Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Lie Theory From Basics to the Heisenberg Lie Group Noah Migoski IU Math DRP April, 2020 Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Group? From Artin [1]: ”A group is a set G together with a law of composition (a map from two elements to another element in the set) that has the following properties”: The law of composition is associative: (ab)c = a(bc) for all a, b, c in G. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Group? From Artin [1]: ”A group is a set G together with a law of composition (a map from two elements to another element in the set) that has the following properties”: The law of composition is associative: (ab)c = a(bc) for all a, b, c in G. G contains an identity element 1, such that 1a = a and a1 = a for all a in G. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Group? From Artin [1]: ”A group is a set G together with a law of composition (a map from two elements to another element in the set) that has the following properties”: The law of composition is associative: (ab)c = a(bc) for all a, b, c in G. G contains an identity element 1, such that 1a = a and a1 = a for all a in G. Every element a of G has an inverse, an element b such that ab = 1 and ba = 1. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples of Groups Examples Z + : The set of integers with additon as its law of composition. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples of Groups Examples Z + : The set of integers with additon as its law of composition. R + , R × , C + , C × : The real or complex numbers under addition or multiplication (with zero removed). Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples of Groups Examples Z + : The set of integers with additon as its law of composition. R + , R × , C + , C × : The real or complex numbers under addition or multiplication (with zero removed). Cyclic Groups: The rotational symmetry of a regular polygon Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples of Groups Examples Z + : The set of integers with additon as its law of composition. R + , R × , C + , C × : The real or complex numbers under addition or multiplication (with zero removed). Cyclic Groups: The rotational symmetry of a regular polygon The Dihedral Groups: The symmetries of a regular polygon including reflections. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples of Groups Examples Z + : The set of integers with additon as its law of composition. R + , R × , C + , C × : The real or complex numbers under addition or multiplication (with zero removed). Cyclic Groups: The rotational symmetry of a regular polygon The Dihedral Groups: The symmetries of a regular polygon including reflections. The symmetric groups: The permutations of a set of n elements Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Lie Group? Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Lie Group? “A Lie group is, roughly speaking, a continuous group” [1]. A more concrete definition is the following: Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Lie Group? “A Lie group is, roughly speaking, a continuous group” [1]. A more concrete definition is the following: Lie Group Definition A group (which is also a manifold) G such that the group product G × G → G and the inverse map G → G are smooth. Meaning they are infinitely differentiable. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples U(n): The unitary groups. (n by n complex matricies satisfying: X ∗ X = XX ∗ = I ) Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples U(n): The unitary groups. (n by n complex matricies satisfying: X ∗ X = XX ∗ = I ) O(n): The orthogonal groups. (n by n real matricies satisfying: X T X = XX T = I ) Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Examples U(n): The unitary groups. (n by n complex matricies satisfying: X ∗ X = XX ∗ = I ) O(n): The orthogonal groups. (n by n real matricies satisfying: X T X = XX T = I ) SU(n) and SO(n): The untitary or orthogonal groups with the added condition that det(X) = 1. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Lie Algebra? Lie Algebra Definition A Lie algebra is a finite-dimensional real or complex vector space g , together with a map[ · , · ] from g × g → g , satisfying: [ · , · ] is bilinear. A Lie algebra can be thought of as the tangent space at the identity of a Lie group. They are connected by the exponential map. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Lie Algebra? Lie Algebra Definition A Lie algebra is a finite-dimensional real or complex vector space g , together with a map[ · , · ] from g × g → g , satisfying: [ · , · ] is bilinear. [ · , · ] is skew symmetric: [ X , Y ] = − [ Y , X ] ∀ X , Y ∈ g A Lie algebra can be thought of as the tangent space at the identity of a Lie group. They are connected by the exponential map. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra What is a Lie Algebra? Lie Algebra Definition A Lie algebra is a finite-dimensional real or complex vector space g , together with a map[ · , · ] from g × g → g , satisfying: [ · , · ] is bilinear. [ · , · ] is skew symmetric: [ X , Y ] = − [ Y , X ] ∀ X , Y ∈ g The Jacobi identity holds: [ X , [ Y , Z ]] + [ Y , [ Z , X ]] + [ Z , [ X , Y ]] = 0 A Lie algebra can be thought of as the tangent space at the identity of a Lie group. They are connected by the exponential map. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra The Exponential Map Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra The Exponential Map Recall the Taylor series for e x : ∞ x m e x = � m ! m =0 Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra The Exponential Map Recall the Taylor series for e x : ∞ x m e x = � m ! m =0 It can be shown that this sum also converges when x is an n × n matrix sufficiently close to the identity. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra The Exponential Map Recall the Taylor series for e x : ∞ x m e x = � m ! m =0 It can be shown that this sum also converges when x is an n × n matrix sufficiently close to the identity. If G is a matrix Lie group with Lie algebra g , then the exponential map for G is the map exp : g → G : X �→ e tX for t ∈ R Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Representations Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Representations A representation can be thought of as a linear action of a group or Lie algebra on a vector space. Representations as matricies allow us to more easily do computations in order to observe the properties of more abstract algebraic structures. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Representations A representation can be thought of as a linear action of a group or Lie algebra on a vector space. Representations as matricies allow us to more easily do computations in order to observe the properties of more abstract algebraic structures. Representation Definition Let G be a group. A representation of G is a homomorphism Π : G → GL ( V ) Where V is a vector space. Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Representations Noah Migoski Lie Theory
Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra Representations There is a natural connection between homomorphisms in Lie groups and homomorphisms in their corresponding Lie algebras. Because of this the representations of a Lie group tell us about the representations of its Lie algebra. Noah Migoski Lie Theory
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