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Abelian extensions of D n in E n 1 Andrew Douglas City University of New York Based on arXiv:1305.6996v1 (J. Pure Appl. Algebra) with Kahrobaei, Repka and arXiv:1305.6946v1 with Repka Groups St Andrews 2013 A. Douglas (CUNY) D n V

  1. Abelian extensions of D n in E n � 1 Andrew Douglas City University of New York Based on arXiv:1305.6996v1 (J. Pure Appl. Algebra) with Kahrobaei, Repka and arXiv:1305.6946v1 with Repka Groups St Andrews 2013 A. Douglas (CUNY) D n � V ã Ñ E n � 1 1 / 22

  2. Introduction Let V be a finite-dimensional representation of the orthogonal Lie algebra D n . Then, we may define an abelian extension of D n by D n � V The representation action of D n on V defines a multiplication between elements of D n and V , and V is regarded as an abelian algebra: r V , V s ✏ 0 (or, more generally, V is solvable). A. Douglas (CUNY) D n � V ã Ñ E n � 1 2 / 22

  3. Introduction Let V be a finite-dimensional representation of the orthogonal Lie algebra D n . Then, we may define an abelian extension of D n by D n � V The representation action of D n on V defines a multiplication between elements of D n and V , and V is regarded as an abelian algebra: r V , V s ✏ 0 (or, more generally, V is solvable). A. Douglas (CUNY) D n � V ã Ñ E n � 1 2 / 22

  4. Introduction Let V be a finite-dimensional representation of the orthogonal Lie algebra D n . Then, we may define an abelian extension of D n by D n � V The representation action of D n on V defines a multiplication between elements of D n and V , and V is regarded as an abelian algebra: r V , V s ✏ 0 (or, more generally, V is solvable). A. Douglas (CUNY) D n � V ã Ñ E n � 1 2 / 22

  5. Introduction Let V be a finite-dimensional representation of the orthogonal Lie algebra D n . Then, we may define an abelian extension of D n by D n � V The representation action of D n on V defines a multiplication between elements of D n and V , and V is regarded as an abelian algebra: r V , V s ✏ 0 (or, more generally, V is solvable). A. Douglas (CUNY) D n � V ã Ñ E n � 1 2 / 22

  6. Objectives 1. Determine all abelian extensions of D n that may be embedded into the exceptional Lie algebra E n � 1 , n ✏ 5 , 6 , and 7 . 2. Classify D n � V ã Ñ E n � 1 , up to inner automorphism ( n ✏ 5 , 6 , 7 ). 3. Application to physics: Examine embedding of the GraviGUT algebra into the quaternionic real form of E 8 , which has been proposed in the physics literature. ( complexification of GraviGUT is D 7 � V , where V is a 64 -dimensional D 7 -irrep. Note: V is not an abelian ideal.) A. Douglas (CUNY) D n � V ã Ñ E n � 1 3 / 22

  7. Objectives 1. Determine all abelian extensions of D n that may be embedded into the exceptional Lie algebra E n � 1 , n ✏ 5 , 6 , and 7 . 2. Classify D n � V ã Ñ E n � 1 , up to inner automorphism ( n ✏ 5 , 6 , 7 ). 3. Application to physics: Examine embedding of the GraviGUT algebra into the quaternionic real form of E 8 , which has been proposed in the physics literature. ( complexification of GraviGUT is D 7 � V , where V is a 64 -dimensional D 7 -irrep. Note: V is not an abelian ideal.) A. Douglas (CUNY) D n � V ã Ñ E n � 1 3 / 22

  8. Objectives 1. Determine all abelian extensions of D n that may be embedded into the exceptional Lie algebra E n � 1 , n ✏ 5 , 6 , and 7 . 2. Classify D n � V ã Ñ E n � 1 , up to inner automorphism ( n ✏ 5 , 6 , 7 ). 3. Application to physics: Examine embedding of the GraviGUT algebra into the quaternionic real form of E 8 , which has been proposed in the physics literature. ( complexification of GraviGUT is D 7 � V , where V is a 64 -dimensional D 7 -irrep. Note: V is not an abelian ideal.) A. Douglas (CUNY) D n � V ã Ñ E n � 1 3 / 22

  9. Objectives 1. Determine all abelian extensions of D n that may be embedded into the exceptional Lie algebra E n � 1 , n ✏ 5 , 6 , and 7 . 2. Classify D n � V ã Ñ E n � 1 , up to inner automorphism ( n ✏ 5 , 6 , 7 ). 3. Application to physics: Examine embedding of the GraviGUT algebra into the quaternionic real form of E 8 , which has been proposed in the physics literature. ( complexification of GraviGUT is D 7 � V , where V is a 64 -dimensional D 7 -irrep. Note: V is not an abelian ideal.) A. Douglas (CUNY) D n � V ã Ñ E n � 1 3 / 22

  10. Objectives 1. Determine all abelian extensions of D n that may be embedded into the exceptional Lie algebra E n � 1 , n ✏ 5 , 6 , and 7 . 2. Classify D n � V ã Ñ E n � 1 , up to inner automorphism ( n ✏ 5 , 6 , 7 ). 3. Application to physics: Examine embedding of the GraviGUT algebra into the quaternionic real form of E 8 , which has been proposed in the physics literature. ( complexification of GraviGUT is D 7 � V , where V is a 64 -dimensional D 7 -irrep. Note: V is not an abelian ideal.) A. Douglas (CUNY) D n � V ã Ñ E n � 1 3 / 22

  11. Background The special orthogonal algebra D n is the Lie algebra of complex 2 n ✂ 2 n matrices N satisfying N tr ✏ ✁ N . The dimension of D n is 2 n 2 ✁ n and its rank is n . Besides the classical Lie algebras, which include D n , there are five exceptional Lie algebras, three of which are E 6 , E 7 , and E 8 . The algebras E 6 , E 7 , and E 8 have ranks 6 , 7 , and 8 , and dimensions 78 , 133 , and 248 , respectively. A. Douglas (CUNY) D n � V ã Ñ E n � 1 4 / 22

  12. Background The special orthogonal algebra D n is the Lie algebra of complex 2 n ✂ 2 n matrices N satisfying N tr ✏ ✁ N . The dimension of D n is 2 n 2 ✁ n and its rank is n . Besides the classical Lie algebras, which include D n , there are five exceptional Lie algebras, three of which are E 6 , E 7 , and E 8 . The algebras E 6 , E 7 , and E 8 have ranks 6 , 7 , and 8 , and dimensions 78 , 133 , and 248 , respectively. A. Douglas (CUNY) D n � V ã Ñ E n � 1 4 / 22

  13. Background The special orthogonal algebra D n is the Lie algebra of complex 2 n ✂ 2 n matrices N satisfying N tr ✏ ✁ N . The dimension of D n is 2 n 2 ✁ n and its rank is n . Besides the classical Lie algebras, which include D n , there are five exceptional Lie algebras, three of which are E 6 , E 7 , and E 8 . The algebras E 6 , E 7 , and E 8 have ranks 6 , 7 , and 8 , and dimensions 78 , 133 , and 248 , respectively. A. Douglas (CUNY) D n � V ã Ñ E n � 1 4 / 22

  14. Background The special orthogonal algebra D n is the Lie algebra of complex 2 n ✂ 2 n matrices N satisfying N tr ✏ ✁ N . The dimension of D n is 2 n 2 ✁ n and its rank is n . Besides the classical Lie algebras, which include D n , there are five exceptional Lie algebras, three of which are E 6 , E 7 , and E 8 . The algebras E 6 , E 7 , and E 8 have ranks 6 , 7 , and 8 , and dimensions 78 , 133 , and 248 , respectively. A. Douglas (CUNY) D n � V ã Ñ E n � 1 4 / 22

  15. Background The special orthogonal algebra D n is the Lie algebra of complex 2 n ✂ 2 n matrices N satisfying N tr ✏ ✁ N . The dimension of D n is 2 n 2 ✁ n and its rank is n . Besides the classical Lie algebras, which include D n , there are five exceptional Lie algebras, three of which are E 6 , E 7 , and E 8 . The algebras E 6 , E 7 , and E 8 have ranks 6 , 7 , and 8 , and dimensions 78 , 133 , and 248 , respectively. A. Douglas (CUNY) D n � V ã Ñ E n � 1 4 / 22

  16. Background Figure: Dynkin diagrams of D n , E 6 , E 7 , and E 8 . ✆ n ✁ 1 ⑤ ✆ 1 ✁✆ 2 ✁☎☎☎✁ n ✁ 2 ✁ ✆ ✆ D n n ✆ 2 ⑤ ✆ 1 ✁✆ 3 ✁ ✆ 4 ✁✆ 5 ✁✆ E 6 6 ✆ 2 ⑤ ✆ 1 ✁✆ 3 ✁ ✆ 4 ✁✆ 5 ✁✆ 6 ✁✆ E 7 7 ✆ 2 ⑤ E 8 ✆ 1 ✁✆ 3 ✁ ✆ 4 ✁✆ 5 ✁✆ 6 ✁✆ 7 ✁✆ 8 A. Douglas (CUNY) D n � V ã Ñ E n � 1 5 / 22

  17. Background Figure: Dynkin diagrams of D n , E 6 , E 7 , and E 8 . ✆ n ✁ 1 ⑤ ✆ 1 ✁✆ 2 ✁☎☎☎✁ n ✁ 2 ✁ ✆ ✆ D n n ✆ 2 ⑤ ✆ 1 ✁✆ 3 ✁ ✆ 4 ✁✆ 5 ✁✆ E 6 6 ✆ 2 ⑤ ✆ 1 ✁✆ 3 ✁ ✆ 4 ✁✆ 5 ✁✆ 6 ✁✆ E 7 7 ✆ 2 ⑤ E 8 ✆ 1 ✁✆ 3 ✁ ✆ 4 ✁✆ 5 ✁✆ 6 ✁✆ 7 ✁✆ 8 A. Douglas (CUNY) D n � V ã Ñ E n � 1 5 / 22

  18. Background Let g be a simple Lie algebra of rank k . We may define g by a set of generators t H i , X i , Y i ✉ 1 ↕ i ↕ k together with the Chevalley-Serre relations: r H i , H j s ✏ 0 , r H j , X i s ✏ M ij X i , r H j , Y i s ✏ ✁ M ij Y i , r X i , Y j s ✏ δ ij H i , ♣ ad X i q 1 ✁ M ji ♣ X j q ✏ 0 , when i ✘ j , ♣ ad Y i q 1 ✁ M ji ♣ Y j q ✏ 0 , when i ✘ j , 1 ↕ i , j ↕ k , and M is the Cartan matrix of g . A. Douglas (CUNY) D n � V ã Ñ E n � 1 6 / 22

  19. Background Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H 1 ,..., H k λ i P H ✝ , λ i ♣ H j q ✏ δ ij For each λ ✏ m 1 λ 1 � ... � m k λ k P H ✝ with nonnegative integers m 1 ,..., m k , there exists a finite-dimensional, irreducible g -module denoted V g ♣ λ q . The representations V g ♣ λ i q for 1 ↕ i ↕ k are the fundamental representations . A. Douglas (CUNY) D n � V ã Ñ E n � 1 7 / 22

  20. Background Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H 1 ,..., H k λ i P H ✝ , λ i ♣ H j q ✏ δ ij For each λ ✏ m 1 λ 1 � ... � m k λ k P H ✝ with nonnegative integers m 1 ,..., m k , there exists a finite-dimensional, irreducible g -module denoted V g ♣ λ q . The representations V g ♣ λ i q for 1 ↕ i ↕ k are the fundamental representations . A. Douglas (CUNY) D n � V ã Ñ E n � 1 7 / 22

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