Graded Lie Algebras and Representations of Supersymmetry Algebras Physics 251 Group Theory and Modern Physics Jaryd Franklin Ulbricht June 14, 2017 J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 1 / 50 Overview 1 Introduction and Definitions Linear Vector Spaces Linear Algebras Lie Algebras 2 Graded Algebras Graded Lie Algebras 3 Supersymmetry Lie Superalgebras Superspace Supersymmetry Algebras 4 Representations of Supersymmetry Algebras Boson and Fermion Number Constructing Massive Representations Constructing Massless Representations J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 2 / 50 How do we construct a graded algebra? It's actually much easier than you think Graded Lie Algebras What is a graded algebra? J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 3 / 50 It's actually much easier than you think Graded Lie Algebras What is a graded algebra? How do we construct a graded algebra? J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 3 / 50 Graded Lie Algebras What is a graded algebra? How do we construct a graded algebra? It's actually much easier than you think J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 3 / 50 Graded Lie Algebras su (2) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 4 / 50 Graded Lie Algebras su (2) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 4 / 50 Graded Lie Algebras so (3) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 5 / 50 Graded Lie Algebras so (3) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 5 / 50 Graded Lie Algebras E6 J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 6 / 50 Graded Lie Algebras E6 J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 6 / 50 ii Field F V is a linear vector space over the field F given the following definitions: Vector addition (+), Abelian operation such that ~vi + ~vj = ~vj + ~vi 2 V Scalar multiplication (·), c 2 F;~v 2 V ! c · ~v 2 V Introduction and Definitions Linear Vector Spaces Linear Vector Spaces i ~v0;~v1;~v2; · · · 2 V J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 7 / 50 V is a linear vector space over the field F given the following definitions: Vector addition (+), Abelian operation such that ~vi + ~vj = ~vj + ~vi 2 V Scalar multiplication (·), c 2 F;~v 2 V ! c · ~v 2 V Introduction and Definitions Linear Vector Spaces Linear Vector Spaces i ~v0;~v1;~v2; · · · 2 V ii Field F J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 7 / 50 Vector addition (+), Abelian operation such that ~vi + ~vj = ~vj + ~vi 2 V Scalar multiplication (·), c 2 F;~v 2 V ! c · ~v 2 V Introduction and Definitions Linear Vector Spaces Linear Vector Spaces i ~v0;~v1;~v2; · · · 2 V ii Field F V is a linear vector space over the field F given the following definitions: J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 7 / 50 Scalar multiplication (·), c 2 F;~v 2 V ! c · ~v 2 V Introduction and Definitions Linear Vector Spaces Linear Vector Spaces i ~v0;~v1;~v2; · · · 2 V ii Field F V is a linear vector space over the field F given the following definitions: Vector addition (+), Abelian operation such that ~vi + ~vj = ~vj + ~vi 2 V J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 7 / 50 Introduction and Definitions Linear Vector Spaces Linear Vector Spaces i ~v0;~v1;~v2; · · · 2 V ii Field F V is a linear vector space over the field F given the following definitions: Vector addition (+), Abelian operation such that ~vi + ~vj = ~vj + ~vi 2 V Scalar multiplication (·), c 2 F;~v 2 V ! c · ~v 2 V J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 7 / 50 ~vi × ~vj 2 g Closure ~vk × (~vi + ~vj) = ~vk × ~vi + ~vk × ~vj Distributive Property Other potential properties of the vector product ~vi × ~vj = ~vj × ~vi Commutativity ~vi × ~vj = −~vj × ~vi Anti-commutativity Introduction and Definitions Linear Algebras Linear Algebra A linear vector space V becomes a linear algebra g if we define the vector product (×) such that J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 8 / 50 ~vk × (~vi + ~vj) = ~vk × ~vi + ~vk × ~vj Distributive Property Other potential properties of the vector product ~vi × ~vj = ~vj × ~vi Commutativity ~vi × ~vj = −~vj × ~vi Anti-commutativity Introduction and Definitions Linear Algebras Linear Algebra A linear vector space V becomes a linear algebra g if we define the vector product (×) such that ~vi × ~vj 2 g Closure J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 8 / 50 Other potential properties of the vector product ~vi × ~vj = ~vj × ~vi Commutativity ~vi × ~vj = −~vj × ~vi Anti-commutativity Introduction and Definitions Linear Algebras Linear Algebra A linear vector space V becomes a linear algebra g if we define the vector product (×) such that ~vi × ~vj 2 g Closure ~vk × (~vi + ~vj) = ~vk × ~vi + ~vk × ~vj Distributive Property J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 8 / 50 ~vi × ~vj = ~vj × ~vi Commutativity ~vi × ~vj = −~vj × ~vi Anti-commutativity Introduction and Definitions Linear Algebras Linear Algebra A linear vector space V becomes a linear algebra g if we define the vector product (×) such that ~vi × ~vj 2 g Closure ~vk × (~vi + ~vj) = ~vk × ~vi + ~vk × ~vj Distributive Property Other potential properties of the vector product J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 8 / 50 ~vi × ~vj = −~vj × ~vi Anti-commutativity Introduction and Definitions Linear Algebras Linear Algebra A linear vector space V becomes a linear algebra g if we define the vector product (×) such that ~vi × ~vj 2 g Closure ~vk × (~vi + ~vj) = ~vk × ~vi + ~vk × ~vj Distributive Property Other potential properties of the vector product ~vi × ~vj = ~vj × ~vi Commutativity J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 8 / 50 Introduction and Definitions Linear Algebras Linear Algebra A linear vector space V becomes a linear algebra g if we define the vector product (×) such that ~vi × ~vj 2 g Closure ~vk × (~vi + ~vj) = ~vk × ~vi + ~vk × ~vj Distributive Property Other potential properties of the vector product ~vi × ~vj = ~vj × ~vi Commutativity ~vi × ~vj = −~vj × ~vi Anti-commutativity J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 8 / 50 [x; [y; z]] 6= [[x; y] ; z] x; y; z 2 g Non-Associative [x; x] = 0 x 2 g Alternating [ax + by; z] = a [x; z] + b [y; z] a; b 2 F x; y; z 2 g Bi-linearity [x; [y; z]] + [z; [x; y]] + [y; [z; x]] = 0 x; y; z 2 g Jacobi Identity Introduction and Definitions Lie Algebras Lie Algebra A linear vector space g over a field F where we define the vector product as a non-associative, alternating bilinear map g × g ! g denoted by the Lie Bracket [:; :] which obeys the Jacobi identity J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 9 / 50 [x; x] = 0 x 2 g Alternating [ax + by; z] = a [x; z] + b [y; z] a; b 2 F x; y; z 2 g Bi-linearity [x; [y; z]] + [z; [x; y]] + [y; [z; x]] = 0 x; y; z 2 g Jacobi Identity Introduction and Definitions Lie Algebras Lie Algebra A linear vector space g over a field F where we define the vector product as a non-associative, alternating bilinear map g × g ! g denoted by the Lie Bracket [:; :] which obeys the Jacobi identity [x; [y; z]] 6= [[x; y] ; z] x; y; z 2 g Non-Associative J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 9 / 50 [ax + by; z] = a [x; z] + b [y; z] a; b 2 F x; y; z 2 g Bi-linearity [x; [y; z]] + [z; [x; y]] + [y; [z; x]] = 0 x; y; z 2 g Jacobi Identity Introduction and Definitions Lie Algebras Lie Algebra A linear vector space g over a field F where we define the vector product as a non-associative, alternating bilinear map g × g ! g denoted by the Lie Bracket [:; :] which obeys the Jacobi identity [x; [y; z]] 6= [[x; y] ; z] x; y; z 2 g Non-Associative [x; x] = 0 x 2 g Alternating J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 9 / 50 [x; [y; z]] + [z; [x; y]] + [y; [z; x]] = 0 x; y; z 2 g Jacobi Identity Introduction and Definitions Lie Algebras Lie Algebra A linear vector space g over a field F where we define the vector product as a non-associative, alternating bilinear map g × g ! g denoted by the Lie Bracket [:; :] which obeys the Jacobi identity [x; [y; z]] 6= [[x; y] ; z] x; y; z 2 g Non-Associative [x; x] = 0 x 2 g Alternating [ax + by; z] = a [x; z] + b [y; z] a; b 2 F x; y; z 2 g Bi-linearity J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 9 / 50 Introduction and Definitions Lie Algebras Lie Algebra A linear vector space g over a field F where we define the vector product as a non-associative, alternating bilinear map g × g ! g denoted by the Lie Bracket [:; :] which obeys the Jacobi identity [x; [y; z]] 6= [[x; y] ; z] x; y; z 2 g Non-Associative [x; x] = 0 x 2 g Alternating [ax + by; z] = a [x; z] + b [y; z] a; b 2 F x; y; z 2 g Bi-linearity [x; [y; z]] + [z; [x; y]] + [y; [z; x]] = 0 x; y; z 2 g Jacobi Identity J.