Roundup on Lie Groups, Lie Algebras and Their Representations

UNIVERSITA` DEGLI STUDI DI TRIESTE DEPARTMENT OF PHYSICS

PH.D. COURSE OF THE 31st CYCLE IN PHYSICS Roundup on Lie groups, Lie algebras and their representations

Supervisor: Ph.D. Student: Bassi Angelo Bacchi Stefano

Academic Year 2016-2017 Matrix Lie group and algebra Lie group and algebra representations

Contents

1. Matrix Lie group and algebra 2. Lie group and algebra representations

Sophus Lie Hermann Weyl

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Matrix Lie group

Deﬁnition 1.1 (Group) A set G is a group (G, ) when: • : G × G → G s.t. a (b c) = (a b) c ∀ a, b, c ∈ G; • ∃ e, a−1 ∈ G s.t. e a = a e = a, a−1 · a = a a−1 = e ∀ a ∈ G.

e.g. (Z, +). Deﬁnition 1.2 (Lie Group) (G, ·) is a Lie group when it is a smooth manifold s.t. (a, b) → a−1 · b is smooth for every a, b ∈ G. (G, ·) is a matrix Lie group if it is a closed subgroup of GL(n, C). e.g. Heisenberg group, Galilean group, Poincar`egroup.

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Matrix Lie algebra

Deﬁnition 1.3 (Algebra)

A vector space (V , ) over the ﬁeld K becomes an algebra (V , , ) over K, where : V × V → V is a bilinear operation. 3 e.g. (R , +, ×). Deﬁnition 1.4 (Lie Algebra)

An algebra (V , +, ·) over the ﬁeld K becomes a Lie algebra over K when · is a Lie bracket, i.e.: • a · a = 0 ∀ a ∈ V or a · b = −b · a ∀ a, b ∈ V; • a · (b · c) + c · (a · b) + b · (a · c) = 0 ∀ a, b, c ∈ V (Jacobi identity). Let G be a matrix Lie group, its Lie algebra g is the set of all matrices M tM s.t. e ∈ G for all t ∈ R. e.g. (su(2), [, ]).

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Connected Lie group

Deﬁnition 1.5 (Connectedness) A matrix Lie group G is connected if every A, B ∈ G are connected by a continuous path in G.

e.g. SO(3).

Deﬁnition 1.6 (Simple connectedness) A matrix Lie group G is simply connected if it is connected and every loop in G can be shrunk continuously to a point in G.

e.g. SU(2).

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Finite dimensional representations

Deﬁnition 2.1 (Representation) matrix Lie group G A ﬁnite-dimensional / representation of a is a R C matrix Lie algebra g

Φ: G → GL(V ) Φ(AB) = Φ(A)Φ(B) continuous map s.t. where V is a φ : g → gl(V ) φ([a, b]) = [φ(a), φ(b)]

ﬁnite dimensional R/C vector space.

Φ(eX ) = eφ(X ) ∀ X ∈ g

⇐======G simply connected Φ ======⇒ φ

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Universal covering group

Deﬁnition 2.2 (Universal covering) Let G be a connected Lie group, a universal covering group of G is a simply connected group H together with a group homomorphism Φ: H → G s.t. the associated Lie algebra homomorphism φ : h → g is a Lie algebra isomorphism.

e.g. SU(2) universal covering of SO(3)

⇓

Φ: SU(2) → SO(3), φ : su(2) → so(3) s.t. X φ(X ) Φ(e ) = e ∀ X ∈ su(2) , su(2) ∼= so(3).

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Inﬁnite dimensional representation

Deﬁnition 2.3 G is a connected Lie group, g, g 0 ∈ G, H is an Hilbert space, 0 ρ : g → Ug ∈ B(H) s.t. Ug Ug 0 = w(g, g )Ugg 0 and Ug is unitary then:

• if |w(g, g 0)| = 1 ∀ g, g 0 then ρ is a PUR;

• if w(g, g 0) = 1 ∀ g, g 0 then ρ is a UR.

Galilean group has only PUR on H.

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Central Extension

Deﬁnition 2.4 (CE)

The central extension of G is Gw = U(1) × G with product:

(φ, g)(φ0, g 0) = (φφ0w(g, g 0), gg 0) φ, φ0 ∈ U(1)

Bargmann UR of CE of Gal. group =====⇒ PUR of Gal. group.

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Take home message

Repres. of conn. Lie group G (Finite transf.).

⇑

Repres. of universal covering group of G.

m Repres. of Lie algebra g (Inﬁnitesimal transf., Generators/Observables).

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