# Constructing Majorana representations Madeleine Whybrow, Imperial - PowerPoint PPT Presentation

## Constructing Majorana representations Madeleine Whybrow, Imperial College London Joint work with M. Pfeiffer, St Andrews The Monster group The Monster group Denoted M , the Monster group is the largest of the 26 sporadic groups in the

1. Constructing Majorana representations Madeleine Whybrow, Imperial College London Joint work with M. Pfeiffer, St Andrews

2. The Monster group

3. The Monster group ◮ Denoted M , the Monster group is the largest of the 26 sporadic groups in the classification of finite simple groups

4. The Monster group ◮ Denoted M , the Monster group is the largest of the 26 sporadic groups in the classification of finite simple groups ◮ It was constructed by R. Griess in 1982 as Aut ( V M ) where V M is a 196 884 - dimensional, real, commutative, non-associative algebra known as the Griess or Monster algebra

5. The Monster group ◮ Denoted M , the Monster group is the largest of the 26 sporadic groups in the classification of finite simple groups ◮ It was constructed by R. Griess in 1982 as Aut ( V M ) where V M is a 196 884 - dimensional, real, commutative, non-associative algebra known as the Griess or Monster algebra ◮ The Monster group contains two conjugacy classes of involutions - denoted 2A and 2B - and M = � 2 A �

6. The Monster group ◮ Denoted M , the Monster group is the largest of the 26 sporadic groups in the classification of finite simple groups ◮ It was constructed by R. Griess in 1982 as Aut ( V M ) where V M is a 196 884 - dimensional, real, commutative, non-associative algebra known as the Griess or Monster algebra ◮ The Monster group contains two conjugacy classes of involutions - denoted 2A and 2B - and M = � 2 A � ◮ If t , s ∈ 2 A then ts is of order at most 6 and belongs to one of nine conjugacy classes: 1 A , 2 A , 2 B , 3 A , 3 C , 4 A , 4 B , 5 A , 6 A .

7. The Monster group

8. The Monster group ◮ In 1984, J. Conway showed that there exists a bijection ψ between the 2A involutions and certain idempotents in the Griess algebra called 2A-axes

9. The Monster group ◮ In 1984, J. Conway showed that there exists a bijection ψ between the 2A involutions and certain idempotents in the Griess algebra called 2A-axes ◮ The 2A-axes generate the Griess algebra i.e. V M = �� ψ ( t ) : t ∈ 2 A ��

10. The Monster group ◮ In 1984, J. Conway showed that there exists a bijection ψ between the 2A involutions and certain idempotents in the Griess algebra called 2A-axes ◮ The 2A-axes generate the Griess algebra i.e. V M = �� ψ ( t ) : t ∈ 2 A �� ◮ If t , s ∈ 2 A then the algebra �� ψ ( t ) , ψ ( s ) �� is called a dihedral subalgebra of V M and has one of nine isomorphism types, depending on the conjugacy class of ts .

11. The Monster group Example

12. The Monster group Example Suppose that t , s ∈ 2 A such that ts ∈ 2 A as well.

13. The Monster group Example Suppose that t , s ∈ 2 A such that ts ∈ 2 A as well. Then the algebra V := �� ψ ( t ) , ψ ( s ) �� is a 2 A dihedral algebra.

14. The Monster group Example Suppose that t , s ∈ 2 A such that ts ∈ 2 A as well. Then the algebra V := �� ψ ( t ) , ψ ( s ) �� is a 2 A dihedral algebra. The algebra V also contains the axis ψ ( ts ). In fact, it is of dimension 3: V = � ψ ( t ) , ψ ( s ) , ψ ( ts ) � R .

15. Monstrous Moonshine and VOAs

16. Monstrous Moonshine and VOAs ◮ In 1992, R. Borcherds famously proved Conway and Norton’s Monstrous Moonshine conjectures, which connect the Monster group to modular forms

17. Monstrous Moonshine and VOAs ◮ In 1992, R. Borcherds famously proved Conway and Norton’s Monstrous Moonshine conjectures, which connect the Monster group to modular forms ◮ The central object in his proof is the Moonshine module, V # = � ∞ n =0 V # n .

18. Monstrous Moonshine and VOAs ◮ In 1992, R. Borcherds famously proved Conway and Norton’s Monstrous Moonshine conjectures, which connect the Monster group to modular forms ◮ The central object in his proof is the Moonshine module, V # = � ∞ n =0 V # n . ◮ It belongs to a class of graded algebras know as vertex operator algebras, or VOA’s

19. Monstrous Moonshine and VOAs ◮ In 1992, R. Borcherds famously proved Conway and Norton’s Monstrous Moonshine conjectures, which connect the Monster group to modular forms ◮ The central object in his proof is the Moonshine module, V # = � ∞ n =0 V # n . ◮ It belongs to a class of graded algebras know as vertex operator algebras, or VOA’s ◮ In particular, we have Aut ( V # ) = M

20. Monstrous Moonshine and VOAs ◮ In 1992, R. Borcherds famously proved Conway and Norton’s Monstrous Moonshine conjectures, which connect the Monster group to modular forms ◮ The central object in his proof is the Moonshine module, V # = � ∞ n =0 V # n . ◮ It belongs to a class of graded algebras know as vertex operator algebras, or VOA’s ◮ In particular, we have Aut ( V # ) = M and V # 2 ∼ = V M .

21. Majorana Theory

22. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have:

23. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w );

24. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ).

25. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that for all a ∈ A we have:

26. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that for all a ∈ A we have: M3 ( a , a ) = 1 and a · a = a ;

27. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that for all a ∈ A we have: M3 ( a , a ) = 1 and a · a = a ; M4 V = V ( a ) ⊕ V ( a ) ⊕ V ( a ) 22 ⊕ V ( a ) where V ( a ) = { v : v ∈ V , a · v = µ v } ; 1 0 1 1 µ 25

28. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that for all a ∈ A we have: M3 ( a , a ) = 1 and a · a = a ; M4 V = V ( a ) ⊕ V ( a ) ⊕ V ( a ) 22 ⊕ V ( a ) where V ( a ) = { v : v ∈ V , a · v = µ v } ; 1 0 1 1 µ 25 M5 V ( a ) = { λ a : λ ∈ R } . 1

29. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that for all a ∈ A we have: M3 ( a , a ) = 1 and a · a = a ; M4 V = V ( a ) ⊕ V ( a ) ⊕ V ( a ) 22 ⊕ V ( a ) where V ( a ) = { v : v ∈ V , a · v = µ v } ; 1 0 1 1 µ 25 M5 V ( a ) = { λ a : λ ∈ R } . 1 Suppose furthermore that V obeys the fusion rules.

30. Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that for all a ∈ A we have: M3 ( a , a ) = 1 and a · a = a ; M4 V = V ( a ) ⊕ V ( a ) ⊕ V ( a ) 22 ⊕ V ( a ) where V ( a ) = { v : v ∈ V , a · v = µ v } ; 1 0 1 1 µ 25 M5 V ( a ) = { λ a : λ ∈ R } . 1 Suppose furthermore that V obeys the fusion rules. Then V is a Majorana algebra with Majorana axes A .

31. Majorana Theory Let V be a Majorana algebra with Majorana axes A .

32. Majorana Theory Let V be a Majorana algebra with Majorana axes A . For each a ∈ A , we can construct an involution τ ( a ) ∈ Aut ( V ) such that

33. Majorana Theory Let V be a Majorana algebra with Majorana axes A . For each a ∈ A , we can construct an involution τ ( a ) ∈ Aut ( V ) such that  for u ∈ V ( a ) ⊕ V ( a ) ⊕ V ( a ) u  1 0  1 τ ( a )( u ) = 22 for u ∈ V ( a ) − u  1  25

34. Majorana Theory Let V be a Majorana algebra with Majorana axes A . For each a ∈ A , we can construct an involution τ ( a ) ∈ Aut ( V ) such that  for u ∈ V ( a ) ⊕ V ( a ) ⊕ V ( a ) u  1 0  1 τ ( a )( u ) = 22 for u ∈ V ( a ) − u  1  25 called a Majorana involution.

35. Majorana Theory Let V be a Majorana algebra with Majorana axes A . For each a ∈ A , we can construct an involution τ ( a ) ∈ Aut ( V ) such that  for u ∈ V ( a ) ⊕ V ( a ) ⊕ V ( a ) u  1 0  1 τ ( a )( u ) = 22 for u ∈ V ( a ) − u  1  25 called a Majorana involution. Given a group G and a normal set of involutions T such that G = � T � , if there exists a Majorana algebra V such that

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