Pontrjagin Duality for Finite Groups

PONTRJAGIN DUALITY FOR FINITE GROUPS

Partha Sarathi Chakraborty The Institute of Mathematical Sciences Chennai e1, ··· , en : basis for V . ∗ V := {φ|φ : V → C linear map }. ∗ hφi , ej i := φi (ej ) := δij , φi , ··· , φn, dual basis for V .

Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗

Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

Motivation: duality for vector spaces

Dual of a vector space V : a ﬁnite dimensional vector space over C. ∗ V := {φ|φ : V → C linear map }. ∗ hφi , ej i := φi (ej ) := δij , φi , ··· , φn, dual basis for V .

Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗

Motivation: duality for vector spaces

Dual of a vector space V : a ﬁnite dimensional vector space over C. e1, ··· , en : basis for V . ∗ hφi , ej i := φi (ej ) := δij , φi , ··· , φn, dual basis for V .

Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗

Motivation: duality for vector spaces

Dual of a vector space V : a ﬁnite dimensional vector space over C. e1, ··· , en : basis for V . ∗ V := {φ|φ : V → C linear map }. ∗ φi , ··· , φn, dual basis for V .

Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗

Motivation: duality for vector spaces

Dual of a vector space V : a ﬁnite dimensional vector space over C. e1, ··· , en : basis for V . ∗ V := {φ|φ : V → C linear map }. hφi , ej i := φi (ej ) := δij , Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗

Motivation: duality for vector spaces

Dual of a vector space V : a ﬁnite dimensional vector space over C. e1, ··· , en : basis for V . ∗ V := {φ|φ : V → C linear map }. ∗ hφi , ej i := φi (ej ) := δij , φi , ··· , φn, dual basis for V . T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗

Motivation: duality for vector spaces

Dual of a vector space V : a ﬁnite dimensional vector space over C. e1, ··· , en : basis for V . ∗ V := {φ|φ : V → C linear map }. ∗ hφi , ej i := φi (ej ) := δij , φi , ··· , φn, dual basis for V .

Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

Motivation: duality for vector spaces

Dual of a map, T : V → W T ∗ : W ∗ → V ∗, T ∗(φ)(v) = φ(T (v)). T : V → W , S : W → U, (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ → W ∗ Proof: hφj , ΦV (ei )i = hφj , ei i = δij . Therefore, ΦV (e1), ··· , ΦV (en) is the dual basis of φ1, ··· , φn. Natural isomorphism

ΦV V / V ∗∗

T T ∗∗

ΦW W / W ∗∗

Proposition ∗∗ ΦV : V → V , ΦV (v)(φ) = φ(v), gives a “natural isomorphism” between V and V ∗∗. Therefore, ΦV (e1), ··· , ΦV (en) is the dual basis of φ1, ··· , φn. Natural isomorphism

ΦV V / V ∗∗

T T ∗∗

ΦW W / W ∗∗

Proposition ∗∗ ΦV : V → V , ΦV (v)(φ) = φ(v), gives a “natural isomorphism” between V and V ∗∗.

Proof: hφj , ΦV (ei )i = hφj , ei i = δij . Natural isomorphism

ΦV V / V ∗∗

T T ∗∗

ΦW W / W ∗∗

Proposition ∗∗ ΦV : V → V , ΦV (v)(φ) = φ(v), gives a “natural isomorphism” between V and V ∗∗.

Proof: hφj , ΦV (ei )i = hφj , ei i = δij . Therefore, ΦV (e1), ··· , ΦV (en) is the dual basis of φ1, ··· , φn. ΦV V / V ∗∗

T T ∗∗

ΦW W / W ∗∗

Proposition ∗∗ ΦV : V → V , ΦV (v)(φ) = φ(v), gives a “natural isomorphism” between V and V ∗∗.

Proof: hφj , ΦV (ei )i = hφj , ei i = δij . Therefore, ΦV (e1), ··· , ΦV (en) is the dual basis of φ1, ··· , φn. Natural isomorphism Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

Proposition ∗∗ ΦV : V → V , ΦV (v)(φ) = φ(v), gives a “natural isomorphism” between V and V ∗∗.

Proof: hφj , ΦV (ei )i = hφj , ei i = δij . Therefore, ΦV (e1), ··· , ΦV (en) is the dual basis of φ1, ··· , φn. Natural isomorphism

ΦV V / V ∗∗

T T ∗∗

ΦW W / W ∗∗ 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)}

Proposition: Gb is a group, called Pontryagin dual of G.

Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Identity φe (g) := 1

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)}

Proposition: Gb is a group, called Pontryagin dual of G.

Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Identity φe (g) := 1

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group Proposition: Gb is a group, called Pontryagin dual of G.

Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Identity φe (g) := 1

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)} Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Identity φe (g) := 1

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)}

Proposition: Gb is a group, called Pontryagin dual of G. Inverse φ−1(g) := φ(g)−1 Identity φe (g) := 1

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)}

Proposition: Gb is a group, called Pontryagin dual of G.

Product (φ1 · φ2)(g) := φ1(g)φ2(g) Identity φe (g) := 1

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)}

Proposition: Gb is a group, called Pontryagin dual of G.

Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

Pontryagin dual of an abelian group

G: a ﬁnite Abelian group 1 S := {z ∈ C||z| = 1}, note that this is also an abelian group 1 Gb = {φ : G → S |φ(e) = 1, φ(g1g2) = φ(g1)φ(g2)}

Proposition: Gb is a group, called Pontryagin dual of G.

Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Identity φe (g) := 1 G: cyclic group of order n G = {0, 1, ··· , n − 1} with addition modulo n. φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)}

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like? G = {0, 1, ··· , n − 1} with addition modulo n. φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)}

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like?

G: cyclic group of order n φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)}

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like?

G: cyclic group of order n G = {0, 1, ··· , n − 1} with addition modulo n. 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)}

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like?

G: cyclic group of order n G = {0, 1, ··· , n − 1} with addition modulo n. φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)}

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like?

G: cyclic group of order n G = {0, 1, ··· , n − 1} with addition modulo n. φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like?

G: cyclic group of order n G = {0, 1, ··· , n − 1} with addition modulo n. φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)} (b) G is isomorphic with G,b in particular |G| = |Gb|.

How does Gb look like?

G: cyclic group of order n G = {0, 1, ··· , n − 1} with addition modulo n. φ ∈ Gb ↔ φ(1) = z ∈ S1, zn = 1 2πik 1 φ 7→ φ(1) ∈ {e n |k = 0, 1, ··· , (n − 1)} ⊂ S proves 2πik Gb =∼ {e n |k = 0, 1, ··· , (n − 1)}

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

How does Gb look like?

Note (a) Given any l ∈ G, l 6= 0 there exists φ ∈ Gb such that φ(l) 6= 1. 2πi More precisely take φ(1) = e n (b) G is isomorphic with G,b in particular |G| = |Gb|. φ ∈ Gb determines a pair of elements φj = φ|Gj ∈ Gbj , j = 1, 2. Conversely, φj ∈ Gbj , j = 1, 2 determine φ ∈ Gb given by φ(g1, g2) = φ1(g1)φ2(g2). ∼ φ 7→ (φ1, φ2) gives an isomorphism Gb = Gc1 × Gc2.

G = G1 × G2, a direct product of two groups G1 and G2 ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2.

Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb|

G = G1 ⊕ G2, a direct sum of two groups G1 and G2 Conversely, φj ∈ Gbj , j = 1, 2 determine φ ∈ Gb given by φ(g1, g2) = φ1(g1)φ2(g2). ∼ φ 7→ (φ1, φ2) gives an isomorphism Gb = Gc1 × Gc2.

G = G1 × G2, a direct product of two groups G1 and G2 ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2.

Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb|

G = G1 ⊕ G2, a direct sum of two groups G1 and G2

φ ∈ Gb determines a pair of elements φj = φ|Gj ∈ Gbj , j = 1, 2. ∼ φ 7→ (φ1, φ2) gives an isomorphism Gb = Gc1 × Gc2.

G = G1 × G2, a direct product of two groups G1 and G2 ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2.

Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb|

G = G1 ⊕ G2, a direct sum of two groups G1 and G2

φ ∈ Gb determines a pair of elements φj = φ|Gj ∈ Gbj , j = 1, 2. Conversely, φj ∈ Gbj , j = 1, 2 determine φ ∈ Gb given by φ(g1, g2) = φ1(g1)φ2(g2). G = G1 × G2, a direct product of two groups G1 and G2 ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2.

Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb|

G = G1 ⊕ G2, a direct sum of two groups G1 and G2

φ ∈ Gb determines a pair of elements φj = φ|Gj ∈ Gbj , j = 1, 2. Conversely, φj ∈ Gbj , j = 1, 2 determine φ ∈ Gb given by φ(g1, g2) = φ1(g1)φ2(g2). ∼ φ 7→ (φ1, φ2) gives an isomorphism Gb = Gc1 × Gc2. ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2.

Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb|

G = G1 ⊕ G2, a direct sum of two groups G1 and G2

G = G1 × G2, a direct product of two groups G1 and G2 Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb|

G = G1 ⊕ G2, a direct sum of two groups G1 and G2

G = G1 × G2, a direct product of two groups G1 and G2 ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2. Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

G = G1 ⊕ G2, a direct sum of two groups G1 and G2

G = G1 × G2, a direct product of two groups G1 and G2 ∼ In this case we have an isomorphism Gb = Gc1 ⊕ Gc2.

Note

|G| = |G1|.|G2| = |Gc1||Gc2| = |Gb| Deﬁne ΦG : G → Gb, as ΦG (g)(φ) = φ(g). Assertion holds if G is cyclic ΦG is one to one: ΦG (g)(φ) = 1, ∀φ ∈ Gb. Then, φ(g) = 1, ∀φ ∈ Gb. This implies g = e. Since |G| = |Gb| = |Gb|, Φ is an isomorphism.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb. ΦG is one to one: ΦG (g)(φ) = 1, ∀φ ∈ Gb. Then, φ(g) = 1, ∀φ ∈ Gb. This implies g = e. Since |G| = |Gb| = |Gb|, Φ is an isomorphism.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Deﬁne ΦG : G → Gb, as ΦG (g)(φ) = φ(g). Assertion holds if G is cyclic Then, φ(g) = 1, ∀φ ∈ Gb. This implies g = e. Since |G| = |Gb| = |Gb|, Φ is an isomorphism.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Deﬁne ΦG : G → Gb, as ΦG (g)(φ) = φ(g). Assertion holds if G is cyclic ΦG is one to one: ΦG (g)(φ) = 1, ∀φ ∈ Gb. Since |G| = |Gb| = |Gb|, Φ is an isomorphism.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Deﬁne ΦG : G → Gb, as ΦG (g)(φ) = φ(g). Assertion holds if G is cyclic ΦG is one to one: ΦG (g)(φ) = 1, ∀φ ∈ Gb. Then, φ(g) = 1, ∀φ ∈ Gb. This implies g = e. Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Deﬁne ΦG : G → Gb, as ΦG (g)(φ) = φ(g). Assertion holds if G is cyclic ΦG is one to one: ΦG (g)(φ) = 1, ∀φ ∈ Gb. Then, φ(g) = 1, ∀φ ∈ Gb. This implies g = e. Since |G| = |Gb| = |Gb|, Φ is an isomorphism. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. Assertion holds for all ﬁnite abelian groups.

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

Pontryagin duality

Theorem: G is “naturally isomorphic” with Gb.

Assertion holds if G = G1 ⊕ G2 where ΦG1 , ΦG2 are isomorphisms. ∼ \ ∼ ∼ Gb = Gc1 × Gc2 = Gc1 ⊕ Gc2 = G1 ⊕ G2 = G. Assertion holds for all ﬁnite abelian groups. Fb : Hb → Gb, Fb(φ)(g) = φ(F (g)).

ΦG G / Gb

F Fb ΦH H / Hb

Meaning of naturality

Dual of a homomorphism: F : G → H ΦG G / Gb

F Fb ΦH H / Hb

Meaning of naturality

Dual of a homomorphism: F : G → H Fb : Hb → Gb, Fb(φ)(g) = φ(F (g)). Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The ﬁnite group case Pontryagin Duality

Meaning of naturality

Dual of a homomorphism: F : G → H Fb : Hb → Gb, Fb(φ)(g) = φ(F (g)).

ΦG G / Gb

F Fb ΦH H / Hb Gb is always abelian Since Gb is always abelian for nonabelian groups one should not hope to have G =∼ Gb.

Pontryagin duality: classical formulation Limitations of the classical formulation Some algebraic structures The ﬁnite group case

What happens if we drop the abelian condition

Gb could be trivial If G is simple then Gb is the trivial group. Pontryagin duality: classical formulation Limitations of the classical formulation Some algebraic structures The ﬁnite group case

What happens if we drop the abelian condition

Gb could be trivial If G is simple then Gb is the trivial group.

Gb is always abelian Since Gb is always abelian for nonabelian groups one should not hope to have G =∼ Gb. Bilinear map : T : U × V → W

T (au1 + bu2, v) = aT (u1, v) + bT (u2, v)

T (u, av1 + bv2) = aT (u, v1) + bT (u, v2)

Example

T : Mn(C) × Mn(C) → Mn(C), (A, B) 7→ AB (A, B) 7→ AB − BA.

Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Prelude on tensor product of vector spaces

U, V , W : Vector spaces over C. T (au1 + bu2, v) = aT (u1, v) + bT (u2, v)

T (u, av1 + bv2) = aT (u, v1) + bT (u, v2)

Example

T : Mn(C) × Mn(C) → Mn(C), (A, B) 7→ AB (A, B) 7→ AB − BA.

U, V , W : Vector spaces over C. Bilinear map : T : U × V → W T (u, av1 + bv2) = aT (u, v1) + bT (u, v2)

Example

T : Mn(C) × Mn(C) → Mn(C), (A, B) 7→ AB (A, B) 7→ AB − BA.

U, V , W : Vector spaces over C. Bilinear map : T : U × V → W

T (au1 + bu2, v) = aT (u1, v) + bT (u2, v) Example

T : Mn(C) × Mn(C) → Mn(C), (A, B) 7→ AB (A, B) 7→ AB − BA.

U, V , W : Vector spaces over C. Bilinear map : T : U × V → W

T (au1 + bu2, v) = aT (u1, v) + bT (u2, v)

T (u, av1 + bv2) = aT (u, v1) + bT (u, v2) (A, B) 7→ AB − BA.

U, V , W : Vector spaces over C. Bilinear map : T : U × V → W

T (au1 + bu2, v) = aT (u1, v) + bT (u2, v)

T (u, av1 + bv2) = aT (u, v1) + bT (u, v2)

Example

T : Mn(C) × Mn(C) → Mn(C), (A, B) 7→ AB Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Prelude on tensor product of vector spaces

U, V , W : Vector spaces over C. Bilinear map : T : U × V → W

T (au1 + bu2, v) = aT (u1, v) + bT (u2, v)

T (u, av1 + bv2) = aT (u, v1) + bT (u, v2)

Example

T : Mn(C) × Mn(C) → Mn(C), (A, B) 7→ AB (A, B) 7→ AB − BA. There exists a vector space U ⊗ V along with a bilinear map ⊗ : U × V → U ⊗ V such that given any other vector space W and a bilinear map T : U × V → W there exists a unique linear map Te : U ⊗ V → W such that T = Te ◦ ⊗. ⊗ U × V / U ⊗ V KK KK KK Te T KKK K% W

U ⊗ V , the tensor product of U and V along with a bilinear map ⊗ : U × V → U ⊗ V such that given any other vector space W and a bilinear map T : U × V → W there exists a unique linear map Te : U ⊗ V → W such that T = Te ◦ ⊗. ⊗ U × V / U ⊗ V KK KK KK Te T KKK K% W

U ⊗ V , the tensor product of U and V There exists a vector space U ⊗ V such that given any other vector space W and a bilinear map T : U × V → W there exists a unique linear map Te : U ⊗ V → W such that T = Te ◦ ⊗. ⊗ U × V / U ⊗ V KK KK KK Te T KKK K% W

U ⊗ V , the tensor product of U and V There exists a vector space U ⊗ V along with a bilinear map ⊗ : U × V → U ⊗ V there exists a unique linear map Te : U ⊗ V → W such that T = Te ◦ ⊗. ⊗ U × V / U ⊗ V KK KK KK Te T KKK K% W

U ⊗ V , the tensor product of U and V There exists a vector space U ⊗ V along with a bilinear map ⊗ : U × V → U ⊗ V such that given any other vector space W and a bilinear map T : U × V → W ⊗ U × V / U ⊗ V KK KK KK Te T KKK K% W

U ⊗ V , the tensor product of U and V There exists a vector space U ⊗ V along with a bilinear map ⊗ : U × V → U ⊗ V such that given any other vector space W and a bilinear map T : U × V → W there exists a unique linear map Te : U ⊗ V → W such that T = Te ◦ ⊗. Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra

σ σe V × U ⊗ / V ⊗ U

Flip: σe : U ⊗ V → V ⊗ U ⊗ U × V / U ⊗ V

σ σe V × U ⊗ / V ⊗ U

Flip: σe : U ⊗ V → V ⊗ U σ :(u, v) 7→ (v, u). Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Working with the concept

Flip: σe : U ⊗ V → V ⊗ U σ :(u, v) 7→ (v, u). ⊗ U × V / U ⊗ V

σ σe V × U ⊗ / V ⊗ U T1 × T2 :(u1, u2) 7→ (T1(u1), T2(u2)), ⊗ U1 × U2 / U1 ⊗ U2

T1×T2 T1⊗T2 V × V V ⊗ V 1 2 ⊗ / 1 2

Tensor product of linear maps

T1 : U1 → V1, T2 : U2 → V2 ⊗ U1 × U2 / U1 ⊗ U2

T1×T2 T1⊗T2 V × V V ⊗ V 1 2 ⊗ / 1 2

Tensor product of linear maps

T1 : U1 → V1, T2 : U2 → V2 T1 × T2 :(u1, u2) 7→ (T1(u1), T2(u2)), Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra

Tensor product of linear maps

T1 : U1 → V1, T2 : U2 → V2 T1 × T2 :(u1, u2) 7→ (T1(u1), T2(u2)), ⊗ U1 × U2 / U1 ⊗ U2

T1×T2 T1⊗T2 V × V V ⊗ V 1 2 ⊗ / 1 2 V : a vector space with basis f1, ··· , fm. U ⊗ V is an nm dimensional vector space with basis {ei ⊗ fj : 1 = 1, ··· , n; j = 1, ··· , m}. φ :(u, λ) 7→ λu, ψ :(λ, u) 7→ λu ⊗ ⊗ U × C / U ⊗ C C ⊗ U o C × U KK s KK ss KK φe ψe ss φ KK ss ψ KK ss % UU ys ∼ ∼ U ⊗ C = U = C ⊗ U

U : a vector space with basis e1, ··· , en. U ⊗ V is an nm dimensional vector space with basis {ei ⊗ fj : 1 = 1, ··· , n; j = 1, ··· , m}. φ :(u, λ) 7→ λu, ψ :(λ, u) 7→ λu ⊗ ⊗ U × C / U ⊗ C C ⊗ U o C × U KK s KK ss KK φe ψe ss φ KK ss ψ KK ss % UU ys ∼ ∼ U ⊗ C = U = C ⊗ U

U : a vector space with basis e1, ··· , en. V : a vector space with basis f1, ··· , fm. φ :(u, λ) 7→ λu, ψ :(λ, u) 7→ λu ⊗ ⊗ U × C / U ⊗ C C ⊗ U o C × U KK s KK ss KK φe ψe ss φ KK ss ψ KK ss % UU ys ∼ ∼ U ⊗ C = U = C ⊗ U

U : a vector space with basis e1, ··· , en. V : a vector space with basis f1, ··· , fm. U ⊗ V is an nm dimensional vector space with basis {ei ⊗ fj : 1 = 1, ··· , n; j = 1, ··· , m}. ∼ ∼ U ⊗ C = U = C ⊗ U

U : a vector space with basis e1, ··· , en. V : a vector space with basis f1, ··· , fm. U ⊗ V is an nm dimensional vector space with basis {ei ⊗ fj : 1 = 1, ··· , n; j = 1, ··· , m}. φ :(u, λ) 7→ λu, ψ :(λ, u) 7→ λu ⊗ ⊗ U × C / U ⊗ C C ⊗ U o C × U KK s KK ss KK φe ψe ss φ KK ss ψ KK ss % UU ys Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Tensor product in concrete terms

U : a vector space with basis e1, ··· , en. V : a vector space with basis f1, ··· , fm. U ⊗ V is an nm dimensional vector space with basis {ei ⊗ fj : 1 = 1, ··· , n; j = 1, ··· , m}. φ :(u, λ) 7→ λu, ψ :(λ, u) 7→ λu ⊗ ⊗ U × C / U ⊗ C C ⊗ U o C × U KK s KK ss KK φe ψe ss φ KK ss ψ KK ss % UU ys ∼ ∼ U ⊗ C = U = C ⊗ U m : A ⊗ A → A, product map η : C → A, unit m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id Often m(a1 ⊗ a2) is denoted by a1 · a2. (a1 · a2) · a3 = a1 · (a2 · a3), a · η(1) = η(1) · a = a

A : a vector space η : C → A, unit m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id Often m(a1 ⊗ a2) is denoted by a1 · a2. (a1 · a2) · a3 = a1 · (a2 · a3), a · η(1) = η(1) · a = a

A : a vector space m : A ⊗ A → A, product map m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id Often m(a1 ⊗ a2) is denoted by a1 · a2. (a1 · a2) · a3 = a1 · (a2 · a3), a · η(1) = η(1) · a = a

A : a vector space m : A ⊗ A → A, product map η : C → A, unit Often m(a1 ⊗ a2) is denoted by a1 · a2. (a1 · a2) · a3 = a1 · (a2 · a3), a · η(1) = η(1) · a = a

A : a vector space m : A ⊗ A → A, product map η : C → A, unit m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id (a1 · a2) · a3 = a1 · (a2 · a3), a · η(1) = η(1) · a = a

A : a vector space m : A ⊗ A → A, product map η : C → A, unit m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id Often m(a1 ⊗ a2) is denoted by a1 · a2. Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Associative algebra (A, m, η)

A : a vector space m : A ⊗ A → A, product map η : C → A, unit m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id Often m(a1 ⊗ a2) is denoted by a1 · a2. (a1 · a2) · a3 = a1 · (a2 · a3), a · η(1) = η(1) · a = a =∼ η⊗η η2 : C / C ⊗ C / A ⊗ A

The algebra (A ⊗ A, m2, η2)

id⊗σe⊗id m⊗m m2 : A ⊗ A ⊗ A ⊗ A / A ⊗ A ⊗ A ⊗ A / A ⊗ A Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra

The algebra (A ⊗ A, m2, η2)

id⊗σe⊗id m⊗m m2 : A ⊗ A ⊗ A ⊗ A / A ⊗ A ⊗ A ⊗ A / A ⊗ A =∼ η⊗η η2 : C / C ⊗ C / A ⊗ A C(G) ⊗ C(G) =∼ C(G × G) C(G) is an associative algebra

Product mG (f1 · f2)(g) := f1(g)f2(g) Unit ηG η(1)(g) := 1, ∀g ∈ G. Associativity (f1 · f2) · f3 := f1 · (f2 · f3)

C(G) = {f |f : G → C} C(G) is an associative algebra

Product mG (f1 · f2)(g) := f1(g)f2(g) Unit ηG η(1)(g) := 1, ∀g ∈ G. Associativity (f1 · f2) · f3 := f1 · (f2 · f3)

C(G) = {f |f : G → C} C(G) ⊗ C(G) =∼ C(G × G) Product mG (f1 · f2)(g) := f1(g)f2(g) Unit ηG η(1)(g) := 1, ∀g ∈ G. Associativity (f1 · f2) · f3 := f1 · (f2 · f3)

C(G) = {f |f : G → C} C(G) ⊗ C(G) =∼ C(G × G) C(G) is an associative algebra Unit ηG η(1)(g) := 1, ∀g ∈ G. Associativity (f1 · f2) · f3 := f1 · (f2 · f3)

C(G) = {f |f : G → C} C(G) ⊗ C(G) =∼ C(G × G) C(G) is an associative algebra

Product mG (f1 · f2)(g) := f1(g)f2(g) Associativity (f1 · f2) · f3 := f1 · (f2 · f3)

C(G) = {f |f : G → C} C(G) ⊗ C(G) =∼ C(G × G) C(G) is an associative algebra

Product mG (f1 · f2)(g) := f1(g)f2(g) Unit ηG η(1)(g) := 1, ∀g ∈ G. Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Algebra of functions

C(G) = {f |f : G → C} C(G) ⊗ C(G) =∼ C(G × G) C(G) is an associative algebra

Product mG (f1 · f2)(g) := f1(g)f2(g) Unit ηG η(1)(g) := 1, ∀g ∈ G. Associativity (f1 · f2) · f3 := f1 · (f2 · f3) ∆ : A → A ⊗ A, coproduct map : A → C, counit ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id

C(G) is a coassociative coalgebra

∆G (f )(g1, g2) := f (g1g2), G (f ) = f (e).

A : a vector space : A → C, counit ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id

C(G) is a coassociative coalgebra

∆G (f )(g1, g2) := f (g1g2), G (f ) = f (e).

A : a vector space ∆ : A → A ⊗ A, coproduct map ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id

C(G) is a coassociative coalgebra

∆G (f )(g1, g2) := f (g1g2), G (f ) = f (e).

A : a vector space ∆ : A → A ⊗ A, coproduct map : A → C, counit C(G) is a coassociative coalgebra

∆G (f )(g1, g2) := f (g1g2), G (f ) = f (e).

A : a vector space ∆ : A → A ⊗ A, coproduct map : A → C, counit ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id ∆G (f )(g1, g2) := f (g1g2), G (f ) = f (e).

A : a vector space ∆ : A → A ⊗ A, coproduct map : A → C, counit ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id

C(G) is a coassociative coalgebra G (f ) = f (e).

C(G) is a coassociative coalgebra

∆G (f )(g1, g2) := f (g1g2), Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Coassociative coalgebra (A, ∆, )

C(G) is a coassociative coalgebra

∆G (f )(g1, g2) := f (g1g2), G (f ) = f (e). m : A ⊗ A → A, product ∆ : A → A ⊗ A, coproduct. m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id (a1 · a2) = (a1)(a2), ∆ ◦ m = m2 ◦ (∆ ⊗ ∆), (η(1)) = 1, ∆(η(1)) = η(1) ⊗ η(1)

A, a vector space; η : C → A, unit : A → C, counit m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id (a1 · a2) = (a1)(a2), ∆ ◦ m = m2 ◦ (∆ ⊗ ∆), (η(1)) = 1, ∆(η(1)) = η(1) ⊗ η(1)

A, a vector space; η : C → A, unit : A → C, counit m : A ⊗ A → A, product ∆ : A → A ⊗ A, coproduct. ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id (a1 · a2) = (a1)(a2), ∆ ◦ m = m2 ◦ (∆ ⊗ ∆), (η(1)) = 1, ∆(η(1)) = η(1) ⊗ η(1)

A, a vector space; η : C → A, unit : A → C, counit m : A ⊗ A → A, product ∆ : A → A ⊗ A, coproduct. m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id (a1 · a2) = (a1)(a2), ∆ ◦ m = m2 ◦ (∆ ⊗ ∆), (η(1)) = 1, ∆(η(1)) = η(1) ⊗ η(1)

A, a vector space; η : C → A, unit : A → C, counit m : A ⊗ A → A, product ∆ : A → A ⊗ A, coproduct. m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id (a1 · a2) = (a1)(a2), ∆ ◦ m = m2 ◦ (∆ ⊗ ∆), Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Bialgebra (A, m, η, ∆, )

A, a vector space; η : C → A, unit : A → C, counit m : A ⊗ A → A, product ∆ : A → A ⊗ A, coproduct. m⊗id id⊗η η⊗id A ⊗ A ⊗ A / A ⊗ A A ⊗ / A ⊗ A o ⊗ A O C C O id⊗m m =∼ m =∼ A ⊗ A / AA / AAo m id id ∆⊗id id⊗ ⊗id A ⊗ A ⊗ AAo ⊗ A A ⊗ o A ⊗ A / ⊗ A O O C O C id⊗∆ ∆ =∼ ∆ =∼ A ⊗ A o A A / A o A ∆ id id (a1 · a2) = (a1)(a2), ∆ ◦ m = m2 ◦ (∆ ⊗ ∆), (η(1)) = 1, ∆(η(1)) = η(1) ⊗ η(1) The group algebra ∗ C (G) = span{ξg , g ∈ G} ∗ mG : ξg ⊗ ξh 7→ ξgh ∗ ∆G : ξg 7→ ξg ⊗ ξg . ∗ ηG : 1 7→ ξe ∗ G : ξg 7→ δg,e .

(C(G), mG , ηG , ∆G , G ) is a bialgebra. ∗ mG : ξg ⊗ ξh 7→ ξgh ∗ ∆G : ξg 7→ ξg ⊗ ξg . ∗ ηG : 1 7→ ξe ∗ G : ξg 7→ δg,e .

(C(G), mG , ηG , ∆G , G ) is a bialgebra. The group algebra ∗ C (G) = span{ξg , g ∈ G} ∗ ∆G : ξg 7→ ξg ⊗ ξg . ∗ ηG : 1 7→ ξe ∗ G : ξg 7→ δg,e .

(C(G), mG , ηG , ∆G , G ) is a bialgebra. The group algebra ∗ C (G) = span{ξg , g ∈ G} ∗ mG : ξg ⊗ ξh 7→ ξgh ∗ ηG : 1 7→ ξe ∗ G : ξg 7→ δg,e .

(C(G), mG , ηG , ∆G , G ) is a bialgebra. The group algebra ∗ C (G) = span{ξg , g ∈ G} ∗ mG : ξg ⊗ ξh 7→ ξgh ∗ ∆G : ξg 7→ ξg ⊗ ξg . ∗ G : ξg 7→ δg,e .

(C(G), mG , ηG , ∆G , G ) is a bialgebra. The group algebra ∗ C (G) = span{ξg , g ∈ G} ∗ mG : ξg ⊗ ξh 7→ ξgh ∗ ∆G : ξg 7→ ξg ⊗ ξg . ∗ ηG : 1 7→ ξe Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra

(C(G), mG , ηG , ∆G , G ) is a bialgebra. The group algebra ∗ C (G) = span{ξg , g ∈ G} ∗ mG : ξg ⊗ ξh 7→ ξgh ∗ ∆G : ξg 7→ ξg ⊗ ξg . ∗ ηG : 1 7→ ξe ∗ G : ξg 7→ δg,e . S : A → A, antipode m ◦ (S ⊗ id) ◦ ∆ = m ◦ (id ⊗ S) ◦ ∆ = η ◦ ∆ ∆ A ⊗ A o A / A ⊗ A

S⊗id ◦η id⊗S A ⊗ A m / A o m A ⊗ A

Hopf algebra (A, m, η, ∆, , S) (A, m, η, ∆, ) : Bialgebra m ◦ (S ⊗ id) ◦ ∆ = m ◦ (id ⊗ S) ◦ ∆ = η ◦ ∆ ∆ A ⊗ A o A / A ⊗ A

S⊗id ◦η id⊗S A ⊗ A m / A o m A ⊗ A

Hopf algebra (A, m, η, ∆, , S) (A, m, η, ∆, ) : Bialgebra S : A → A, antipode Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra Hopf algebra

Hopf algebra (A, m, η, ∆, , S) (A, m, η, ∆, ) : Bialgebra S : A → A, antipode m ◦ (S ⊗ id) ◦ ∆ = m ◦ (id ⊗ S) ◦ ∆ = η ◦ ∆ ∆ A ⊗ A o A / A ⊗ A

S⊗id ◦η id⊗S A ⊗ A m / A o m A ⊗ A −1 SG : C(G) → C(G), SG (f )(g) = f (g ).

∗ ∗ ∗ ∗ ∗ ∗ (C (G), mG , ηG , ∆G , G , SG ) ∗ ∗ ∗ ∗ SG : C (G) → C (G), SG (ξg ) = ξg −1 .

(C(G), mG , ηG , ∆G , G , SG ) ∗ ∗ ∗ ∗ ∗ ∗ (C (G), mG , ηG , ∆G , G , SG ) ∗ ∗ ∗ ∗ SG : C (G) → C (G), SG (ξg ) = ξg −1 .

(C(G), mG , ηG , ∆G , G , SG ) −1 SG : C(G) → C(G), SG (f )(g) = f (g ). ∗ ∗ ∗ ∗ SG : C (G) → C (G), SG (ξg ) = ξg −1 .

(C(G), mG , ηG , ∆G , G , SG ) −1 SG : C(G) → C(G), SG (f )(g) = f (g ).

∗ ∗ ∗ ∗ ∗ ∗ (C (G), mG , ηG , ∆G , G , SG ) Tensor product of vector spaces Associative algebra Pontryagin duality: classical formulation Associative algebra of functions Limitations of the classical formulation Coassociative coalgebra of functions Some algebraic structures Bialgebra The ﬁnite group case Examples of bialgebras Hopf algebra

(C(G), mG , ηG , ∆G , G , SG ) −1 SG : C(G) → C(G), SG (f )(g) = f (g ).

∗ ∗ ∗ ∗ ∗ ∗ (C (G), mG , ηG , ∆G , G , SG ) ∗ ∗ ∗ ∗ SG : C (G) → C (G), SG (ξg ) = ξg −1 . ∗ ∗ ∗ ∗ ∗ m : H ⊗ H → H , m (φ1 ⊗ φ2)(a) = (φ1 ⊗ φ2)(∆(a)); ∗ ∗ ∗ ∗ ∗ ∆ : H → H ⊗ H , ∆ (φ)(a1 ⊗ a2) = φ(m(a1 ⊗ a2)); ∗ ∗ ∗ η : C → H , η (1)(a) = (a); ∗ ∗ ∗ : H → C, (φ) = φ(η(1)); S∗ : H∗ → H∗, S∗(φ)(a) = φ(S(a)).

∗ H = {φ|φ : H → C linear }