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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 := fφjφ : V ! C linear map g. ∗ 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 := fφjφ : V ! C linear map g. ∗ 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. 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 ∗ 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. e1; ··· ; en : basis for V . ∗ V := fφjφ : V ! C linear map g. ∗ 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. e1; ··· ; en : basis for V . ∗ V := fφjφ : V ! C linear map g. 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 ∗ 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. e1; ··· ; en : basis for V . ∗ V := fφjφ : V ! C linear map g. ∗ hφi ; ej i := φi (ej ) := δij , φi ; ··· ; φn, dual basis for 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. e1; ··· ; en : basis for V . ∗ V := fφjφ : V ! C linear map g. ∗ 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 vector space V : a finite dimensional vector space over C. e1; ··· ; en : basis for V . ∗ V := fφjφ : V ! C linear map g. ∗ 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 ∗ 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 ∗∗ 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 ∗∗. Therefore, ΦV (e1); ··· ; ΦV (en) is the dual basis of φ1; ··· ; φn. Natural isomorphism ΦV V / V ∗∗ T T ∗∗ ΦW W / 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 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 ∗∗ 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. ΦV V / V ∗∗ T T ∗∗ ΦW W / 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 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 := fz 2 Cjjzj = 1g, note that this is also an abelian group 1 Gb = fφ : G ! S jφ(e) = 1; φ(g1g2) = φ(g1)φ(g2)g 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 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 Gb = fφ : G ! S jφ(e) = 1; φ(g1g2) = φ(g1)φ(g2)g 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 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 := fz 2 Cjjzj = 1g, 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 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 := fz 2 Cjjzj = 1g, note that this is also an abelian group 1 Gb = fφ : G ! S jφ(e) = 1; φ(g1g2) = φ(g1)φ(g2)g Product (φ1 · φ2)(g) := φ1(g)φ2(g) Inverse φ−1(g) := φ(g)−1 Identity φe (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 := fz 2 Cjjzj = 1g, note that this is also an abelian group 1 Gb = fφ : G ! S jφ(e) = 1; φ(g1g2) = φ(g1)φ(g2)g Proposition: Gb is a group, called Pontryagin dual of G. Inverse φ−1(g) := φ(g)−1 Identity φe (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 := fz 2 Cjjzj = 1g, note that this is also an abelian group 1 Gb = fφ : G ! S jφ(e) = 1; φ(g1g2) = φ(g1)φ(g2)g Proposition: Gb is a group, called Pontryagin dual of G.
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