Constructions Freudenthal magic square

1 constructions

1.1 tits approach
1.2 vinberg s symmetric method
1.3 triality

constructions

see history context , motivation. these constructed circa 1958 freudenthal , tits, more elegant formulations following in later years.

tits approach

tits approach, discovered circa 1958 , published in (tits 1966), follows.

associated normed real division algebra (i.e., r, c, h or o) there jordan algebra, j3(a), of 3 × 3 a-hermitian matrices. pair (a, b) of such division algebras, 1 can define lie algebra

l
=

(


d
e
r


(
a
)
⊕


d
e
r


(

j

3


(
b
)
)
)

⊕

(

a

0


⊗

j

3


(
b

)

0


)



{\displaystyle l=\left({\mathfrak {der}}(a)\oplus {\mathfrak {der}}(j_{3}(b))\right)\oplus \left(a_{0}\otimes j_{3}(b)_{0}\right)}

where





d
e
r




{\displaystyle {\mathfrak {der}}}

denotes lie algebra of derivations of algebra, , subscript 0 denotes trace-free part. lie algebra l has





d
e
r


(
a
)
⊕


d
e
r


(

j

3


(
b
)
)


{\displaystyle {\mathfrak {der}}(a)\oplus {\mathfrak {der}}(j_{3}(b))}

subalgebra, , acts naturally on




a

0


⊗

j

3


(
b

)

0




{\displaystyle a_{0}\otimes j_{3}(b)_{0}}

. lie bracket on




a

0


⊗

j

3


(
b

)

0




{\displaystyle a_{0}\otimes j_{3}(b)_{0}}

(which not subalgebra) not obvious, tits showed how defined, , produced following table of compact lie algebras.

note construction, row of table a=r gives





d
e
r


(

j

3


(
b
)
)


{\displaystyle {\mathfrak {der}}(j_{3}(b))}

, , vice versa.

vinberg s symmetric method

the magic of freudenthal magic square constructed lie algebra symmetric in , b. not obvious tits construction. ernest vinberg gave construction manifestly symmetric, in (vinberg 1966). instead of using jordan algebra, uses algebra of skew-hermitian trace-free matrices entries in ⊗ b, denoted






s
a



3


(
a
⊗
b
)


{\displaystyle {\mathfrak {sa}}_{3}(a\otimes b)}

. vinberg defines lie algebra structure on

d
e
r


(
a
)
⊕


d
e
r


(
b
)
⊕



s
a



3


(
a
⊗
b
)
.


{\displaystyle {\mathfrak {der}}(a)\oplus {\mathfrak {der}}(b)\oplus {\mathfrak {sa}}_{3}(a\otimes b).}

when , b have no derivations (i.e., r or c), lie (commutator) bracket on






s
a



3


(
a
⊗
b
)


{\displaystyle {\mathfrak {sa}}_{3}(a\otimes b)}

. in presence of derivations, these form subalgebra acting naturally on






s
a



3


(
a
⊗
b
)


{\displaystyle {\mathfrak {sa}}_{3}(a\otimes b)}

in tits construction, , tracefree commutator bracket on






s
a



3


(
a
⊗
b
)


{\displaystyle {\mathfrak {sa}}_{3}(a\otimes b)}

modified expression values in





d
e
r


(
a
)
⊕


d
e
r


(
b
)


{\displaystyle {\mathfrak {der}}(a)\oplus {\mathfrak {der}}(b)}

.

triality

a more recent construction, due pierre ramond (ramond 1976) , bruce allison (allison 1978) , developed chris barton , anthony sudbery, uses triality in form developed john frank adams; presented in (barton & sudbery 2000), , in streamlined form in (barton & sudbery 2003). whereas vinberg s construction based on automorphism groups of division algebra (or rather lie algebras of derivations), barton , sudbery use group of automorphisms of corresponding triality. triality trilinear map

a

1


×

a

2


×

a

3


→

r



{\displaystyle a_{1}\times a_{2}\times a_{3}\to \mathbf {r} }

obtained taking 3 copies of division algebra a, , using inner product on dualize multiplication. automorphism group subgroup of so(a1) × so(a2) × so(a3) preserving trilinear map. denoted tri(a). following table compares lie algebra lie algebra of derivations.

barton , sudbery identify magic square lie algebra corresponding (a,b) lie algebra structure on vector space

t
r
i


(
a
)
⊕


t
r
i


(
b
)
⊕
(

a

1


⊗

b

1


)
⊕
(

a

2


⊗

b

2


)
⊕
(

a

3


⊗

b

3


)
.


{\displaystyle {\mathfrak {tri}}(a)\oplus {\mathfrak {tri}}(b)\oplus (a_{1}\otimes b_{1})\oplus (a_{2}\otimes b_{2})\oplus (a_{3}\otimes b_{3}).}

the lie bracket compatible z2 × z2 grading, tri(a) , tri(b) in degree (0,0), , 3 copies of ⊗ b in degrees (0,1), (1,0) , (1,1). bracket preserves tri(a) , tri(b) , these act naturally on 3 copies of ⊗ b, in other constructions, brackets between these 3 copies more constrained.

for instance when , b octonions, triality of spin(8), double cover of so(8), , barton-sudbery description yields

e



8


≅



s
o



8


⊕





s
o

^




8


⊕
(
v
⊗



v
^



)
⊕
(

s

+


⊗




s
^




+


)
⊕
(

s

−


⊗




s
^




−


)


{\displaystyle {\mathfrak {e}}_{8}\cong {\mathfrak {so}}_{8}\oplus {\widehat {\mathfrak {so}}}_{8}\oplus (v\otimes {\widehat {v}})\oplus (s_{+}\otimes {\widehat {s}}_{+})\oplus (s_{-}\otimes {\widehat {s}}_{-})}

where v, s+ , s− 3 8 dimensional representations of






s
o



8




{\displaystyle {\mathfrak {so}}_{8}}

(the fundamental representation , 2 spin representations), , hatted objects isomorphic copy.

with respect 1 of z2 gradings, first 3 summands combine give






s
o



16




{\displaystyle {\mathfrak {so}}_{16}}

, last 2 form 1 of spin representations Δ+ (the superscript denotes dimension). known symmetric decomposition of e8.

the barton-sudbery construction extends other lie algebras in magic square. in particular, exceptional lie algebras in last row (or column), symmetric decompositions are:

f



4


≅



s
o



9


⊕

Δ

16




{\displaystyle {\mathfrak {f}}_{4}\cong {\mathfrak {so}}_{9}\oplus \delta ^{16}}









e



6


≅
(



s
o



10


⊕



u



1


)
⊕

Δ

32




{\displaystyle {\mathfrak {e}}_{6}\cong ({\mathfrak {so}}_{10}\oplus {\mathfrak {u}}_{1})\oplus \delta ^{32}}









e



7


≅
(



s
o



12


⊕



s
p



1


)
⊕

Δ

+


64




{\displaystyle {\mathfrak {e}}_{7}\cong ({\mathfrak {so}}_{12}\oplus {\mathfrak {sp}}_{1})\oplus \delta _{+}^{64}}









e



8


≅



s
o



16


⊕

Δ

+


128


.


{\displaystyle {\mathfrak {e}}_{8}\cong {\mathfrak {so}}_{16}\oplus \delta _{+}^{128}.}






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