Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
Magic squares of Lie groups Tevian Dray Department of Mathematics Oregon State University http://math.oregonstate.edu/~tevian
(supported by FQXi and the John Templeton Foundation) Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
Joshua Kinkaid Department of Mathematics Oregon State University
Corinne Manogue Department of Physics Oregon State University
John Huerta Centro de An´ alise Matem´ atica, Geometria e Sistemas Dinˆ amicos Instituto Superior T´ecnico (Lisboa)
Aaron Wangberg Dept of Mathematics & Statistics Winona State University Robert Wilson School of Mathematical Sciences Queen Mary, University of London
(supported by FQXi and the John Templeton Foundation)
Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
Book
The Geometry of the Octonions Tevian Dray and Corinne A. Manogue World Scientific 2015 ISBN: 978-981-4401-81-4 http://octonions.geometryof.org/GO
Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
Lie Algebras History
The Freudenthal–Tits Magic Square Freudenthal (1964), Tits (1966): R C H O Vinberg (1966):
Goal:
R a1 a2 c3 f4
C a2 a2 ⊕ a2 a5 e6
H c3 a5 d6 e7
O f4 e6 e7 e8
sa(3, A ⊗ B) ⊕ der(A) ⊕ der(B) der(H) = so(3); der(O) = g2 Description as symmetry groups
[Wangberg (PhD 2007), Wangberg & Dray (JMP 2013, JAA 2014), Dray, Manogue, and Wilson (CMUC 2014)] Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
Lie Algebras History
History Barton & Sudbery (2003): Well-understood in terms of Lie algebras. Satisfactory group description not yet known. Rosenfeld (1956/1997): Isometry groups of projective planes over A ⊗ B. Cayley-Moufang plane:
F4 ←→ OP2
Baez (2002): OK for E6 ; not for E7 , E8 . In short, more work must be done before we can claim to fully understand the geometrical meaning of the Lie groups E6 , E7 and E8 . Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
R C H O
R su(3, R) sl(3, R) sp(6, R) f4(4)
Real Forms The Structure of E6 Symplectic Groups Conformal Groups
C su(3, C) sl(3, C) su(3, 3, C) e6(2)
Tevian Dray
H su(3, H) sl(3, H) d6(−6) e7(−5)
O f4 e6(−26) e7(−25) e8(−24)
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
R C H O
R su(3, R) sl(3, R) sp(6, R) f4(4)
Real Forms The Structure of E6 Symplectic Groups Conformal Groups
C su(3, C) sl(3, C) su(3, 3, C) e6(2)
H su(3, H) sl(3, H) d6(−6) e7(−5)
O su(3, O) sl(3, O) e7(−25) e8(−24)
Dray & Manogue (2010):
F4 ∼ = SO(9, 1) ⊂ E6 = SL(3, O) using SL(2, O) ∼ = SU(3, O), E6 ∼ ! ! X θ M 0 X = M= 0 1 θ† n Triality!
Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
R C H O
R su(3, R) sl(3, R) sp(6, R) ??
Real Forms The Structure of E6 Symplectic Groups Conformal Groups
C su(3, C) sl(3, C) sp(6, C) ??
H su(3, H) sl(3, H) sp(6, H) ??
O su(3, O) sl(3, O) sp(6, O) ??
Dray & Manogue (2010):
F4 ∼ = SU(3, O), E6 ∼ = SL(3, O) using SL(2, O) ∼ = SO(9, 1) ⊂ E6 ! ! X θ M 0 X = M= 0 1 θ† n Triality!
Tevian Dray
Magic squares of Lie groups
Introduction 3 × 3 Magic Square 2 × 2 Magic Square Clifford Algebras
Real Forms The Structure of E6 Symplectic Groups Conformal Groups
The Subgroup Structure of E6 164
116 1
sl(3, O) m6 [E6 ] hQQQQ O Q mmm
m (2) Btz mmmm m mmm mmm
so(9, 1) = sl(2, O) [D5 ] O
(3)
O
so(9, 1) = sl(2, O) [D5 ] O
(3)
Btz
so(9) = su(2, O) [B4 ]
Btz
so(9) = su(2, O) [B4 ] O
hQQQ QQQ QQQ QQQ QQQ QQ
m6 mmm mmm mmm mmm mmm so(8) m hQ 6 mmm [DO 4 ] QQQQQ (1) QQQ R(1) mmm Rxℓ mmmm QQQxℓ (1) (1) (2) m QQ Rxℓ Sℓ →Sℓ mmm (1) QQQ (3) Sℓ →Sℓ mmm
so(6) [D3 ]
so(6) [D3 ]
(1)
O
so(5) [B2 ]
O
so(4) = so(3) ⊕ so(3) [D2 ]
O
so(4) = so(3) ⊕ so(3) [D2 ]
hQQQ QQGQℓ +2Sℓ(1) QQQ QQQ QQQ QQ
O
so(4) = so(3) ⊕ so(3) [D2 ]
mm6 mmm mmm (3) mmm mmm Gℓ +2Sℓ mmm (2)
Gℓ +Sℓ
so(3) [B1 ] O
Gℓ +Sℓ1
su(3, 1, H)1 [C ]
4 £@ @ O O ££ ££ ££ ££Gℓ +S 1 Aℓ £ ℓ £ ££ ££ GF
^
Gℓ −2Sℓ
so(5) [B2 ]
O
su(3, O) [F4 ] G O
`
Aℓ
Aℓ
Â?
6V
sl(3, H) ⊕ su(2, C)C [A5 ⊕O A1 ]
sl(2, O) = so(9, 1) [D5 ] \
}> }} }} }} }} }} }} ED } } }} 1 Btz
2 Aℓ ,Btz
su(2, 1, H)1 [C3 ]
(3)
Gℓ −2Sℓ
so(5) [B2 ]
@A
O
(2)
Gℓ −2Sℓ
H
/O
so(6) [D3 ]
O
O O
2 Btz
so(7) = su(1, O) [B3 ]
O
su(2, 1, O) [F4(36,16) ]
sl(2, 1, H) [A5 ]
1 Btz
so(7) = su(1, O) [B3 ] O
1 Btz 2 Btz
Gℓ +Sℓ1
so(9) = su(2, O) [B4 ]
so(7) = su(1, O) [B3 ] O
B1 2 tz Btz
O
Btz →Btz
(2)
(1)
Btz
sl(2, 1, H) ⊕ su(2, C)2 [A5 ⊕ A1 ]
QQQ B (2) QQQtz QQ (1) QQQ
(1)
Btz
so(9, 1) = sl(2, O) [D5 ]
. sl(3, O) 2 [E6 ] h O > }} }} }} } } 2 1 Btz Btz }} } 2 Btz }} } } }}
#¤
su(2, O) = so(9) o [B4 ]
Aℓ
sl(3, H) [A5 ] [
O
1 Btz
? _ so(8)
[D4 ]
Gℓ +Sℓ1
2 Btz
Gℓ +Sℓ1
Aℓ }> H ±G O }} µµ ±± }} µµ Aℓ ± 1 Rxℓ }} µ G1 +Sℓ1 ±± 1 }} Rxℓ µµ ±± BC }} µµ }} ±±± } µ } }} µµ ±± @Aµµ ±± su(3, H)1 sl(2, H) ±± su(1, O) = so(7) µµ Aℓ µ ±±± [C3 ] [B3 ] [A = D ] µ 3 3 V.. Gℓ +Sℓ1 µµ e ± > O ] ]
