Shuffle algebra perspective on operator valued probability theory - - PowerPoint PPT Presentation

1/25

Shuffle algebra perspective on operator valued probability theory

30 mars 2020

2/25

Operator valued probability theory

3/25

Operator valued probability theory

Definition (Operator valued space)

An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆,

3/25

Operator valued probability theory

Definition (Operator valued space)

An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆, A ⋆-algeba (A, ⋆), which is a B-B bimodule over B : b1 · (a · b2) = (b1 · a) · b2, (a1 · b)a2 = a1(b · a2).

3/25

Operator valued probability theory

Definition (Operator valued space)

An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆, A ⋆-algeba (A, ⋆), which is a B-B bimodule over B : b1 · (a · b2) = (b1 · a) · b2, (a1 · b)a2 = a1(b · a2). A positive B-B module map E : A → B : E(b1ab2) = b1E(a)b2, E(aa⋆) ∈ BB⋆.

3/25

Operator valued probability theory

Definition (Operator valued space)

An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆, A ⋆-algeba (A, ⋆), which is a B-B bimodule over B : b1 · (a · b2) = (b1 · a) · b2, (a1 · b)a2 = a1(b · a2). A positive B-B module map E : A → B : E(b1ab2) = b1E(a)b2, E(aa⋆) ∈ BB⋆.

Speicher, R. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory.

3/25

Operator valued probability theory

Definition (Operator valued space)

An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆, A ⋆-algeba (A, ⋆), which is a B-B bimodule over B : b1 · (a · b2) = (b1 · a) · b2, (a1 · b)a2 = a1(b · a2). A positive B-B module map E : A → B : E(b1ab2) = b1E(a)b2, E(aa⋆) ∈ BB⋆.

Speicher, R. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Speicher, R. Operator-valued free probability and block random matrices.

4/25

Operator valued probability theory

Definition (Distribution of random variables)

Let a1, . . . , an ∈ A. The distribution of a1, . . . , an is the collection of elements in B defined by : E(b1a1b2 · · · anbn+1), b1, . . . , bn+1 ∈ B.

4/25

Operator valued probability theory

Definition (Distribution of random variables)

Let a1, . . . , an ∈ A. The distribution of a1, . . . , an is the collection of elements in B defined by : E(b1a1b2 · · · anbn+1), b1, . . . , bn+1 ∈ B.

Definition (Free Multiplicative extension on NC.)

Eπ(b1, . . . , b10) = E(b1ab2aE(b3aE(b4ab5aE(b6ab7))ab8)ab9ab10)

5/25

Operator valued probability theory

Definition (Boolean multiplicative extension)

Let IP be the poset of interval partitions, and write I = I1 · · · Ip for I = {I1, . . . , Ip} ∈ IP. EI(b1, . . . , b|I|) =

• i∈1,...,p

E(b···+Ij−1+1 · · · b···+Ij)

5/25

Operator valued probability theory

Definition (Boolean multiplicative extension)

Let IP be the poset of interval partitions, and write I = I1 · · · Ip for I = {I1, . . . , Ip} ∈ IP. EI(b1, . . . , b|I|) =

• i∈1,...,p

E(b···+Ij−1+1 · · · b···+Ij)

Definition (Boolean and Free cumulants)

E(b1, . . . , bn+1) =

• π∈NC(n)

κπ(b1, . . . , bn+1) =

• β∈IP(n)

βπ(b1, . . . , bn+1).

5/25

Operator valued probability theory

Definition (Boolean multiplicative extension)

Let IP be the poset of interval partitions, and write I = I1 · · · Ip for I = {I1, . . . , Ip} ∈ IP. EI(b1, . . . , b|I|) =

• i∈1,...,p

E(b···+Ij−1+1 · · · b···+Ij)

Definition (Boolean and Free cumulants)

E(b1, . . . , bn+1) =

• π∈NC(n)

κπ(b1, . . . , bn+1) =

• β∈IP(n)

βπ(b1, . . . , bn+1). Free and Boolean cumulants linearize Free and Boolean operator valued independance.

6/25

Shuffle approach to scalar probability theory

7/25

Double bar construction

H = ¯ T(T(A)). ∅, a1 · · · an, a1

1 · · · a1 n1 | a2 1 · · · a2 m1

∆✁(·) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆(·) = ∅ ⊗ · + · ⊗ ∅ + ∆≺(·) + ∆≻(·). ✁ ✁

7/25

Double bar construction

H = ¯ T(T(A)). ∅, a1 · · · an, a1

1 · · · a1 n1 | a2 1 · · · a2 m1

∆✁(·) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆(·) = ∅ ⊗ · + · ⊗ ∅ + ∆≺(·) + ∆≻(·).

Proposition

HomVectC (H, ✁) is a monoid and G = HomAlg(H, ✁) is a group.

7/25

Double bar construction

H = ¯ T(T(A)). ∅, a1 · · · an, a1

1 · · · a1 n1 | a2 1 · · · a2 m1

∆✁(·) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆(·) = ∅ ⊗ · + · ⊗ ∅ + ∆≺(·) + ∆≻(·).

Proposition

HomVectC (H, ✁) is a monoid and G = HomAlg(H, ✁) is a group. exp≺(k) = 1⋆ +

• n≥1

k≺n, exp≻(k) = 1⋆ +

• n≥1

k≻n

7/25

Double bar construction

H = ¯ T(T(A)). ∅, a1 · · · an, a1

1 · · · a1 n1 | a2 1 · · · a2 m1

∆✁(·) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆(·) = ∅ ⊗ · + · ⊗ ∅ + ∆≺(·) + ∆≻(·).

Proposition

HomVectC (H, ✁) is a monoid and G = HomAlg(H, ✁) is a group. exp≺(k) = 1⋆ +

• n≥1

k≺n, exp≻(k) = 1⋆ +

• n≥1

k≻n exp≺(k)−1 = exp≻(−k).

8/25

Shuffle and non-commutative probability theory

A ⋆-algebra A and an expectation E : A → C. M ∈ G, M(a1 ⊗ · · · ⊗ an) = E(a1 ·A · · · ·A an) k ∈ Lie(G), k(a1 ⊗ · · · ⊗ an) = κ(a1, . . . , an) b ∈ Lie(G), b(a1, . . . , an) = β(a1 ⊗ · · · ⊗ an)

8/25

Shuffle and non-commutative probability theory

A ⋆-algebra A and an expectation E : A → C. M ∈ G, M(a1 ⊗ · · · ⊗ an) = E(a1 ·A · · · ·A an) k ∈ Lie(G), k(a1 ⊗ · · · ⊗ an) = κ(a1, . . . , an) b ∈ Lie(G), b(a1, . . . , an) = β(a1 ⊗ · · · ⊗ an) M = ε + k ≺ M, M = ε + M ≻ b M = exp≺(k) = exp≻(b)

8/25

Shuffle and non-commutative probability theory

A ⋆-algebra A and an expectation E : A → C. M ∈ G, M(a1 ⊗ · · · ⊗ an) = E(a1 ·A · · · ·A an) k ∈ Lie(G), k(a1 ⊗ · · · ⊗ an) = κ(a1, . . . , an) b ∈ Lie(G), b(a1, . . . , an) = β(a1 ⊗ · · · ⊗ an) M = ε + k ≺ M, M = ε + M ≻ b M = exp≺(k) = exp≻(b) Ebrahimi-Fard, K., Patras, F. Cumulants, free cumulants and half-shuffles.

Ebrahimi-Fard, K., Patras, F. Monotone, free, and boolean cumulants : a shuffle algebra approach.

9/25

Relation between Möbius inversion and Shuffles

Ebrahimi-Fard, K., Foissy, L., Kock, J., Patras, F. Operads of (noncrossing)

partitions, interacting bialgebras, and moment-cumulant relations.

Shuffle Approach = ⇒ Gap insertion operad of non-crossing partitions

Operad NC − → incidence bi-algebra (N, ∆) on words on non-crossing partitions : ∆(π) =

• π=q◦(p1,...,pn)

q ⊗ (p1 ⊗ . . . ⊗ pn) = ∆+

≺(π) + ∆+ ≻(π).

f = (E(an))n≥1 F : NC → C, multiplicative F : N → C, F = εN + f ≺ F.

9/25

Relation between Möbius inversion and Shuffles

Ebrahimi-Fard, K., Foissy, L., Kock, J., Patras, F. Operads of (noncrossing)

partitions, interacting bialgebras, and moment-cumulant relations.

Möbius inversion = ⇒ Block substitution operad

10/25

Shuffle operadic approach to operator valued cumulants and moments

11/25

⊠ Express multiplicativity of {Eπ, π ∈ NC}. Define a decomposition map ∆ that presevers linear order of the "legs" of a non-crossing partition. ⊗ ⊗ ⊗ Give a Lie theoretic perspective, with a group of morphisms and a Lie algebra of infinitesimal morphisms and a fixed point equation for {Eπ, π ∈ NC}.

12/25

⊗

12/25

13/25

Duoidal Category of bigraded collections

n, m ≥ 0, Cn,m ∈ VectC, C C C = (Cn,m)n,m

13/25

Duoidal Category of bigraded collections

n, m ≥ 0, Cn,m ∈ VectC, C C C = (Cn,m)n,m Horizontal product ⊗ ⊗ ⊗ and Vertical product ⊠

13/25

Duoidal Category of bigraded collections

n, m ≥ 0, Cn,m ∈ VectC, C C C = (Cn,m)n,m Horizontal product ⊗ ⊗ ⊗ and Vertical product ⊠ (C⊗ ⊗ ⊗D)n,m =

• nc+nd=n

mc+md=m

Cnc,mc ⊗ Dnd,md, (C⊠D)n,m =

• k

Cn,k ⊗ Dk,m

13/25

Duoidal Category of bigraded collections

n, m ≥ 0, Cn,m ∈ VectC, C C C = (Cn,m)n,m Horizontal product ⊗ ⊗ ⊗ and Vertical product ⊠ (C⊗ ⊗ ⊗D)n,m =

• nc+nd=n

mc+md=m

Cnc,mc ⊗ Dnd,md, (C⊠D)n,m =

• k

Cn,k ⊗ Dk,m ⊗ ⊗ ⊗ ⊠

13/25

Duoidal Category of bigraded collections

n, m ≥ 0, Cn,m ∈ VectC, C C C = (Cn,m)n,m Horizontal product ⊗ ⊗ ⊗ and Vertical product ⊠ (C⊗ ⊗ ⊗D)n,m =

• nc+nd=n

mc+md=m

Cnc,mc ⊗ Dnd,md, (C⊠D)n,m =

• k

Cn,k ⊗ Dk,m (C C C⊠)n,m = δn=mC, (C C C⊗

⊗ ⊗)n,m = δn=m=0C

13/25

Duoidal Category of bigraded collections

n, m ≥ 0, Cn,m ∈ VectC, C C C = (Cn,m)n,m Horizontal product ⊗ ⊗ ⊗ and Vertical product ⊠ (C⊗ ⊗ ⊗D)n,m =

• nc+nd=n

mc+md=m

Cnc,mc ⊗ Dnd,md, (C⊠D)n,m =

• k

Cn,k ⊗ Dk,m Vallette, B. A Koszul duality for props. Transactions of the American

Mathematical Society.

14/25

Lax property

⊠ ⊠ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

14/25

Lax property

15/25

Consequences :

The category Alg⊗

⊗ ⊗ of horizontal algebras endowed with ⊠ is monoidal

with unit C C C⊠. The category CoAlg⊠ of vertical co-algebras endowed with ⊗ ⊗ ⊗ is monoidal with unit C C C⊗

⊗ ⊗.

(A, mA

⊗ ⊗ ⊗), (B, mB ⊗ ⊗ ⊗),

mA⊠B

⊗ ⊗ ⊗

= (mA

⊗ ⊗ ⊗ ⊠ mB ⊗ ⊗ ⊗) ◦ R

(A, ∆⊠), (B, ∆⊠), ∆⊠

A⊗ ⊗ ⊗B = R ◦ (∆⊠ A⊗

⊗ ⊗∆⊠

B )

16/25

⊠⊗ ⊗ ⊗ - bialgebras

Proposition

A bi-graded collection C is a (⊠-co|⊗ ⊗ ⊗-al)gebra if and only if it is a (⊗ ⊗ ⊗-al|⊠-co)gebra. ∆⊠ : C → C⊠C, m⊗

⊗ ⊗ : C⊗

⊗ ⊗C → C, ε : C → C C C⊠.

16/25

⊠⊗ ⊗ ⊗ - bialgebras

Proposition

A bi-graded collection C is a (⊠-co|⊗ ⊗ ⊗-al)gebra if and only if it is a (⊗ ⊗ ⊗-al|⊠-co)gebra. ∆⊠ : C → C⊠C, m⊗

⊗ ⊗ : C⊗

⊗ ⊗C → C, ε : C → C C C⊠. C⊗ ⊗ ⊗C (C⊠C)⊗ ⊗ ⊗(C⊠C) (C⊗ ⊗ ⊗C)⊠(C⊗ ⊗ ⊗C) C C⊠C

m⊗

⊗ ⊗

∆⊗ ⊗ ⊗∆ R m⊗

⊗ ⊗⊠m⊗ ⊗ ⊗

∆

C⊠C C⊠⊠C C C

ε⊠id ∆⊠ id

17/25

⊠⊗ ⊗ ⊗ - Hopf algebras

Definition

An algebra (C, m⊗

⊗ ⊗ : C⊗

⊗ ⊗C → C, ) and maps in Alg⊗

⊗ ⊗ :

∆⊠ : C → C⊠C, ε : C → C C C⊠

17/25

⊠⊗ ⊗ ⊗ - Hopf algebras

Definition

An algebra (C, m⊗

⊗ ⊗ : C⊗

⊗ ⊗C → C, ) and maps in Alg⊗

⊗ ⊗ :

∆⊠ : C → C⊠C, ε : C → C C C⊠ ∇⊠ : C⊠C → C, S : C → C, η : C C C⊠ → C ∇⊠ ◦ (S⊠idC) ◦ ∆⊠ = ε ◦ η, ∇⊠ ◦ (idC⊠S) ◦ ∆⊠ = ε ◦ η

18/25

An unshuffle (⊠-co)(⊗ ⊗ ⊗-al)gebra

Definition

A bigraded collection C with Cn,m = δn=mCn,m.

18/25

An unshuffle (⊠-co)(⊗ ⊗ ⊗-al)gebra

Definition

A bigraded collection C with Cn,m = δn=mCn,m. A (⊠-co)(⊗ ⊗ ⊗-al)gebra ( ¯ C ¯ C ¯ C = C C C ⊕ C C C⊠, ∆⊠, m⊗

⊗ ⊗, ∇⊠)

∆(c) = ¯ ∆(c) + c ⊠ 1m + 1n ⊠ c, ¯ ∆ = ∆⊠

≺ + ∆⊠ ≻,

18/25

An unshuffle (⊠-co)(⊗ ⊗ ⊗-al)gebra

Definition

A bigraded collection C with Cn,m = δn=mCn,m. A (⊠-co)(⊗ ⊗ ⊗-al)gebra ( ¯ C ¯ C ¯ C = C C C ⊕ C C C⊠, ∆⊠, m⊗

⊗ ⊗, ∇⊠)

∆(c) = ¯ ∆(c) + c ⊠ 1m + 1n ⊠ c, ¯ ∆ = ∆⊠

≺ + ∆⊠ ≻,

C C C⊠ C, ∆⊠

≺,≻(C

C C⊠ ) = C C C⊠ ∆⊠

≺,≻

18/25

An unshuffle (⊠-co)(⊗ ⊗ ⊗-al)gebra

Definition

A bigraded collection C with Cn,m = δn=mCn,m. A (⊠-co)(⊗ ⊗ ⊗-al)gebra ( ¯ C ¯ C ¯ C = C C C ⊕ C C C⊠, ∆⊠, m⊗

⊗ ⊗, ∇⊠)

∆(c) = ¯ ∆(c) + c ⊠ 1m + 1n ⊠ c, ¯ ∆ = ∆⊠

≺ + ∆⊠ ≻,

C C C⊠ C, ∆⊠

≺,≻(C

C C⊠ ) = C C C⊠ ∆⊠

≺,≻

(∆⊠

≺,≻ ◦ ρ)(p⊗

⊗ ⊗q) = mC⊠C

⊗ ⊗ ⊗

• (∆⊠

≺,≻⊗

⊗ ⊗∆)(p⊗ ⊗ ⊗q), p ∈ C C C⊠, q ∈ C.

19/25

Monoid of Horizontal morphisms

Let B an algebra.

• Hom(B⊗

⊗ ⊗n, B)

• n≥1 its endomorphisms operad.

19/25

Monoid of Horizontal morphisms

Let B an algebra.

• Hom(B⊗

⊗ ⊗n, B)

• n≥1 its endomorphisms operad.

Hom(B) =

• n≥0

Hom(B⊗

⊗ ⊗n, B), T⊗ ⊗ ⊗(Hom(B)) ⊂

• n,m≥0

Hom(B⊗

⊗ ⊗n, B⊗ ⊗ ⊗m)

19/25

Monoid of Horizontal morphisms

Let B an algebra.

• Hom(B⊗

⊗ ⊗n, B)

• n≥1 its endomorphisms operad.

Hom(B) =

• n≥0

Hom(B⊗

⊗ ⊗n, B), T⊗ ⊗ ⊗(Hom(B)) ⊂

• n,m≥0

Hom(B⊗

⊗ ⊗n, B⊗ ⊗ ⊗m)

α, β ∈ HomAlg⊗

⊗ ⊗(C, T⊗

⊗ ⊗(Hom(B))).

α ⋆ β = T⊗

⊗ ⊗(∇⊠ Hom(B)) ◦ (α ⊠ β) ◦ ∆⊠ ∈ HomAlg⊗

⊗ ⊗

19/25

Monoid of Horizontal morphisms

Let B an algebra.

• Hom(B⊗

⊗ ⊗n, B)

• n≥1 its endomorphisms operad.

Hom(B) =

• n≥0

Hom(B⊗

⊗ ⊗n, B), T⊗ ⊗ ⊗(Hom(B)) ⊂

• n,m≥0

Hom(B⊗

⊗ ⊗n, B⊗ ⊗ ⊗m)

α, β ∈ HomAlg⊗

⊗ ⊗(C, T⊗

⊗ ⊗(Hom(B))).

α ⋆ β = T⊗

⊗ ⊗(∇⊠ Hom(B)) ◦ (α ⊠ β) ◦ ∆⊠ ∈ HomAlg⊗

⊗ ⊗

If C is Hopf, α ∈ HomAlg⊠ ⊂ HomAlg⊗

⊗ ⊗ then α−1 = α ◦ S.

α−1 ∈ HomAlg⊠

19/25

Monoid of Horizontal morphisms

Let B an algebra.

• Hom(B⊗

⊗ ⊗n, B)

• n≥1 its endomorphisms operad.

Hom(B) =

• n≥0

Hom(B⊗

⊗ ⊗n, B), T⊗ ⊗ ⊗(Hom(B)) ⊂

• n,m≥0

Hom(B⊗

⊗ ⊗n, B⊗ ⊗ ⊗m)

α, β ∈ HomAlg⊗

⊗ ⊗(C, T⊗

⊗ ⊗(Hom(B))).

α ⋆ β = T⊗

⊗ ⊗(∇⊠ Hom(B)) ◦ (α ⊠ β) ◦ ∆⊠ ∈ HomAlg⊗

⊗ ⊗

S2 = idC, (α−1)−1 = α−1 ◦ S.

20/25

Example

= + + + +

∆

• ⊠

∅ ⊠ ∅ ∅ ∅ ∅ ∅ ∅ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ∅ ⊠ ⊗ ⊗ ⊗ ∅ ⊠ ∅ ∅ ∅ ∅ ∅ ∅ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊠ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ∅ ⊗ ⊗ ⊗ ∅ ⊠ ∅ ∅ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ + ⊗ ⊗ ⊗ ∅ ∅

21/25

Unshuffling the gap insertion operad

= + ⊠ ∅ ∅ ∅ ∅ ∅ ∅ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

∆≺

• ⊠

∅ + ⊠ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ∅ ⊗ ⊗ ⊗ ∅ + ⊠ ∅ ∅ ∅ ∅ ∅ ∅ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ∅ ⊠ ⊗ ⊗ ⊗ ∅ + ⊠ ∅ ∅ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ∅ ⊗ ⊗ ⊗ ∅

∆≻

• =

22/25

T⊗

⊗ ⊗(NC) is a ⊠⊗

⊗ ⊗ unshuffle Hopf algebra.

The space of words on non-crossing partitions T⊗

⊗ ⊗(NC) is bigraded.

22/25

T⊗

⊗ ⊗(NC) is a ⊠⊗

⊗ ⊗ unshuffle Hopf algebra.

The space of words on non-crossing partitions T⊗

⊗ ⊗(NC) is bigraded.

T⊗

⊗ ⊗(NC) is an horizontal algebra,

22/25

T⊗

⊗ ⊗(NC) is a ⊠⊗

⊗ ⊗ unshuffle Hopf algebra.

The space of words on non-crossing partitions T⊗

⊗ ⊗(NC) is bigraded.

T⊗

⊗ ⊗(NC) is an horizontal algebra,

∇ : T⊗

⊗ ⊗(NC) ⊠ T⊗ ⊗ ⊗(NC) → T⊗ ⊗ ⊗(NC).

22/25

T⊗

⊗ ⊗(NC) is a ⊠⊗

⊗ ⊗ unshuffle Hopf algebra.

The space of words on non-crossing partitions T⊗

⊗ ⊗(NC) is bigraded.

T⊗

⊗ ⊗(NC) is an horizontal algebra,

∇ : T⊗

⊗ ⊗(NC) ⊠ T⊗ ⊗ ⊗(NC) → T⊗ ⊗ ⊗(NC).

Two ⊠⊗ ⊗ ⊗ half unshuffles ∆≺ and ∆≻ : ∆≺,≻ : T⊗

⊗ ⊗(NC) → T⊗ ⊗ ⊗(NC)⊠T⊗ ⊗ ⊗(NC)

22/25

T⊗

⊗ ⊗(NC) is a ⊠⊗

⊗ ⊗ unshuffle Hopf algebra.

The space of words on non-crossing partitions T⊗

⊗ ⊗(NC) is bigraded.

T⊗

⊗ ⊗(NC) is an horizontal algebra,

∇ : T⊗

⊗ ⊗(NC) ⊠ T⊗ ⊗ ⊗(NC) → T⊗ ⊗ ⊗(NC).

Two ⊠⊗ ⊗ ⊗ half unshuffles ∆≺ and ∆≻ : ∆≺,≻ : T⊗

⊗ ⊗(NC) → T⊗ ⊗ ⊗(NC)⊠T⊗ ⊗ ⊗(NC)

An antipode S : T⊗

⊗ ⊗(NC) → T⊗ ⊗ ⊗(NC).

S(π) = (−1)nb(π)δπ∈PI π. η(1m) = ∅m.

23/25

24/25

Half shuffle exponentials

Infinitesimal character

k : T⊗

⊗ ⊗(NC) → T⊗ ⊗ ⊗(Hom(B)),

k(∅pπ∅q) = ∅pk(π)∅q.

24/25

Half shuffle exponentials

Infinitesimal character

k : T⊗

⊗ ⊗(NC) → T⊗ ⊗ ⊗(Hom(B)),

k(∅pπ∅q) = ∅pk(π)∅q.

Proposition (Free morphisms)

K = η ◦ ε + k ≺ K. K(α ◦ (β1, . . . , β|α|)) = K(α) ◦ (K(β1), . . . , K(β|α|)) ⇔ k(π) = δ♯π=1k(π), k(1n) ◦n+1 k(1m) = k(1m) ◦1 k(1n)

24/25

Half shuffle exponentials

Infinitesimal character

k : T⊗

⊗ ⊗(NC) → T⊗ ⊗ ⊗(Hom(B)),

k(∅pπ∅q) = ∅pk(π)∅q.

Proposition (Boolean morphisms)

B = η ◦ ε + B ≻ k B(π) = 0, π ∈ IP. B(I ◦ (I1, . . . , Ip)) = B(I) ◦ B(I1), . . . , B(Ip). ⇔ k(π) = δ♯π=1k(π), k(1n) ◦n+1 k(1m) = k(1m) ◦1 k(1n)

25/25

Takk skal du ha !