Structures Preserved by Exceptional Lie Algebras

15 Pages·0.15 MB·English

by  T. A. Larsson

Structures Preserved by Exceptional Lie Algebras
T. A. Larsson
Vanadisv¨agen 29, S-113 23 Stockholm, Sweden,
email: [email protected]
February 7, 2008
3
0
0
2
Abstract

n
a Forsp(n+2)andeachexceptionalLiealgebraarealizationofdepth
J
2 preserving the spaces spanned by a contact one-form and a bilinear
7
form is given. For e7 and e6 a realization of depth 1 preserving a

lightcone and the space spanned by a bilinear form is also presented.
1
v This makes the origin of the exceptions clear.
6
0
0 1 Introduction
1
0
The exceptional simple Lie algebras e , e , e , f and g were discovered
3 8 7 6 4 2
0 over a century ago. Although many things are known about them, they still
/
h remainsomewhatofamystery. Thepurposeofthisworkistodescribethem
p in a way that makes their existence obvious and inevitable.
-
h In [2] the exceptions were presented as algebras of vector fields, i.e. sub-
t algebras of vect(n) for some n. e and e can be described as a graded Lie
a 7 6
m algebra of depth 1 (three-graded structure, conformal realization):
:
v
g = g +g +g ,
i −1 0 1
X
r and every exception can be described as a graded Lie algebra of depth 2
a
(five-graded structure, quasiconformal realization):1
g = g +g +g +g +g .
−2 −1 0 1 2
Unfortunately,vectorfieldsofpositivedegreearenon-linearlyrealized,which
makes the description in [2] quite opaque. However, it is known that the
classical simple Lie algebras, both the finite- and infinite-dimensional ones,
are best described as vector fields preserving some structure, be it a differ-
ential form, a bilinear form, or a subspace spanned by certain forms. In the
subsequent sections similar descriptions for the exceptions are given.
1Unfortunately,standard notation for thedegree 2subspacecoincides with theexcep-
tion g2.
1
Themainideaisthatthepreservedstructuresarecompletelydetermined
by vector fields in g + g + g . This is the basis for the method of
−2 −1 0
Cartan prolongation, extensively developed by Leites and Shchepochkina in
the context of infinite-dimensional Lie superalgebras [1, 4]. The strategy
used in the present paper can thus be summarized as follows:
1. Find a realization of g + g + g on Cn; in most cases, this has
−2 −1 0
already been done in [2].
2. Determine the structures preserved by these vector fields.
3. Define g as the subalgebra of vect(n) preserving the same structures.
This strategy was employed to find realizations of three exceptional Lie
superalgebras in [3].
This method has the drawback that one can not be completely certain
that the resulting algebra is the right one. However, vector fields preserving
somestructureautomatically defineaclosedsubalgebra,andthissubalgebra
is not empty since it contains the right vector fields at non-positive degree.
Moreover, and unlike the algebras themselves, the preserved structures have
very simple and natural descriptions, which makes me believe that this is
the best way to understand the exceptional Lie algebras.
All infinite-dimensional simple Lie algebras of vector fields can also be
described as graded algebras, i.e.
g = g +...+g +g +g +g +...,
−d −1 0 1 2
where the depth d ≤ 2; there are two Lie superalgebras of depth 3. Re-
garding simple Lie algebras as algebra of vector fields thus gives a unified
description of the finite- and infinite-dimensional cases. In fact, the distinc-
tion between finite and infinite dimension seems somewhat artificial. The
Poincar´e and Hamiltonian algebras are examples of the two types, and yet
they preserve very similar structures; the square length element and the
symplectic two-form only differ in their symmetries.
All results inthis paperare formulated on theLie algebra level, butthey
shouldreadilygeneralizetothecorrespondinggroupsE ,E ,E ,F andG .
8 7 6 4 2
Namely, if g ⊂ vect(n) preserves some structure, G should be the subgroup
of the diffeomorphism group Diff(Cn) that preserves the same structure.
Moreover, I expect that the results can be globalized, i.e. that manifolds
with exceptional structures exist in certain dimensions, just like symplectic
and contact manifolds correspond to algebras of Hamiltonian and contact
vector fields.
Tensor calculus notation is used throughout this paper. Repeated in-
dices, one up and one down, are implicitly summed over. Symmetrization
is denoted by parentheses, a(ibj) ≡ aibj +ajbi and anti-symmetrization by
2
brackets, a[ibj] ≡ aibj −ajbi. The Kronecker delta δi and the totally anti-
j
symmetricconstantǫi1i2..in withinverseǫ aretheonlyinvarianttensors
i1i2..in
under vect(n).
The base field is C.
2 Classical simple Lie algebras
Consider Cn with coordinates xi and let ∂ = ∂/∂xi. The algebra of general
i
vector fieldsvect(n), generated byvector fieldsξ = ξi(x)∂ ,has thefollowing
i
infinite-dimensional simple subalgebras:
vect(n): Setting degxi = 1 gives vect(n) a grading of depth 1. The
grading operator Z = xi∂ can be identified with dilatations.
i
svect(n): Divergence-free vector fields which preserve the volume form
V = dx1dx2...dxn = ǫ dxidxj...dxk. Thus L V = 0 for all X ∈ svect(n).
ij..k X
Setting degxi = 1 gives svect(n) a grading of depth 1. Thegradingoperator
is Z = xi∂ .
i
h(n), n even: Hamiltonian vector fields which preserve the symplectic
two-form ω = ω dxidxj, where the symplectic metric ω = −ω and its
ij ij ji
inverse ωij are structure constants. Thus L ω = 0 for all X ∈ h(n). A
X
Hamiltonian vector field is of the form H = ωij∂ f∂ where f is a function,
f i j
andthebracketinh(n)readsexplicitly[H ,H ]= H ,wherethePoisson
f g {f,g}
bracket reads {f,g} = ωij∂ f∂ g. Setting degxi = 1 gives h(n) a grading of
i j
depth 1. The grading operator is Z = xi∂ .
i
k(n+1), n even: Denote the coordinates of Cn+1 by t and xi, i =
1,2,...,n, and let ∂ and ∂ denote the corresponding derivatives. Con-
0 i
tact vector fields preserve the space spanned by the contact one-form α =
dt+ω xidxj, where ω are the same structure constants as in h(n). Thus
ij ij
L α = f α, where f is some function, for all X ∈ k(n + 1). Setting
X X X
degxi = 1, degt = 2 gives k(n+1) a grading of depth 2. The generators at
non-positive degree are
f K
f
g : t H = ∂ ,
−2 0
g : x P = ∂ −x ∂ , (2.1)
−1 i i i i 0
g : x x J = J = x ∂ +x ∂ ,
0 i j ij ji i j j i
t Z = 2t∂ +xi∂ ,
0 i
where x = ω xj. J generate sp(n) and Z computes the grading.
i ij ij
Finite-dimensional simple algebras are subalgebras of gl(n), which in
turn is the subalgebra of vect(n) that preserves the grading operator Z =
xi∂ , i.e. X ∈ gl(n) iff [Z,X] = 0.
i
3
sl(n): X ∈ sl(n) if X ∈ svect(n) and [Z,X] = 0. dimsl(n) = n2 −1.
ranksl(n) = n−1.
sp(n), n even: X ∈ sp(n) if X ∈ h(n) and [Z,X] = 0. dimsp(n) =
n(n+1)/2. ranksp(n) = n/2.
so(n): The non-simple Poincar´e algebra poin(n) consists of vector fields
which preserve the length element ds2 = g dxidxj, where g = g are
ij ij ji
symmetric structureconstants. so(n) is the subalgebra at degree 0, i.e. X ∈
so(n) if X ∈ poin(n) and [Z,X] = 0. dimso(n) = n(n−1)/2. rankso(n) =
n/2 if n even and rankso(n) = (n−1)/2 if n odd.
However, it is more interesting to describe these algebras together with
a non-trivial grading.
sl(n+1): Thisalgebraadmitsagradingofdepth1oftheformsl(n+1)=
n∗ + (sl(n) + gl(1)) + n, where g are described as sl(n) modules. The
±1
generators are
g : P = ∂ ,
−1 i i
g : Ji = xi∂ − 1δixk∂ ,
0 j j n j k (2.2)
Z = xi∂ ,
i
g : Ki = xixk∂ .
1 k
µ
The sl(n+1) generators J are identified as
ν
1
J0 = Z, Ji → Ji− δiZ,
0 j j n j
(2.3)
J0 = P , Ji = −Ki.
i i 0
dimsl(n+1) = n+n2 +n = (n+1)2 −1. ranksl(n+1) = ranksl(n)+
rankgl(1) = n.
Note that sl(n+1) contains all vector fields from vect(n) of degree 0 and
−1, so non-trivial conditions only appear in g . X = Xi(x)∂ ∈ sl(n+1) iff
1 i
(n+1)∂ ∂ Xi = δi∂ ∂ Xl +δi∂ ∂ Xl. (2.4)
j k j l k k j l
Sincethis isasecond-orderequation, sl(n+1)does notpreservesomemulti-
linear form, which would yield a system of first-order equations. Phrased
differently, it is a partial prolong rather than a full prolong. Let Γi be (the
jk
inhomogeneous part of) a connection, which transforms under vect(n) as
L Γi = −Xl∂ Γi +∂ XiΓl
X jk l jk l jk
(2.5)
−∂ XlΓi −∂ XlΓi +∂ ∂ Xi.
j lk k jl j k
sl(n+1) preserves the subspace spanned by the traceless part of Γi ,
jk
1
γi = Γi − (δiΓl +δiΓl ). (2.6)
jk jk n+1 j lk k jl
4
Hence X ∈sl(n+1) ⊂ vect(n) iff L γi = fi|mnγl for some X-dependent
X jk l|jk mn
i|mn
functions f .
l|jk
co(n)= so(n+2): The conformal algebra co(n) admits a grading of
depth 1 of the form so(n + 2) = n∗ + (so(n) +gl(1)) + n, where g are
±1
described as so(n) modules. The generators are
g : P =∂ ,
−1 i i
g : J = x ∂ −x ∂ ,
0 ij i j j i
(2.7)
Z = xi∂ ,
i
g : K = x xk∂ − 1xkx ∂ ,
1 i i k 2 k i
wherex = g xj. co(n) preserves the space generated by the squared length
i ij
element ds2 = g dxidxj, i.e. it preserves the lightcone ds2 = 0. The
ij
so(n+2) generators J are identified as
µν
J = Z, J = J ,
0¯0 ij ij
(2.8)
J = P J = −K .
0i i ¯0i i
dimso(n+2) = n+(n(n−1)/2+1)+n = (n+2)(n+1)/2. rankso(n+2) =
rankso(n)+rankgl(1) = n/2+1 if n even.
sp(n+2), n even: This algebra admits a grading of depth 2 of the form
sp(n+2) = 1+n+(sp(n)+gl(1))+n+1, where g and g are described
±1 ±2
as sp(n) modules. More precisely, X ∈ sp(n+2) if X ∈ k(n+1) and
L βij = fijβkl, (2.9)
X kl
ij
for some X-dependent functions f , where
kl
βij = dx[idxj]+ωijω dxkdxl. (2.10)
kl
The generators are
g : H = ∂ ,
−2 0
g : P = ∂ −x ∂ ,
−1 i i i 0
g : J = x ∂ +x ∂ ,
0 ij i j j i
(2.11)
Z = 2t∂ +xi∂ ,
0 i
g : K = x t∂ +x xj∂ −t∂ ,
1 i i 0 i j i
g : ∆ = t2∂ +txi∂ ,
2 0 i
where x = ω xj. Note that the extra condition (2.10) does not restrict the
i ij
g +g +g subalgebra, which is the same as (2.1). One verifies that the
−2 −1 0
5
generators in (2.11) satisfy e.g.
[P ,P ] = 2ω H, [∆,P ]= K , [H,∆] = Z,
i j ij i i
(2.12)
[K ,K ] = −2ω ∆ [H,K ]= −P , [P ,K ] = J −ω Z.
i j ij i i i j ij ij
The sp(n+2) generators J are identified as
µν
J = H, J = P , J = Z,
00 0i i 0¯0
(2.13)
J = ∆, J = K , J = J .
¯0¯0 ¯0i i ij ij
dimsp(n + 2) = 1 + n + (n(n + 1)/2 + 1) + n + 1 = (n + 2)(n + 3)/2.
ranksp(n+2) = ranksp(n)+rankgl(1) =n/2+1.
3 General construction
Consider Cn+1, n even, with coordinates t and xµ, where µ = 1,2,...,n.
Denote the corresponding derivatives by ∂ = ∂/∂t and ∂ = ∂/∂xµ. A
0 µ
general vector field can be written as X = Pµ∂ +Q∂ for some functions
µ 0
Pµ and Q.
Since n is even we can introduce a non-degenerate constant symplectic
metric Ω = −Ω . The contact algebra k(n + 1) is the subalgebra of
µν νµ
vect(n+1) preserving the space spanned by the contact form
α = dt+Ω xµdxν. (3.1)
µν
In other words, L α = f α for some functions f . This condition leads to
X X X
Pµ = ΩµνDe f, Q = 2f −xµDe f, (3.2)
ν µ
where f is some function and
De = ∂ +Ω xµ∂ . (3.3)
µ µ µν 0
Alternatively, k(n+1) can be defined as the subalgebra of vect(n+1) pre-
e
serving the space spanned by D .
µ
Now assume that the contact vector field X also preserves the subspace
spanned by the bilinear form
βab = βabdxµdxν. (3.4)
µν
Hence we require that L βab = gabβcd for some X-dependent functions gab.
X cd cd
One checks that this leads to the additional conditions
βab∂ Pµ = 0,
µν 0
(3.5)
βab∂ Pρ+βab∂ Pρ = gabβcd.
ρν µ µρ ν cd µν
6
In terms of the function f, this becomes
Ωµρβab∂ De f = 0,
µν 0 ρ
(3.6)
Ωρσβab ∂ De f = gabβcd.
ρ(µ ν) σ cd µν
The question is now whether a non-trivial bilinear form (3.4) exists. It
is clear that the structureconstants Ωµν and βab must bebuiltfrom natural
µν
constants only. If g can be described as sl(n) plus some sl(n) modules, the
0
only such constants are ǫi1i2..in and ǫ ; if g includes sp(n), we may also
i1i2..in 0
use the symplectic metric ω and its inverse ωij.
ij
One bilinear form can always be constructed:
βµν = dx[µdxν]+ΩµνΩ dxρdxσ. (3.7)
ρσ
The algebra which preserves the spaces spanned by α and this βµν can be
identified with sp(n+2), cf. (2.10). In the next five sections similar bilin-
ear forms corresponding to the five exceptional Lie algebras are described.
Although I only prove that bilinear forms yield the correct g +g +g
−2 −1 0
subalgebras, I have no doubt that the exceptions can be described in this
way; that it is possible to write down an exceptional bilinear form for each
exceptional Lie algebra is highly non-trivial. Moreover, vector fields that
preserve some structure do generate a subalgebra automatically.
One can also look for exceptional subalgebras of a conformal algebra
co(n) = so(n+2) instead of a contact algebra. Rather than preserving the
contact form (3.1) up to a function, such vector fields preserve a lightcone
ds2 ≡ G dxµdxν = 0, (3.8)
µν
where G = G is a symmetric metric. Such realizations, necessarily of
µν νµ
depth 1, are given for e and e in Sections 9 and 10. It is known that e , f
7 6 8 4
and g do not admit gradings of depth 1.
2
4 e , depth 2
8
According to [2], e admits a grading of depth 2:
8
e = 1+56+(e +gl(1))+56+1
8 7
(4.1)
= 1+(28+28∗)+(sl(8)+70+gl(1))+(28+28∗)+1,
where the subspaces are described by their decomposition as e and sl(8)
7
modules, respectively. Note that dime = 133 and dimsl(8) = 63, so
7
dime = 248. ranke = ranke + rankgl(1) = 7 + 1 = ranksl(8) +
8 8 7
rankgl(1) = 7+1= 8.
7
ConsiderC57 with coordinates t,xij = −xji andx¯ = −x¯ ,wherei,j =
ij ji
1,2,...,8. Denote the corresponding derivatives by ∂ = ∂/∂t, ∂ = ∂/∂xij
0 ij
and ∂¯ij = ∂/∂x¯ , where ∂ xij = ∂¯ijx¯ = δij ≡ δiδj −δjδi.
ij kl kl kl k l k l
The generators of g +g +g are
−2 −1 0
g : H = ∂
−2 0
g : D = ∂ +x¯ ∂ ,
−1 ij ij ij 0
Eij = ∂¯ij −xij∂ ,
0
(4.2)
g : Ji = xik∂ −x¯ ∂¯ik − 1δi(xkl∂ −x¯ ∂¯kl),
0 j jk jk 8 j kl kl
Gijkl = x[ij∂¯kl]−ǫijklmnpqx¯ ∂¯ ,
mn pq
Z = 2t∂ + 1xij∂ + 1x¯ ∂¯ij.
0 2 ij 2 ij
The subspace g + g is described in [2]; the explicit description is very
1 2
tedious. Fortunately, it suffices to know g + g + g to determine the
−2 −1 0
preserved structures. Define
α = dt+ 1xijdx¯ − 1x¯ dxij,
2 ij 2 ij
(4.3)
βijkl = dx[ijdxkl]+ǫijklmnpqdx¯ dx¯ .
mn pq
α is the contact one-form and the bilinear form βijkl = βklij is totally
antisymmetric. By direct calculation one shows that
L α = fα,
X
(4.4)
L βijkl = gijkl βmnpq,
X mnpq
ijkl
for some X-dependent functions f and g . e can thus be described as
mnpq 8
the subalgebra of vect(57) which preserves the spaces spanned by α and
βijkl.
Moreover, the generators in (4.2) are the only vector fields of degree ≤ 0
with this property. To see this, note that contact algebra k(57) consists of
vector fields that preserve the Pfaff equation α = 0. All such vector fields
can be written in the form
X = K = EeijfDe −De fEeij +4fH, (4.5)
f ij ij
where f is a function and
De = ∂ −x¯ ∂ , Eeij = ∂¯ij +xij∂ . (4.6)
ij ij ij 0 0
In particular, vector fields of degree ≤ −1 correspond to functions of degree
≤ 1, i.e.
K = 4H, K = −2Eij, K = 2D . (4.7)
1 xij x¯ij ij
It remains to ensure that the vector fields of degree zero, corresponding to
the functions xijxkl, xijx¯ , x¯ x¯ , and t, are restricted to Ji, Gijkl, and Z.
kl ij kl j
This is accompished by the second condition in (4.4).
8
5 e , depth 2
7
Weobtainarealization ofe fromtheprevioussection byrestriction sl(8) →
7
sl(4)+sl(4):
e = 1+(16+16∗)+(sl(4)+sl(4)+36+gl(1))+(16+16∗)+1,
7
where the subspaces are described by their decomposition as sl(4) + sl(4)
modules; see [2], Eq. (32). Note that dimsl(4) = 15, so dime = 133.
7
ranke = ranksl(4)+ranksl(4)+rankgl(1) = 3+3+1 = 7.
7
Consider C33 with coordinates t, xi and x¯a, where i,j = 1,2,3,4 and
a i
a,b = 1,2,3,4 are two different sets of indices. Denote the corresponding
derivatives by ∂ = ∂/∂t, ∂a = ∂/∂xi and ∂¯i = ∂/∂x¯a, where ∂axi =
0 i a a i j b
∂¯ix¯a = δiδa.
b j j b
The generators of g +g +g are
−2 −1 0
g : H = ∂
−2 0
g : Da =∂a +x¯a∂ ,
−1 i i i 0
Ei = ∂¯i −xi∂ ,
a a a 0
g : Ii = xi∂a−x¯a∂¯i − 1δi(xk∂a−x¯a∂¯k), (5.1)
0 j a j j a 4 j a k k a
Ja = x¯a∂¯i −xi∂a − 1δa(x¯c∂¯i −xi∂c),
b i b b i 4 b i c c i
Gij = x[i∂¯j]−ǫijklǫ x¯cx¯d,
ab [a b] abcd k l
Z = 2t∂ +xi∂a+x¯a∂¯i.
0 a i i a
Again, it suffices to know g +g +g to determine the preserved struc-
−2 −1 0
tures. Define
α = dt+xidx¯a −x¯adxi,
a i i a
(5.2)
βij = dx[idxj]+ǫijklǫ dx¯cdx¯d.
ab [a b] abcd k l
ij ji
α is the contact one-form and the bilinear form β = β is symmetric. By
ab ba
direct calculation one shows that
L α = fα,
X
(5.3)
L βij = gij|cdβkl,
X ab kl|ab cd
ij|cd
for some X-dependent functions f and g . e can thus be described as
kl|ab 7
ij
the subalgebra of vect(33) which preserves the spaces spanned by α and β .
ab
The proof that (5.3) uniquely singles out (5.1) among all vector fields of
degree ≤ 0 is completely analogous to the e case.
8
9
6 e , depth 2
6
According to [2], Eq. (33), e admits a grading of depth 2:
6
e = 1+20∗+(sl(6)+gl(1))+20+1,
6
where the subspaces are described by their decomposition as sl(6) modules,
respectively. Notethatdimsl(6) = 35,sodime = 78. ranke = ranksl(6)+
6 6
rankgl(1) = 5+1= 6.
Consider C21 with coordinates t, xijk = −xjik = xjki, where i,j,k =
1,2,...,6. Denote the corresponding derivatives by ∂ = ∂/∂t and ∂ =
0 ijk
∂/∂xijk, where ∂ xijk = δijk ≡ δ[iδjδk].
lmn lmn l l m
The generators of g +g +g are
−2 −1 0
g : H = ∂
−2 0
g : D = ∂ +ǫ xlmn∂ ,
−1 ijk ijk ijklmn 0
(6.1)
g : Ji = xikl∂ − 1δixklm∂ ,
0 j jkl 6 j klm
Z = 2t∂ + 1xijk∂ .
0 6 ijk
Let
α = dt+ǫ xijkdxlmn,
ijklmn
βijk|lmn = dxijkdxlmn+dxlmndxijk,
1
(6.2)
βijk|lmn = ǫ (ǫijlmnqdxkrsdxntu +[ijk]+[lmn]),
2 pqrstu
βijk|lmn = βijk|lmn+βijk|lmn,
1 2
where[ijk]and[lmn]standfortermsneededforproperantisymmetrization.
The bilinear form βijk|lmn has the symmetries
βijk|lmn = −βjik|lmn = βjki|lmn = βlmn|ijk. (6.3)
One shows that
L α = fα,
X
(6.4)
L βijk|lmn = gijk|lmnβpqr|stu,
X pqr|stu
ijk|lmn
for some X-dependent functions f and g . e can thus be described
pqr|stu 6
as the subalgebra of vect(21) which preserves the spaces spanned by α and
βijk|lmn.
10