[PPT] - Introduction Let V be a finite-dimensional representation of the PowerPoint Presentation

SLIDE 1

Abelian extensions of Dn in En1

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)

Dn V ã Ñ En1 1 / 22

SLIDE 2

Introduction

Let V be a finite-dimensional representation of the orthogonal Lie algebra Dn. Then, we may define an abelian extension of Dn by Dn V The representation action of Dn on V defines a multiplication between elements of Dn and V, and V is regarded as an abelian algebra: rV,Vs ✏ 0 (or, more generally, V is solvable).

A. Douglas (CUNY)

Dn V ã Ñ En1 2 / 22

SLIDE 3

Introduction

Let V be a finite-dimensional representation of the orthogonal Lie algebra Dn. Then, we may define an abelian extension of Dn by Dn V The representation action of Dn on V defines a multiplication between elements of Dn and V, and V is regarded as an abelian algebra: rV,Vs ✏ 0 (or, more generally, V is solvable).

A. Douglas (CUNY)

Dn V ã Ñ En1 2 / 22

SLIDE 4

Introduction

Let V be a finite-dimensional representation of the orthogonal Lie algebra Dn. Then, we may define an abelian extension of Dn by Dn V The representation action of Dn on V defines a multiplication between elements of Dn and V, and V is regarded as an abelian algebra: rV,Vs ✏ 0 (or, more generally, V is solvable).

A. Douglas (CUNY)

Dn V ã Ñ En1 2 / 22

SLIDE 5

Introduction

Let V be a finite-dimensional representation of the orthogonal Lie algebra Dn. Then, we may define an abelian extension of Dn by Dn V The representation action of Dn on V defines a multiplication between elements of Dn and V, and V is regarded as an abelian algebra: rV,Vs ✏ 0 (or, more generally, V is solvable).

A. Douglas (CUNY)

Dn V ã Ñ En1 2 / 22

SLIDE 6

Objectives

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7).

3. Application to physics: Examine embedding of the GraviGUT

algebra into the quaternionic real form of E8, which has been proposed in the physics literature. ( complexification of GraviGUT is D7 V, where V is a 64-dimensional D7-irrep. Note: V is not an abelian ideal.)

A. Douglas (CUNY)

Dn V ã Ñ En1 3 / 22

SLIDE 7

Objectives

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7).

3. Application to physics: Examine embedding of the GraviGUT

algebra into the quaternionic real form of E8, which has been proposed in the physics literature. ( complexification of GraviGUT is D7 V, where V is a 64-dimensional D7-irrep. Note: V is not an abelian ideal.)

A. Douglas (CUNY)

Dn V ã Ñ En1 3 / 22

SLIDE 8

Objectives

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7).

3. Application to physics: Examine embedding of the GraviGUT

algebra into the quaternionic real form of E8, which has been proposed in the physics literature. ( complexification of GraviGUT is D7 V, where V is a 64-dimensional D7-irrep. Note: V is not an abelian ideal.)

A. Douglas (CUNY)

Dn V ã Ñ En1 3 / 22

SLIDE 9

Objectives

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7).

3. Application to physics: Examine embedding of the GraviGUT

algebra into the quaternionic real form of E8, which has been proposed in the physics literature. ( complexification of GraviGUT is D7 V, where V is a 64-dimensional D7-irrep. Note: V is not an abelian ideal.)

A. Douglas (CUNY)

Dn V ã Ñ En1 3 / 22

SLIDE 10

Objectives

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7).

3. Application to physics: Examine embedding of the GraviGUT

algebra into the quaternionic real form of E8, which has been proposed in the physics literature. ( complexification of GraviGUT is D7 V, where V is a 64-dimensional D7-irrep. Note: V is not an abelian ideal.)

A. Douglas (CUNY)

Dn V ã Ñ En1 3 / 22

SLIDE 11

Background

The special orthogonal algebra Dn is the Lie algebra of complex 2n✂2n matrices N satisfying Ntr ✏ ✁N. The dimension of Dn is 2n2 ✁n and its rank is n. Besides the classical Lie algebras, which include Dn, there are five exceptional Lie algebras, three of which are E6, E7, and E8. The algebras E6, E7, and E8 have ranks 6, 7, and 8, and dimensions 78, 133, and 248, respectively.

A. Douglas (CUNY)

Dn V ã Ñ En1 4 / 22

SLIDE 12

Background

The special orthogonal algebra Dn is the Lie algebra of complex 2n✂2n matrices N satisfying Ntr ✏ ✁N. The dimension of Dn is 2n2 ✁n and its rank is n. Besides the classical Lie algebras, which include Dn, there are five exceptional Lie algebras, three of which are E6, E7, and E8. The algebras E6, E7, and E8 have ranks 6, 7, and 8, and dimensions 78, 133, and 248, respectively.

A. Douglas (CUNY)

Dn V ã Ñ En1 4 / 22

SLIDE 13

Background

The special orthogonal algebra Dn is the Lie algebra of complex 2n✂2n matrices N satisfying Ntr ✏ ✁N. The dimension of Dn is 2n2 ✁n and its rank is n. Besides the classical Lie algebras, which include Dn, there are five exceptional Lie algebras, three of which are E6, E7, and E8. The algebras E6, E7, and E8 have ranks 6, 7, and 8, and dimensions 78, 133, and 248, respectively.

A. Douglas (CUNY)

Dn V ã Ñ En1 4 / 22

SLIDE 14

Background

The special orthogonal algebra Dn is the Lie algebra of complex 2n✂2n matrices N satisfying Ntr ✏ ✁N. The dimension of Dn is 2n2 ✁n and its rank is n. Besides the classical Lie algebras, which include Dn, there are five exceptional Lie algebras, three of which are E6, E7, and E8. The algebras E6, E7, and E8 have ranks 6, 7, and 8, and dimensions 78, 133, and 248, respectively.

A. Douglas (CUNY)

Dn V ã Ñ En1 4 / 22

SLIDE 15

Background

The special orthogonal algebra Dn is the Lie algebra of complex 2n✂2n matrices N satisfying Ntr ✏ ✁N. The dimension of Dn is 2n2 ✁n and its rank is n. Besides the classical Lie algebras, which include Dn, there are five exceptional Lie algebras, three of which are E6, E7, and E8. The algebras E6, E7, and E8 have ranks 6, 7, and 8, and dimensions 78, 133, and 248, respectively.

A. Douglas (CUNY)

Dn V ã Ñ En1 4 / 22

SLIDE 16

Background

Figure: Dynkin diagrams of Dn, E6, E7, and E8.

Dn ✆

1 ✁✆ 2 ✁☎☎☎✁

✆n✁1 ⑤ ✆

n✁2✁ ✆ n

E6 ✆

1 ✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁✆ 6

E7 ✆

1 ✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁✆ 6 ✁✆ 7

E8 ✆

1 ✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁✆ 6 ✁✆ 7 ✁✆ 8

A. Douglas (CUNY)

Dn V ã Ñ En1 5 / 22

SLIDE 17

Background

Figure: Dynkin diagrams of Dn, E6, E7, and E8.

Dn ✆

1 ✁✆ 2 ✁☎☎☎✁

✆n✁1 ⑤ ✆

n✁2✁ ✆ n

E6 ✆

1 ✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁✆ 6

E7 ✆

1 ✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁✆ 6 ✁✆ 7

E8 ✆

1 ✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁✆ 6 ✁✆ 7 ✁✆ 8

A. Douglas (CUNY)

Dn V ã Ñ En1 5 / 22

SLIDE 18

Background

Let g be a simple Lie algebra of rank k. We may define g by a set of generators tHi,Xi,Yi✉1↕i↕k together with the Chevalley-Serre relations: rHi,Hjs ✏ 0, rHj,Xis ✏ MijXi, rHj,Yis ✏ ✁MijYi, rXi,Yjs ✏ δijHi, ♣ad Xiq1✁Mji♣Xjq ✏ 0, when i ✘ j, ♣ad Yiq1✁Mji♣Yjq ✏ 0, when i ✘ j, 1 ↕ i,j ↕ k, and M is the Cartan matrix of g.

A. Douglas (CUNY)

Dn V ã Ñ En1 6 / 22

SLIDE 19

Background

Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H1,...,Hk λi P H✝, λi♣Hjq ✏ δij For each λ ✏ m1λ1 ...mkλk P H✝ with nonnegative integers m1,...,mk, there exists a finite-dimensional, irreducible g-module denoted Vg♣λq. The representations Vg♣λiq for 1 ↕ i ↕ k are the fundamental representations.

A. Douglas (CUNY)

Dn V ã Ñ En1 7 / 22

SLIDE 20

Background

Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H1,...,Hk λi P H✝, λi♣Hjq ✏ δij For each λ ✏ m1λ1 ...mkλk P H✝ with nonnegative integers m1,...,mk, there exists a finite-dimensional, irreducible g-module denoted Vg♣λq. The representations Vg♣λiq for 1 ↕ i ↕ k are the fundamental representations.

A. Douglas (CUNY)

Dn V ã Ñ En1 7 / 22

SLIDE 21

Background

Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H1,...,Hk λi P H✝, λi♣Hjq ✏ δij For each λ ✏ m1λ1 ...mkλk P H✝ with nonnegative integers m1,...,mk, there exists a finite-dimensional, irreducible g-module denoted Vg♣λq. The representations Vg♣λiq for 1 ↕ i ↕ k are the fundamental representations.

A. Douglas (CUNY)

Dn V ã Ñ En1 7 / 22

SLIDE 22

Background

Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H1,...,Hk λi P H✝, λi♣Hjq ✏ δij For each λ ✏ m1λ1 ...mkλk P H✝ with nonnegative integers m1,...,mk, there exists a finite-dimensional, irreducible g-module denoted Vg♣λq. The representations Vg♣λiq for 1 ↕ i ↕ k are the fundamental representations.

A. Douglas (CUNY)

Dn V ã Ñ En1 7 / 22

SLIDE 23

Background

Finite-dimensional, irreducible representations of g H the Cartan subalgebra with basis H1,...,Hk λi P H✝, λi♣Hjq ✏ δij For each λ ✏ m1λ1 ...mkλk P H✝ with nonnegative integers m1,...,mk, there exists a finite-dimensional, irreducible g-module denoted Vg♣λq. The representations Vg♣λiq for 1 ↕ i ↕ k are the fundamental representations.

A. Douglas (CUNY)

Dn V ã Ñ En1 7 / 22

SLIDE 24

Recall Objectives 1 and 2:

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7). We first consider Dn ã Ñ En1. Dynkin, Losev, Maltsev, Minchenko.

A. Douglas (CUNY)

Dn V ã Ñ En1 8 / 22

SLIDE 25

Recall Objectives 1 and 2:

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7). We first consider Dn ã Ñ En1. Dynkin, Losev, Maltsev, Minchenko.

A. Douglas (CUNY)

Dn V ã Ñ En1 8 / 22

SLIDE 26

Recall Objectives 1 and 2:

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7). We first consider Dn ã Ñ En1. Dynkin, Losev, Maltsev, Minchenko.

A. Douglas (CUNY)

Dn V ã Ñ En1 8 / 22

SLIDE 27

Recall Objectives 1 and 2:

1. Determine all abelian extensions of Dn that may be embedded into

the exceptional Lie algebra En1, n ✏ 5,6, and 7.

2. Classify Dn V ã

Ñ En1, up to inner automorphism (n ✏ 5,6,7). We first consider Dn ã Ñ En1. Dynkin, Losev, Maltsev, Minchenko.

A. Douglas (CUNY)

Dn V ã Ñ En1 8 / 22

SLIDE 28

Dn ã Ñ En1

Example 1: ϕn : Dn ã Ñ En1 Dn ✆

n ✁

✆n✁1 ⑤ ✆

n✁2✁ ☎☎☎✁✆ 2 ✁✆ 1

En1 ✆

1✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁☎☎☎✁✆ n1

A. Douglas (CUNY)

Dn V ã Ñ En1 9 / 22

SLIDE 29

Dn ã Ñ En1

Example 2: ρn : Dn ã Ñ En1 Dn ✆

n ✁

✆n✁1 ⑤ ✆

n✁2✁ ☎☎☎✁✆ 2 ✁✆ 1

Dn ✆

n✁1

✁ ✆n ⑤ ✆

n✁2✁ ☎☎☎✁✆ 2 ✁✆ 1

En1 ✆

1✁✆ 3 ✁

✆2 ⑤ ✆

4 ✁✆ 5 ✁☎☎☎✁✆ n1

A. Douglas (CUNY)

Dn V ã Ñ En1 10 / 22

SLIDE 30

Theorem

Dn ã Ñ En1 Embeddings D7 ã Ñ E8 ϕ7 D6 ã Ñ E7 ϕ6, ρ6 D5 ã Ñ E6 ϕ5, ρ5

A. Douglas (CUNY)

Dn V ã Ñ En1 11 / 22

SLIDE 31

We now need to determine Dn V in En1. This is done with respect to each embedding of Dn ã Ñ En1.

A. Douglas (CUNY)

Dn V ã Ñ En1 12 / 22

SLIDE 32

We now need to determine Dn V in En1. This is done with respect to each embedding of Dn ã Ñ En1.

A. Douglas (CUNY)

Dn V ã Ñ En1 12 / 22

SLIDE 33

D7 V in E8

E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q ✕ϕ7♣D7q rX✶sϕ7♣D7q ❵ rX✸sϕ7♣D7q ❵ rY1sϕ7♣D7q ❵ rX✷sϕ7♣D7q ❵ rY✶sϕ7♣D7q ❵ rHsϕ7♣D7q where X✶ ✏ X4,5,6,7,8,2,3,4,5,6,7, X✷ ✏ ✁X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2, X✸ ✏ X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8, Y✶ ✏ ✁Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1, H ✏ 4H1 5H2 7H3 10H4 8H5 6H6 4H7 2H8.

A. Douglas (CUNY)

Dn V ã Ñ En1 13 / 22

SLIDE 34

D7 V in E8

E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q ✕ϕ7♣D7q rX✶sϕ7♣D7q ❵ rX✸sϕ7♣D7q ❵ rY1sϕ7♣D7q ❵ rX✷sϕ7♣D7q ❵ rY✶sϕ7♣D7q ❵ rHsϕ7♣D7q where X✶ ✏ X4,5,6,7,8,2,3,4,5,6,7, X✷ ✏ ✁X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2, X✸ ✏ X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8, Y✶ ✏ ✁Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1, H ✏ 4H1 5H2 7H3 10H4 8H5 6H6 4H7 2H8.

A. Douglas (CUNY)

Dn V ã Ñ En1 13 / 22

SLIDE 35

D7 V in E8

E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q ✕ϕ7♣D7q rX✶sϕ7♣D7q ❵ rX✸sϕ7♣D7q ❵ rY1sϕ7♣D7q ❵ rX✷sϕ7♣D7q ❵ rY✶sϕ7♣D7q ❵ rHsϕ7♣D7q where X✶ ✏ X4,5,6,7,8,2,3,4,5,6,7, X✷ ✏ ✁X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2, X✸ ✏ X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8, Y✶ ✏ ✁Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1, H ✏ 4H1 5H2 7H3 10H4 8H5 6H6 4H7 2H8.

A. Douglas (CUNY)

Dn V ã Ñ En1 13 / 22

SLIDE 36

D7 V in E8

E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q ✕ϕ7♣D7q rX✶sϕ7♣D7q ❵ rX✸sϕ7♣D7q ❵ rY1sϕ7♣D7q ❵ rX✷sϕ7♣D7q ❵ rY✶sϕ7♣D7q ❵ rHsϕ7♣D7q where X✶ ✏ X4,5,6,7,8,2,3,4,5,6,7, X✷ ✏ ✁X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2, X✸ ✏ X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8, Y✶ ✏ ✁Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1, H ✏ 4H1 5H2 7H3 10H4 8H5 6H6 4H7 2H8.

A. Douglas (CUNY)

Dn V ã Ñ En1 13 / 22

SLIDE 37

D7 V in E8

E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q ✕ϕ7♣D7q rX✶sϕ7♣D7q ❵ rX✸sϕ7♣D7q ❵ rY1sϕ7♣D7q ❵ rX✷sϕ7♣D7q ❵ rY✶sϕ7♣D7q ❵ rHsϕ7♣D7q where X✶ ✏ X4,5,6,7,8,2,3,4,5,6,7, X✷ ✏ ✁X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2, X✸ ✏ X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8, Y✶ ✏ ✁Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1, H ✏ 4H1 5H2 7H3 10H4 8H5 6H6 4H7 2H8.

A. Douglas (CUNY)

Dn V ã Ñ En1 13 / 22

SLIDE 38

D7 V in E8

E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q ✕ϕ7♣D7q rX✶sϕ7♣D7q ❵ rX✸sϕ7♣D7q ❵ rY1sϕ7♣D7q ❵ rX✷sϕ7♣D7q ❵ rY✶sϕ7♣D7q ❵ rHsϕ7♣D7q where X✶ ✏ X4,5,6,7,8,2,3,4,5,6,7, X✷ ✏ ✁X3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2, X✸ ✏ X8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3,4,5,6,7,8, Y✶ ✏ ✁Y5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1, H ✏ 4H1 5H2 7H3 10H4 8H5 6H6 4H7 2H8.

A. Douglas (CUNY)

Dn V ã Ñ En1 14 / 22

SLIDE 39

D7 V in E8

Theorem

The only abelian extensions of D7 that may be embedded into E8 are the following: D7 VD7♣0q and D7 VD7♣λ1q D7 C and D7 C14

A. Douglas (CUNY)

Dn V ã Ñ En1 15 / 22

SLIDE 40

D7 V in E8

Theorem

The only abelian extensions of D7 that may be embedded into E8 are the following: D7 VD7♣0q and D7 VD7♣λ1q D7 C and D7 C14

A. Douglas (CUNY)

Dn V ã Ñ En1 15 / 22

SLIDE 41

Dn V in En1

The only abelian extensions of D7 that may be embedded into E8 are the following: D7 C and D7 C14 The only abelian extension of D6 that may be embedded into E7 is the following: D6 C The only abelian extensions of D5 that may be embedded into E6 are the following: D5 C, D5 VD5♣λ4q, and D5 VD5♣λ5q

A. Douglas (CUNY)

Dn V ã Ñ En1 16 / 22

SLIDE 42

Dn V in En1

The only abelian extensions of D7 that may be embedded into E8 are the following: D7 C and D7 C14 The only abelian extension of D6 that may be embedded into E7 is the following: D6 C The only abelian extensions of D5 that may be embedded into E6 are the following: D5 C, D5 VD5♣λ4q, and D5 VD5♣λ5q

A. Douglas (CUNY)

Dn V ã Ñ En1 16 / 22

SLIDE 43

Dn V in En1

The only abelian extensions of D7 that may be embedded into E8 are the following: D7 C and D7 C14 The only abelian extension of D6 that may be embedded into E7 is the following: D6 C The only abelian extensions of D5 that may be embedded into E6 are the following: D5 C, D5 VD5♣λ4q, and D5 VD5♣λ5q

A. Douglas (CUNY)

Dn V ã Ñ En1 16 / 22

SLIDE 44

Dn V in En1

The only abelian extensions of D7 that may be embedded into E8 are the following: D7 C and D7 C14 The only abelian extension of D6 that may be embedded into E7 is the following: D6 C The only abelian extensions of D5 that may be embedded into E6 are the following: D5 C, D5 VD5♣λ4q, and D5 VD5♣λ5q

A. Douglas (CUNY)

Dn V ã Ñ En1 16 / 22

SLIDE 45

D7 V ã Ñ E8

All possible lifts of the embedding ϕ7 to D7 C14 or D7 C, resp.: ⑨ ϕ7

λ1,α :

D7 C14 ã Ñ E8 u ÞÑ αX✸ ⑨ ϕ7

λ ✶

1,α :

D7 C14 ã Ñ E8 u ÞÑ αY✶ ⑨ ϕ7

0,α :

D7 C ã Ñ E8 u ÞÑ α H

Table: Classification of embeddings of abelian extensions of D7 into E8.

Abelian Extension D7 V Embedding D7 V ã Ñ E8 D7 C14 ⑨ ϕ7

λ1,1,

⑨ ϕ7

λ ✶

1,1

D7 C ⑨ ϕ7

0,α,

α P C✝

A. Douglas (CUNY)

Dn V ã Ñ En1 17 / 22

SLIDE 46

D7 V ã Ñ E8

All possible lifts of the embedding ϕ7 to D7 C14 or D7 C, resp.: ⑨ ϕ7

λ1,α :

D7 C14 ã Ñ E8 u ÞÑ αX✸ ⑨ ϕ7

λ ✶

1,α :

D7 C14 ã Ñ E8 u ÞÑ αY✶ ⑨ ϕ7

0,α :

D7 C ã Ñ E8 u ÞÑ α H

Table: Classification of embeddings of abelian extensions of D7 into E8.

Abelian Extension D7 V Embedding D7 V ã Ñ E8 D7 C14 ⑨ ϕ7

λ1,1,

⑨ ϕ7

λ ✶

1,1

D7 C ⑨ ϕ7

0,α,

α P C✝

A. Douglas (CUNY)

Dn V ã Ñ En1 17 / 22

SLIDE 47

D7 V ã Ñ E8

All possible lifts of the embedding ϕ7 to D7 C14 or D7 C, resp.: ⑨ ϕ7

λ1,α :

D7 C14 ã Ñ E8 u ÞÑ αX✸ ⑨ ϕ7

λ ✶

1,α :

D7 C14 ã Ñ E8 u ÞÑ αY✶ ⑨ ϕ7

0,α :

D7 C ã Ñ E8 u ÞÑ α H

Table: Classification of embeddings of abelian extensions of D7 into E8.

Abelian Extension D7 V Embedding D7 V ã Ñ E8 D7 C14 ⑨ ϕ7

λ1,1,

⑨ ϕ7

λ ✶

1,1

D7 C ⑨ ϕ7

0,α,

α P C✝

A. Douglas (CUNY)

Dn V ã Ñ En1 17 / 22

SLIDE 48

D7 V ã Ñ E8

All possible lifts of the embedding ϕ7 to D7 C14 or D7 C, resp.: ⑨ ϕ7

λ1,α :

D7 C14 ã Ñ E8 u ÞÑ αX✸ ⑨ ϕ7

λ ✶

1,α :

D7 C14 ã Ñ E8 u ÞÑ αY✶ ⑨ ϕ7

0,α :

D7 C ã Ñ E8 u ÞÑ α H

Table: Classification of embeddings of abelian extensions of D7 into E8.

Abelian Extension D7 V Embedding D7 V ã Ñ E8 D7 C14 ⑨ ϕ7

λ1,1,

⑨ ϕ7

λ ✶

1,1

D7 C ⑨ ϕ7

0,α,

α P C✝

A. Douglas (CUNY)

Dn V ã Ñ En1 17 / 22

SLIDE 49

Summary Dn V ã Ñ En1

Abelian Extension D7 V Embedding D7 V ã Ñ E8 D7 C14 ⑨ ϕ7

λ1,1,

⑨ ϕ7

λ ✶

1,1

D7 C ⑨ ϕ7

0,α,

α P C✝

Theorem

The only abelian extension of D6 that may be embedded into E7 is D6 VD6 ♣0q ✕ D6 C. Any embedding of D6 C into E7 is equivalent to ⑨ ϕ6

0,♣α,β,γq or ⑨

ρ6

0,♣α,β,γq for some α,β,γ P C, not all zero. For α,β,γ P C, not all zero, the embeddings are classified

according to the following rules: (a) ⑨ ϕ6

0,♣α,β,γq ✚ ⑨

ρ6

0,♣α,β,γq for all α,β,γ.

(b) ⑨ ϕ6

0,♣α,β,γq ✒ ⑨

ϕ6

0,♣α✶,β✶,γ✶q ô

♣α✶,β✶,γ✶q ✏ ♣αc2 ✁βb2 ✁2γbc,✁αd2 βa2 2γad,✁αcdβabγ♣acbdqq, for some a,b,c,d P C, such that ac✁bd ✏ 1. (c) ⑨ ρ6

0,♣α,β,γq ✒ ⑨

ρ6

0,♣α✶,β✶,γ✶q ô

♣α✶,β✶,γ✶q ✏ ♣αc2 ✁βb2 ✁2γbc,✁αd2 βa2 2γad,✁αcdβabγ♣acbdqq, for some a,b,c,d P C, such that ac✁bd ✏ 1.

Abelian Extension D5 V Embedding D5 V ã Ñ E6 D5 VD5♣λ5q ⑨ ϕ5

λ5,1,

r ρ5

λ5,1

D5 VD5♣λ4q ⑨ ϕ5

λ4,1,

r ρ5

λ4,1

D5 VD5♣0q ✕ D5 C ⑨ ϕ5

0,α,

r ρ5

0,α,

α P C✝

A. Douglas (CUNY)

Dn V ã Ñ En1 18 / 22

SLIDE 50

Recall Objective 3:

3. Application to physics: Examine proposed embeddings the

GraviGUT algebra ( whose complexification is D7 V, where V is a 64-dimensional D7-irrep) into E8. Note: V is not an abelian ideal.

A. Douglas (CUNY)

Dn V ã Ñ En1 19 / 22

SLIDE 51

Recall Objective 3:

3. Application to physics: Examine proposed embeddings the

GraviGUT algebra ( whose complexification is D7 V, where V is a 64-dimensional D7-irrep) into E8. Note: V is not an abelian ideal.

A. Douglas (CUNY)

Dn V ã Ñ En1 19 / 22

SLIDE 52

Recall Objective 3:

3. Application to physics: Examine proposed embeddings the

GraviGUT algebra ( whose complexification is D7 V, where V is a 64-dimensional D7-irrep) into E8. Note: V is not an abelian ideal.

A. Douglas (CUNY)

Dn V ã Ñ En1 19 / 22

SLIDE 53

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 54

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 55

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 56

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 57

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 58

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 59

The Standard Model of particle physics, with gauge group U♣1q✂SU♣2q✂SU♣3q, attempts to describe all particles and all forces, except gravity. Grand Unified Theories (GUT) attempt to unify the forces and particles of the Standard Model. The three main GUTs are Georgi and Glashow’s SU♣5q theory, Georgi’s Spin♣10q theory, and the Pati-Salam model based on the Lie group SU♣2q✂SU♣2q✂SU♣4q. Lisi attempted to construct a unification which includes gravity. In this construction, Lisi first embeds gravity and the standard model into spin♣11,3q. He then embeds spin♣11,3q together with the positive chirality 64-dimensional spin♣11,3q irrep into the quaternionic real form of E8. Lisi refers to the embedded Lie algebra as the GraviGUT algebra.

A. Douglas (CUNY)

Dn V ã Ñ En1 20 / 22

SLIDE 60

Theorem (D, Repka)

The GraviGUT algebra cannot be embedded into the quaternionic real form of E8, or any other real form of E8. GraviGUTC ✕ D7 V, V is a 64-dimensional D7-irreps, non-abelian. We actually showed that you cannot embed D7 V into E8. Recall: There is just one embedding of ϕ : D7 ã Ñ E8. Recall also: E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q However, neither VD7♣λ6q nor VD7♣λ7q is a subalgebra of E8.

A. Douglas (CUNY)

Dn V ã Ñ En1 21 / 22

SLIDE 61

Theorem (D, Repka)

The GraviGUT algebra cannot be embedded into the quaternionic real form of E8, or any other real form of E8. GraviGUTC ✕ D7 V, V is a 64-dimensional D7-irreps, non-abelian. We actually showed that you cannot embed D7 V into E8. Recall: There is just one embedding of ϕ : D7 ã Ñ E8. Recall also: E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q However, neither VD7♣λ6q nor VD7♣λ7q is a subalgebra of E8.

A. Douglas (CUNY)

Dn V ã Ñ En1 21 / 22

SLIDE 62

Theorem (D, Repka)

The GraviGUT algebra cannot be embedded into the quaternionic real form of E8, or any other real form of E8. GraviGUTC ✕ D7 V, V is a 64-dimensional D7-irreps, non-abelian. We actually showed that you cannot embed D7 V into E8. Recall: There is just one embedding of ϕ : D7 ã Ñ E8. Recall also: E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q However, neither VD7♣λ6q nor VD7♣λ7q is a subalgebra of E8.

A. Douglas (CUNY)

Dn V ã Ñ En1 21 / 22

SLIDE 63

Theorem (D, Repka)

The GraviGUT algebra cannot be embedded into the quaternionic real form of E8, or any other real form of E8. GraviGUTC ✕ D7 V, V is a 64-dimensional D7-irreps, non-abelian. We actually showed that you cannot embed D7 V into E8. Recall: There is just one embedding of ϕ : D7 ã Ñ E8. Recall also: E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q However, neither VD7♣λ6q nor VD7♣λ7q is a subalgebra of E8.

A. Douglas (CUNY)

Dn V ã Ñ En1 21 / 22

SLIDE 64

Theorem (D, Repka)

The GraviGUT algebra cannot be embedded into the quaternionic real form of E8, or any other real form of E8. GraviGUTC ✕ D7 V, V is a 64-dimensional D7-irreps, non-abelian. We actually showed that you cannot embed D7 V into E8. Recall: There is just one embedding of ϕ : D7 ã Ñ E8. Recall also: E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q However, neither VD7♣λ6q nor VD7♣λ7q is a subalgebra of E8.

A. Douglas (CUNY)

Dn V ã Ñ En1 21 / 22

SLIDE 65

Theorem (D, Repka)

The GraviGUT algebra cannot be embedded into the quaternionic real form of E8, or any other real form of E8. GraviGUTC ✕ D7 V, V is a 64-dimensional D7-irreps, non-abelian. We actually showed that you cannot embed D7 V into E8. Recall: There is just one embedding of ϕ : D7 ã Ñ E8. Recall also: E8 ✕ϕ7♣D7q VD7♣λ2q ❵ VD7♣λ1q ❵ VD7♣λ6q ❵ VD7♣λ7q ❵ VD7♣λ1q ❵ VD7♣0q However, neither VD7♣λ6q nor VD7♣λ7q is a subalgebra of E8.

A. Douglas (CUNY)

Dn V ã Ñ En1 21 / 22

SLIDE 66

Thank you!

A. Douglas (CUNY)

Dn V ã Ñ En1 22 / 22