ ⊗ z},

5) {xy ⊗ z} =

• (x)(y)(z)

{x′, y′ ad z′} ∂1(x′′) ρ {y′′, z′′} , 6) {∂2(k′) ⊗ x′}{x′′ ⊗ ∂2(k′′)} =

(k)

k′ ∂1(x) ρ S(k′′) ,

2-Crossed Modules of Hopf Algebras 2-Crossed Modules

2-Crossed Module

Remark 17 Here we put the action: x ′

ρ k .

=

• (k)

k′ {∂2(S(k′′)) ⊗ x} in the condition (4), makes ∂2 a crossed module. However ∂1 is just a precrossed module. Any 2-crossed module of cocommutative Hopf algebras will be denoted by (K, H, I, ∂1, ∂2). We denote the category of 2-crossed modules of cocommutative Hopf algebras by X2Hopf which the morphisms of this category can be defined in a natural way.

2-Crossed Modules of Hopf Algebras 2-Crossed Modules

2-Crossed Module

Theorem 18 The functor Prim: Hopf → Lie preserves 2-crossed modules.

2-Crossed Modules of Hopf Algebras 2-Crossed Modules

2-Crossed Module

Theorem 18 The functor Prim: Hopf → Lie preserves 2-crossed modules. Theorem 19 The functor ( )gl : Hopf → Grp preserves 2-crossed modules.

2-Crossed Modules of Hopf Algebras 2-Crossed Modules

More relations?

2-Crossed Modules of Hopf Algebras Functorial Relations for 2-Crossed Modules The Functor Simp≤2 → X2 Hopf

The Functor Simp≤2 → X2Hopf

Theorem 20 Let be a simplicial cocommutative Hopf algebra with Moore complex of length 2. Take: K . = NL2

∂2

− → H . = NL1

∂1

− → I . = L0 If we define the action of I on H with s0 (n) ad m and also of I on K with s1s0 (n) ad l with the Peiffer lifting { , }: H × H → K such that: {x, y} =

• (x)(y)
• s1(x′) ad s1(y′)

S s0(x′′) ad s1(y′′) then we have a 2-crossed module (K, H, I, ∂1, ∂2) where ∂i is the restriction of di. Thus we have the functor: X2 : Simp≤2 → X2Hopf.

2-Crossed Modules of Hopf Algebras Functorial Relations for 2-Crossed Modules The Functor Simp≤2 → X2 Hopf

The Functor Simp≤2 → X2Hopf

Theorem 20 Let be a simplicial cocommutative Hopf algebra with Moore complex of length 2. Take: K . = NL2

∂2

− → H . = NL1

∂1

− → I . = L0 If we define the action of I on H with s0 (n) ad m and also of I on K with s1s0 (n) ad l with the Peiffer lifting { , }: H × H → K such that: {x, y} =

• (x)(y)
• s1(x′) ad s1(y′)

S s0(x′′) ad s1(y′′) then we have a 2-crossed module (K, H, I, ∂1, ∂2) where ∂i is the restriction of di. Thus we have the functor: X2 : Simp≤2 → X2Hopf. Proof. Length of the Moore complex + 6 “three dimensional Peiffer pairings" prove 6 conditions required!

2-Crossed Modules of Hopf Algebras Functorial Relations for 2-Crossed Modules The Functor X2 Hopf → Simp≤2

The Functor: X2Hopf → Simp≤2

By using the similar construction to case of crossed modules, we will have the following: Theorem 21 The category of 2-crossed modules of cocommutative Hopf algebras is equivalent to the category of Simplicial cocommutative Hopf algebras with Moore complex length 2. This can be pictured as: Tr3Simp

cos3

• Simp≤2

X2

X2Hopf

( )3

2-Crossed Modules of Hopf Algebras Functorial Relations for 2-Crossed Modules The Functor X2 Hopf → Simp≤2

2-Crossed Modules of Hopf Algebras Functorial Relations for 2-Crossed Modules The Functor X2 Hopf → Simp≤2

References I

[1]

• N. Andruskiewitsch and J. Devoto.

Extensions of Hopf algebras.

• St. Petersbg. Math. J., 7(1):22–61, 1995.

[2]

• P. Carrasco and A.M. Cegarra.

Group-theoretic algebraic models for homotopy types. Journal of Pure and Applied Algebra, 75(3):195 – 235, 1991. [3]

• J. Duskin.

Simplicial methods and the interpretation of ”triple” cohomology.

• Mem. Am. Math. Soc., 163:135, 1975.

[4]

• J. Faria Martins.

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy.

• J. Geom. Phys., 99:68–110, 2016.

[5] Shahn Majid. Strict quantum 2-groups. 2012. [6]

• A. Mutlu and T. Porter.

Iterated Peiffer pairings in the Moore complex of a simplicial group.

• Appl. Categ. Struct., 9(2):111–130, 2001.

[7] Hans-E. Porst. Limits and colimits of Hopf algebras.

• J. Algebra, 328(1):254–267, 2011.