Quantum Inverse Semigroups Examples Quantum Groupoids and Inverse Semigroups

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum Inverse Semigroups

Marat Aukhadiev Joint work with A. Buss and T. Timmermann

University of M¨unster

Warsaw, November 2016

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Contents

1 Classical inverse semigroups

2 Quantum inverse semigroups

3 Examples

4 Quantum groupoids and inverse semigroups

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Idempotents in S are of the form xx∗, all commute. S is a group iﬀ it has only one idempotent. There is a natural quotient of S which is a group.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroup

[V. V. Vagner 1952]: A semigroup S is an inverse semigroup if for any x ∈ S there exists a unique x∗ ∈ S such that

xx∗x = x, x∗xx∗ = x∗.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups S is a group iﬀ it has only one idempotent. There is a natural quotient of S which is a group.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroup

[V. V. Vagner 1952]: A semigroup S is an inverse semigroup if for any x ∈ S there exists a unique x∗ ∈ S such that

xx∗x = x, x∗xx∗ = x∗.

Idempotents in S are of the form xx∗, all commute.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups There is a natural quotient of S which is a group.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroup

[V. V. Vagner 1952]: A semigroup S is an inverse semigroup if for any x ∈ S there exists a unique x∗ ∈ S such that

xx∗x = x, x∗xx∗ = x∗.

Idempotents in S are of the form xx∗, all commute. S is a group iﬀ it has only one idempotent.

Inverse semigroup

[V. V. Vagner 1952]: A semigroup S is an inverse semigroup if for any x ∈ S there exists a unique x∗ ∈ S such that

xx∗x = x, x∗xx∗ = x∗.

Idempotents in S are of the form xx∗, all commute. S is a group iﬀ it has only one idempotent. There is a natural quotient of S which is a group.

Examples

Example 1. Let X be a set. A partial bijection of X is a bijection

α: Y → Z such that Y , Z ⊂ X .

The set of partial bijections I(X ) is the symmetric inverse semigroup of X .

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of Münster Quantum Inverse Semigroups n n-dimensional tiling – infinite set of tiles covering R . Pattern – finite subset of a tiling, s.t. the union is connected. Let T tiling, C = {(p2, P, p1): P pattern in T , p1, p2 tiles in P} For x = (p2, P, p1) set d(x) = p1, r(x) = p2. n G all translations of R . S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 2. Tiling semigroup, [J. Kellendonk 1997]. n Tile – connected bounded subset of R , closure of its interior.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Pattern – ﬁnite subset of a tiling, s.t. the union is connected. Let T tiling, C = {(p2, P, p1): P pattern in T , p1, p2 tiles in P} For x = (p2, P, p1) set d(x) = p1, r(x) = p2. n G all translations of R . S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 2. Tiling semigroup, [J. Kellendonk 1997]. n Tile – connected bounded subset of R , closure of its interior. n n-dimensional tiling – inﬁnite set of tiles covering R .

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Let T tiling, C = {(p2, P, p1): P pattern in T , p1, p2 tiles in P} For x = (p2, P, p1) set d(x) = p1, r(x) = p2. n G all translations of R . S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups C = {(p2, P, p1): P pattern in T , p1, p2 tiles in P} For x = (p2, P, p1) set d(x) = p1, r(x) = p2. n G all translations of R . S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups For x = (p2, P, p1) set d(x) = p1, r(x) = p2. n G all translations of R . S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups n G all translations of R . S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 2. Tiling semigroup, [J. Kellendonk 1997]. n Tile – connected bounded subset of R , closure of its interior. n n-dimensional tiling – inﬁnite set of tiles covering R . Pattern – ﬁnite subset of a tiling, s.t. the union is connected. Let T tiling, C = {(p2, P, p1): P pattern in T , p1, p2 tiles in P} For x = (p2, P, p1) set d(x) = p1, r(x) = p2.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups S set of equivalence classes under: x ∼ y if ∃g ∈ G : gx = y.

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups [x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Examples

[x0y 0] if ∃x0 ∼ x, y 0 ∼ y : d(x0) = r(y 0), [x] · [y] = 0 otherwise

Here (p2, P, p1)(p1, Q, q1) = (p2, P ∪ Q, q1).

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups S(G) is a semigroup generated by elements tg for g ∈ G satisfying:

tg −1 tg th = tg −1 tgh (1)

tg thth−1 = tghth−1 (2)

tg t1 = tg (3) ∗ S(G) is an inverse semigroup with unit t1 and involution tg = tg −1 . [R. Exel, V. Vieira 1998] Partial actions (partial representations) of G are in one-to-one correspondence with actions (*-representations) of S(G).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 3. [N. Sieben 1997] Let G be a group.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups tg −1 tg th = tg −1 tgh (1)

tg thth−1 = tghth−1 (2)

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 3. [N. Sieben 1997] Let G be a group. S(G) is a semigroup generated by elements tg for g ∈ G satisfying:

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups ∗ S(G) is an inverse semigroup with unit t1 and involution tg = tg −1 . [R. Exel, V. Vieira 1998] Partial actions (partial representations) of G are in one-to-one correspondence with actions (*-representations) of S(G).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 3. [N. Sieben 1997] Let G be a group. S(G) is a semigroup generated by elements tg for g ∈ G satisfying:

tg −1 tg th = tg −1 tgh (1)

tg thth−1 = tghth−1 (2)

tg t1 = tg (3)

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups [R. Exel, V. Vieira 1998] Partial actions (partial representations) of G are in one-to-one correspondence with actions (*-representations) of S(G).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 3. [N. Sieben 1997] Let G be a group. S(G) is a semigroup generated by elements tg for g ∈ G satisfying:

tg −1 tg th = tg −1 tgh (1)

tg thth−1 = tghth−1 (2)

tg t1 = tg (3) ∗ S(G) is an inverse semigroup with unit t1 and involution tg = tg −1 .

Examples

Example 3. [N. Sieben 1997] Let G be a group. S(G) is a semigroup generated by elements tg for g ∈ G satisfying:

tg −1 tg th = tg −1 tgh (1)

tg thth−1 = tghth−1 (2)

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups −1 Deﬁne λa : P → P, λa(b) = ab. Then {λa, λa } in I (P) generates an inverse semigroup, called the left inverse hull of P.

Example 5. There exists embedding P ,→ P∗ in the universal inverse semigroup ∗ generated by elements {vp, vq : p, q ∈ P} under conditions ∗ vpvq = vpq, vp vp = 1

Theorem [A. 2016] ∗ ∗ ∗ There exist surjective *-homomorphisms C (P ) → C (Il (P)), ∗ ∗ ∗ Cr (P ) → Cr (P); there is a 1-1 correspondence between injective actions and crossed products of P and P∗; amenability of P connected to nuclearity of all the C*-algebras above.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 4. P a left cancellative semigroup: ab = ac ⇒ b = c.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups −1 Then {λa, λa } in I (P) generates an inverse semigroup, called the left inverse hull of P.

Example 5. There exists embedding P ,→ P∗ in the universal inverse semigroup ∗ generated by elements {vp, vq : p, q ∈ P} under conditions ∗ vpvq = vpq, vp vp = 1

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 4. P a left cancellative semigroup: ab = ac ⇒ b = c. Deﬁne λa : P → P, λa(b) = ab.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Example 5. There exists embedding P ,→ P∗ in the universal inverse semigroup ∗ generated by elements {vp, vq : p, q ∈ P} under conditions ∗ vpvq = vpq, vp vp = 1

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 4. P a left cancellative semigroup: ab = ac ⇒ b = c. −1 Deﬁne λa : P → P, λa(b) = ab. Then {λa, λa } in I (P) generates an inverse semigroup, called the left inverse hull of P.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups ∗ generated by elements {vp, vq : p, q ∈ P} under conditions ∗ vpvq = vpq, vp vp = 1

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 4. P a left cancellative semigroup: ab = ac ⇒ b = c. −1 Deﬁne λa : P → P, λa(b) = ab. Then {λa, λa } in I (P) generates an inverse semigroup, called the left inverse hull of P.

Example 5. There exists embedding P ,→ P∗ in the universal inverse semigroup

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Theorem [A. 2016] ∗ ∗ ∗ There exist surjective *-homomorphisms C (P ) → C (Il (P)), ∗ ∗ ∗ Cr (P ) → Cr (P); there is a 1-1 correspondence between injective actions and crossed products of P and P∗; amenability of P connected to nuclearity of all the C*-algebras above.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 4. P a left cancellative semigroup: ab = ac ⇒ b = c. −1 Deﬁne λa : P → P, λa(b) = ab. Then {λa, λa } in I (P) generates an inverse semigroup, called the left inverse hull of P.

Example 5. There exists embedding P ,→ P∗ in the universal inverse semigroup ∗ generated by elements {vp, vq : p, q ∈ P} under conditions ∗ vpvq = vpq, vp vp = 1

Examples

Example 4. P a left cancellative semigroup: ab = ac ⇒ b = c. −1 Deﬁne λa : P → P, λa(b) = ab. Then {λa, λa } in I (P) generates an inverse semigroup, called the left inverse hull of P.

Example 5. There exists embedding P ,→ P∗ in the universal inverse semigroup ∗ generated by elements {vp, vq : p, q ∈ P} under conditions ∗ vpvq = vpq, vp vp = 1

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups such that φγ,βφβ,α = φγ,α if α > β > γ. Then S = tα∈X Gα is an inverse semigroup with product: ab = φγ,α(a)φγ,β(b).

where a ∈ Gα, b ∈ Gβ, γ = αβ. The simplest example is R = R ∪ {∞} under addition. In fact, it is a compact semitopological inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 6. Inverse Cliﬀord semigroup. Let Gα, α ∈ X be a semi-lattice of groups, with morphisms φβ,α : Gα → Gβ for α > β,

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Then S = tα∈X Gα is an inverse semigroup with product: ab = φγ,α(a)φγ,β(b).

where a ∈ Gα, b ∈ Gβ, γ = αβ. The simplest example is R = R ∪ {∞} under addition. In fact, it is a compact semitopological inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 6. Inverse Cliﬀord semigroup. Let Gα, α ∈ X be a semi-lattice of groups, with morphisms φβ,α : Gα → Gβ for α > β, such that φγ,βφβ,α = φγ,α if α > β > γ.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups The simplest example is R = R ∪ {∞} under addition. In fact, it is a compact semitopological inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples

Example 6. Inverse Cliﬀord semigroup. Let Gα, α ∈ X be a semi-lattice of groups, with morphisms φβ,α : Gα → Gβ for α > β, such that φγ,βφβ,α = φγ,α if α > β > γ. Then S = tα∈X Gα is an inverse semigroup with product: ab = φγ,α(a)φγ,β(b).

where a ∈ Gα, b ∈ Gβ, γ = αβ.

Examples

where a ∈ Gα, b ∈ Gβ, γ = αβ. The simplest example is R = R ∪ {∞} under addition. In fact, it is a compact semitopological inverse semigroup.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Deﬁne S = G ∪ {0}.

For a, b not composable in G set a · b = 0 in S, otherw. a · b = ab.

For a ∈ G set a∗ = a−1, and 0∗ = 0. ⇒ aa∗a = a For any a, b ∈ S either a∗b = 0 or aa∗ = bb∗.

⇒ idempotents in S are mutually orthogonal. ⇒ S is an inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroups from groupoids

Let G be groupoid.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups For a, b not composable in G set a · b = 0 in S, otherw. a · b = ab.

For a ∈ G set a∗ = a−1, and 0∗ = 0. ⇒ aa∗a = a For any a, b ∈ S either a∗b = 0 or aa∗ = bb∗.

⇒ idempotents in S are mutually orthogonal. ⇒ S is an inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroups from groupoids

Let G be groupoid. Deﬁne S = G ∪ {0}.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups For a ∈ G set a∗ = a−1, and 0∗ = 0. ⇒ aa∗a = a For any a, b ∈ S either a∗b = 0 or aa∗ = bb∗.

⇒ idempotents in S are mutually orthogonal. ⇒ S is an inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroups from groupoids

Let G be groupoid. Deﬁne S = G ∪ {0}.

For a, b not composable in G set a · b = 0 in S, otherw. a · b = ab.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups ⇒ aa∗a = a For any a, b ∈ S either a∗b = 0 or aa∗ = bb∗.

⇒ idempotents in S are mutually orthogonal. ⇒ S is an inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroups from groupoids

Let G be groupoid. Deﬁne S = G ∪ {0}.

For a, b not composable in G set a · b = 0 in S, otherw. a · b = ab.

For a ∈ G set a∗ = a−1, and 0∗ = 0.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups For any a, b ∈ S either a∗b = 0 or aa∗ = bb∗.

⇒ idempotents in S are mutually orthogonal. ⇒ S is an inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Inverse semigroups from groupoids

Let G be groupoid. Deﬁne S = G ∪ {0}.

For a, b not composable in G set a · b = 0 in S, otherw. a · b = ab.

For a ∈ G set a∗ = a−1, and 0∗ = 0. ⇒ aa∗a = a

Inverse semigroups from groupoids

Let G be groupoid. Deﬁne S = G ∪ {0}.

For a, b not composable in G set a · b = 0 in S, otherw. a · b = ab.

For a ∈ G set a∗ = a−1, and 0∗ = 0. ⇒ aa∗a = a For any a, b ∈ S either a∗b = 0 or aa∗ = bb∗.

⇒ idempotents in S are mutually orthogonal. ⇒ S is an inverse semigroup.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups A unital C ∗-algebra, comultiplication ∆: A → A ⊗ A, dense A ⊂ A *-bialgebra with ∆, bijective antihomomorphism κ: A → A id ∗ κ ∗ id = id, κ ∗ id ∗ κ = κ Here f ∗ g = m(f ⊗ g)∆.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum inverse semigroups

Deﬁnition. A compact quantum inverse semigroup (CQIS) is a pair (A, ∆)

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups dense A ⊂ A *-bialgebra with ∆, bijective antihomomorphism κ: A → A id ∗ κ ∗ id = id, κ ∗ id ∗ κ = κ Here f ∗ g = m(f ⊗ g)∆.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum inverse semigroups

Deﬁnition. A compact quantum inverse semigroup (CQIS) is a pair (A, ∆) A unital C ∗-algebra, comultiplication ∆: A → A ⊗ A,

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups bijective antihomomorphism κ: A → A id ∗ κ ∗ id = id, κ ∗ id ∗ κ = κ Here f ∗ g = m(f ⊗ g)∆.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum inverse semigroups

Deﬁnition. A compact quantum inverse semigroup (CQIS) is a pair (A, ∆) A unital C ∗-algebra, comultiplication ∆: A → A ⊗ A, dense A ⊂ A *-bialgebra with ∆,

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups id ∗ κ ∗ id = id, κ ∗ id ∗ κ = κ Here f ∗ g = m(f ⊗ g)∆.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum inverse semigroups

Deﬁnition. A compact quantum inverse semigroup (CQIS) is a pair (A, ∆) A unital C ∗-algebra, comultiplication ∆: A → A ⊗ A, dense A ⊂ A *-bialgebra with ∆, bijective antihomomorphism κ: A → A

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Here f ∗ g = m(f ⊗ g)∆.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum inverse semigroups

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups 1 X ∗ ` (S) = { as δs : as ∈ C}, δs δt = δst , δs = δs∗ s

∗ ∗ ∗ 1 The full C -algebra C (S) = Cenv (` (S)) with supremum norm over all *-representations of S. Deﬁne ∆: C ∗(S) → C ∗(S) ⊗ C ∗(S) and κ: C ∗(S) → C ∗(S): P P ∆(δs ) = δs ⊗ δs , κ( s as δs ) = s as δs∗ . Then κ is an anti-isomorphism on `1(S) and id ∗ κ ∗ id = id. So, ∗ ∗ C (S) is a CQIS. The same true for Cr (S).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 1. Let S be an inverse semigroup.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups ∗ ∗ ∗ 1 The full C -algebra C (S) = Cenv (` (S)) with supremum norm over all *-representations of S. Deﬁne ∆: C ∗(S) → C ∗(S) ⊗ C ∗(S) and κ: C ∗(S) → C ∗(S): P P ∆(δs ) = δs ⊗ δs , κ( s as δs ) = s as δs∗ . Then κ is an anti-isomorphism on `1(S) and id ∗ κ ∗ id = id. So, ∗ ∗ C (S) is a CQIS. The same true for Cr (S).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 1. Let S be an inverse semigroup.

1 X ∗ ` (S) = { as δs : as ∈ C}, δs δt = δst , δs = δs∗ s

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Deﬁne ∆: C ∗(S) → C ∗(S) ⊗ C ∗(S) and κ: C ∗(S) → C ∗(S): P P ∆(δs ) = δs ⊗ δs , κ( s as δs ) = s as δs∗ . Then κ is an anti-isomorphism on `1(S) and id ∗ κ ∗ id = id. So, ∗ ∗ C (S) is a CQIS. The same true for Cr (S).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 1. Let S be an inverse semigroup.

1 X ∗ ` (S) = { as δs : as ∈ C}, δs δt = δst , δs = δs∗ s

∗ ∗ ∗ 1 The full C -algebra C (S) = Cenv (` (S)) with supremum norm over all *-representations of S.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Then κ is an anti-isomorphism on `1(S) and id ∗ κ ∗ id = id. So, ∗ ∗ C (S) is a CQIS. The same true for Cr (S).

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 1. Let S be an inverse semigroup.

1 X ∗ ` (S) = { as δs : as ∈ C}, δs δt = δst , δs = δs∗ s

Examples of quantum inverse semigroups

Example 1. Let S be an inverse semigroup.

1 X ∗ ` (S) = { as δs : as ∈ C}, δs δt = δst , δs = δs∗ s

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Functions πij : S → C generate a C*-subalgebra Cπ(S) in Cb(S). Deﬁne ∆(πij )(x, y) = πij (x · y) = (π(x)π(y))ij P ⇒ ∆(πij ) = k πik ⊗ πkj .

With κ(πij ) = πji ,(Cπ(S), ∆) is a CQIS. Proposition If S is a compact inverse semigroup with a totally disconnected space of idempotents, C(S) is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 2. Let π be a f.d. *-representation of an inverse semigroup S.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Deﬁne ∆(πij )(x, y) = πij (x · y) = (π(x)π(y))ij P ⇒ ∆(πij ) = k πik ⊗ πkj .

With κ(πij ) = πji ,(Cπ(S), ∆) is a CQIS. Proposition If S is a compact inverse semigroup with a totally disconnected space of idempotents, C(S) is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 2. Let π be a f.d. *-representation of an inverse semigroup S. Functions πij : S → C generate a C*-subalgebra Cπ(S) in Cb(S).

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups P ⇒ ∆(πij ) = k πik ⊗ πkj .

With κ(πij ) = πji ,(Cπ(S), ∆) is a CQIS. Proposition If S is a compact inverse semigroup with a totally disconnected space of idempotents, C(S) is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 2. Let π be a f.d. *-representation of an inverse semigroup S. Functions πij : S → C generate a C*-subalgebra Cπ(S) in Cb(S). Deﬁne ∆(πij )(x, y) = πij (x · y) = (π(x)π(y))ij

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups With κ(πij ) = πji ,(Cπ(S), ∆) is a CQIS. Proposition If S is a compact inverse semigroup with a totally disconnected space of idempotents, C(S) is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 2. Let π be a f.d. *-representation of an inverse semigroup S. Functions πij : S → C generate a C*-subalgebra Cπ(S) in Cb(S). Deﬁne ∆(πij )(x, y) = πij (x · y) = (π(x)π(y))ij P ⇒ ∆(πij ) = k πik ⊗ πkj .

Examples of quantum inverse semigroups

With κ(πij ) = πji ,(Cπ(S), ∆) is a CQIS. Proposition If S is a compact inverse semigroup with a totally disconnected space of idempotents, C(S) is a compact quantum inverse semigroup.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups π(g)π(h)π(h−1) = π(gh)π(h−1),

π(h−1)π(h)π(g) = π(h−1)π(hg),

π(g −1) = π(g)∗, π(1) = 1

Due to [R. Exel 1998], π factors through a representation of S(G). Hence, (Cπ(S(G)), ∆) from Example 2 is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 3. [R. Exel]: π : G → B(H) is a partial representation of a group G if

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups π(h−1)π(h)π(g) = π(h−1)π(hg),

π(g −1) = π(g)∗, π(1) = 1

Due to [R. Exel 1998], π factors through a representation of S(G). Hence, (Cπ(S(G)), ∆) from Example 2 is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 3. [R. Exel]: π : G → B(H) is a partial representation of a group G if

π(g)π(h)π(h−1) = π(gh)π(h−1),

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups π(g −1) = π(g)∗, π(1) = 1

Due to [R. Exel 1998], π factors through a representation of S(G). Hence, (Cπ(S(G)), ∆) from Example 2 is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 3. [R. Exel]: π : G → B(H) is a partial representation of a group G if

π(g)π(h)π(h−1) = π(gh)π(h−1),

π(h−1)π(h)π(g) = π(h−1)π(hg),

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Due to [R. Exel 1998], π factors through a representation of S(G). Hence, (Cπ(S(G)), ∆) from Example 2 is a compact quantum inverse semigroup.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 3. [R. Exel]: π : G → B(H) is a partial representation of a group G if

π(g)π(h)π(h−1) = π(gh)π(h−1),

π(h−1)π(h)π(g) = π(h−1)π(hg),

π(g −1) = π(g)∗, π(1) = 1

Examples of quantum inverse semigroups

Example 3. [R. Exel]: π : G → B(H) is a partial representation of a group G if

π(g)π(h)π(h−1) = π(gh)π(h−1),

π(h−1)π(h)π(g) = π(h−1)π(hg),

π(g −1) = π(g)∗, π(1) = 1

Due to [R. Exel 1998], π factors through a representation of S(G). Hence, (Cπ(S(G)), ∆) from Example 2 is a compact quantum inverse semigroup.

Examples of quantum inverse semigroups

Example 4. Deﬁnition ([T. Banica, A. Skalski, 2015]) ˜+ ∗ N C(SN ) universal C -algebra generated by (uij )i,j=1 s.t. ∗ 2 uij = uij = uij , uij uik = 0, uji uki = 0. P Comultiplication ∆(uij ) = k uik ⊗ ukj , Counit ε(uij ) = δij . P P Add requirement: i uij , i uji lie in the centre. Algebra A satisfying these relations and with κ(uij ) = uji is a CQIS.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups with morphisms φβ,α : Aβ → Aα if α > β such that φβ,αφγ,β = φγ,α if α > β > γ.

L Then A = α∈X Aα is a compact quantum inverse semigroup with the coproduct M ∆|Aγ = (φγ,α ⊗ φγ,β)∆γ αβ=γ

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 5. Let (Aα, ∆α), α ∈ X be a semi-lattice of compact quantum groups

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups φβ,αφγ,β = φγ,α if α > β > γ.

L Then A = α∈X Aα is a compact quantum inverse semigroup with the coproduct M ∆|Aγ = (φγ,α ⊗ φγ,β)∆γ αβ=γ

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 5. Let (Aα, ∆α), α ∈ X be a semi-lattice of compact quantum groups with morphisms φβ,α : Aβ → Aα if α > β such that

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups L Then A = α∈X Aα is a compact quantum inverse semigroup with the coproduct M ∆|Aγ = (φγ,α ⊗ φγ,β)∆γ αβ=γ

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Examples of quantum inverse semigroups

Example 5. Let (Aα, ∆α), α ∈ X be a semi-lattice of compact quantum groups with morphisms φβ,α : Aβ → Aα if α > β such that φβ,αφγ,β = φγ,α if α > β > γ.

Examples of quantum inverse semigroups

Example 5. Let (Aα, ∆α), α ∈ X be a semi-lattice of compact quantum groups with morphisms φβ,α : Aβ → Aα if α > β such that φβ,αφγ,β = φγ,α if α > β > γ.

L Then A = α∈X Aα is a compact quantum inverse semigroup with the coproduct M ∆|Aγ = (φγ,α ⊗ φγ,β)∆γ αβ=γ

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Deﬁne A˜ = A ⊕ Cp,

∆(˜ a + λp) = ∆(a) + λ∆(˜ p),

∆(˜ p) = 1A ⊗ 1A − ∆(1A) + 1A ⊗ p + p ⊗ 1A + p ⊗ p. ⇒ ∆˜ is unital *-homomorphism.

κ(a + λp) = S(a) + λp

Then A˜ is a compact quantum inverse semigroup with ∆˜ and κ.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum groupoids and inverse semigroups

Let (A, ∆, , S) be a weak C ∗-Hopf algebra [G. B¨ohm,N. Florian, K. Szlachnyi 1999].

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups ∆(˜ a + λp) = ∆(a) + λ∆(˜ p),

∆(˜ p) = 1A ⊗ 1A − ∆(1A) + 1A ⊗ p + p ⊗ 1A + p ⊗ p. ⇒ ∆˜ is unital *-homomorphism.

κ(a + λp) = S(a) + λp

Then A˜ is a compact quantum inverse semigroup with ∆˜ and κ.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum groupoids and inverse semigroups

Let (A, ∆, , S) be a weak C ∗-Hopf algebra [G. B¨ohm,N. Florian, K. Szlachnyi 1999]. Deﬁne A˜ = A ⊕ Cp,

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups ∆(˜ p) = 1A ⊗ 1A − ∆(1A) + 1A ⊗ p + p ⊗ 1A + p ⊗ p. ⇒ ∆˜ is unital *-homomorphism.

κ(a + λp) = S(a) + λp

Then A˜ is a compact quantum inverse semigroup with ∆˜ and κ.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum groupoids and inverse semigroups

Let (A, ∆, , S) be a weak C ∗-Hopf algebra [G. B¨ohm,N. Florian, K. Szlachnyi 1999]. Deﬁne A˜ = A ⊕ Cp,

∆(˜ a + λp) = ∆(a) + λ∆(˜ p),

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups κ(a + λp) = S(a) + λp

Then A˜ is a compact quantum inverse semigroup with ∆˜ and κ.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum groupoids and inverse semigroups

Let (A, ∆, , S) be a weak C ∗-Hopf algebra [G. B¨ohm,N. Florian, K. Szlachnyi 1999]. Deﬁne A˜ = A ⊕ Cp,

∆(˜ a + λp) = ∆(a) + λ∆(˜ p),

∆(˜ p) = 1A ⊗ 1A − ∆(1A) + 1A ⊗ p + p ⊗ 1A + p ⊗ p. ⇒ ∆˜ is unital *-homomorphism.

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups Then A˜ is a compact quantum inverse semigroup with ∆˜ and κ.

Classical inverse semigroups Quantum inverse semigroups Examples Quantum groupoids and inverse semigroups

Quantum groupoids and inverse semigroups

Let (A, ∆, , S) be a weak C ∗-Hopf algebra [G. B¨ohm,N. Florian, K. Szlachnyi 1999]. Deﬁne A˜ = A ⊕ Cp,

∆(˜ a + λp) = ∆(a) + λ∆(˜ p),

∆(˜ p) = 1A ⊗ 1A − ∆(1A) + 1A ⊗ p + p ⊗ 1A + p ⊗ p. ⇒ ∆˜ is unital *-homomorphism.

κ(a + λp) = S(a) + λp

Quantum groupoids and inverse semigroups

Let (A, ∆, , S) be a weak C ∗-Hopf algebra [G. B¨ohm,N. Florian, K. Szlachnyi 1999]. Deﬁne A˜ = A ⊕ Cp,

∆(˜ a + λp) = ∆(a) + λ∆(˜ p),

∆(˜ p) = 1A ⊗ 1A − ∆(1A) + 1A ⊗ p + p ⊗ 1A + p ⊗ p. ⇒ ∆˜ is unital *-homomorphism.

κ(a + λp) = S(a) + λp

Then A˜ is a compact quantum inverse semigroup with ∆˜ and κ.

Last slide

Thank you!

Marat Aukhadiev Joint work with A. Buss and T. Timmermann University of M¨unster Quantum Inverse Semigroups