On Lie-algebras and their representations Part 1 : Part 2

On Lie-algebras and their representations
Mashhour Bani-Ata
PAAET-Kuwait
Abstract :
Simple Lie-algebras can be characterized by their root systems and turns out that the irreducible root systems have only finitely many types : Al (l ≥ 1), Bl (l ≥ 2), Cl (l ≥ 3), Dl (l ≥ 4), E6,E7, E8, F4 and G2. In this poster we will
discuss the construction of E6 and its representation. The 27-dimensional module for E6 was investigated by Aschbacher [1]. For more information about these Lie-algebras one can see [2], [4] and [3].
Part 2 :
Part 1 :
Proposition 1.1
For a vector space A over a field K, we denote by End(A) the Lie-algebra consisting of
all linear transformations of A with Lie-product [X, Y ] = XY − Y X.
Proposition 1.1
Let Q be a quadratic form on a 6-dimensional vector space V over F2, β is the
set of all non-zero vectors of V such that Q(v) = 0 and a bilinear form on V,
(x|y) = Q(x + y) − Q(x) − Q(y)
Then :
(i) The normed-subsets Γ of β has at most size six, where (x|y) = 1 for all x, y ∈ Γ.
b = {g ∈ GL(V )|g
(ii) β contains exactly 72-normed-subsets of size six and the group H
is preserved by Q} acts transitively on these normed-subsets with stabilizer S6.
Proof :
P
Let Γ = {b1, b2, · · · , bn} be a normed-set of size n. For I ⊆ {1, · · · , n}, set b1 =
bi .
i∈I
If i ∈ I, then (bi|bI ) = |I| − 1, and (bj |bI ) = |I| for all j ∈
/ I. Hence each proper subset
of Γ consists of linearly independent vectors, and |Γ| ≤ 7.
|I|
Moreover, Q(b1) = 2 . Hence. If |Γ| ≥ 6, then every subset of size six forms a base of
P
7
V. If |Γ| = 7, then s =
b = 0 and Q(s) = 2 = 1, a contradiction. Hence |Γ| ≥ 6.
b∈Γ
There do exist E-sets of size 6. For example, choose a, b ∈ β with (a|b) = 1. Then
Xa,b = {a} ∪ {x ∈ β|(x|a) = 1, (x|b) = 0} is a E-set of size six.
If Γ = {b1, · · · , b6} is a E-set of size six, then Γ is a base, as shown above, and
P
P
b is transitive on E-sets of size six, with
Q( xibi) =
xixj , for xi ∈ F2. Hence, H
i
i<j
stabilizer S6. More precisely, if ε = {c1, c2, · · · , c6} is another E-set of size six, then the
b
transformation sending bi to ci preserves Q and is thus contained in H.
Definition 1.2.
E-sets of size six are called M-sets.
Remark 1. If Γ is a M-set, then s =
P
Corollary 1.1.
Let Γ, ε two M-sets corresponding two distinct reflections σ and τ . Then
(1) If [σ, τ ] = 1, then |Γ ∩ ε| = |Γ ∩ εσ | = 1.
(2) If [σ, τ ] 6= 1, then |Γ ∩ ε| = 3 and Γ ∩ εσ = φ or |Γ ∩ εσ | = 3 and Γ ∩ ε = φ.
Let A be a 27-dimensional vector space over F2 with base {ex|x ∈ β}. We are
going to construct a Lie-subalgebra of End(A) using M-subsets of β. For v ∈ V define
v
Hv ∈ End(A) by eH
x = (x|v)ex (x ∈ β)
For a M-set
Γ with corresponding reflection σ, define M (Γ) ∈ End(A) by
ex ,
x∈Γ
M (Γ)
=
(x ∈ β)
ex
0, otherwise
Proposition 1.4.
Let Γi be M-sets with corresponding reflections σi, i = 1, 2. Γ2 6= Γ1, Γσ1 1 . In particular,
σ1 6= σ2.
(1) If [σ1, σ2] = 1, then [M (Γ1), M (Γ2)] = 0.
(2) If [σ1, σ2] 6= 1 and Γσ2 1 6= Γσ1 2 , then M (Γ1)M (Γ2) = [M (Γ1), M (Γ2)] = 0.
(3) If [σ1, σ2] 6= 1 and Γσ2 1 = Γσ1 2 , then [M (Γ1), M (Γ2)] = M (Γσ1 2 ).
Proposition 1.5.
Let Γi be M-sets with corresponding reflections σi, i = 1, 2. If σ1 6= σ2, then
(1) M (Γ1)M (Γ2)M (Γ1) = 0.
(2) [M (Γ1), M (Γ2)] = 0 or [M (Γ1), M (Γσ2 2 )] = 0.
Lemma 1.1.
The transformations Hv , v ∈ V , together with the transformations M [Γ], Γ a M-set in
β, generate a Lie-algebra E of dimension 78. E is of type E6. H is a Cartan sub-algebra
of E, the transformations M [Γ] form a set of 72 roots.
x is a non-singular vector. We say
x∈Γ
that s and σs correspond to Γ. Each non-singular vector, or each reflection, corresponds
to exactly two M-sets. This is obvious, as W is transitive on M-sets and non-singular
b
vectors. Also, if Γ corresponds to s and σ, then Γg corresponds to sg and σ g for all g ∈ H.
Proposition 1.2.
Let Γ be a M-set with corresponding non-singular vector s and reflection σ = σs. Let
τ 6= σ be a reflection.
(1) The M-sets corresponding to s are Γ and Γσ .
Γ ∪ Γσ = {x ∈ β|(x|s) = 1}.
(2) If [σ, τ ] = 1, then Γτ = Γ and Γστ = Γσ .
(3) If στ has order 3, then Γ ∩ Γτ = Γ ∩ CV (τ ) has size 3 and Γσ ∩ Γτ = φ.
Γ ∩ Γστ σ = Γ ∩ CV (τ σ ) has size 3 and Γσ ∩ Γστ σ = φ. Hence also
Γ ∩ Γστ = Γ ∩ Γτ σ = φ.
GeToPhyMa-2014
http ://algtop.net/geto14
June 3-6, 2014, Rabat,
Morocco
Part 3 :
b and x ∈ β.
b acts monomially on A by egx = exg , g ∈ H
Remark 3. The group H
b and M-sets Γ, H
b induces automorphisms of the
As M (Γ)g = M (Γg ) for all g ∈ H
Lie-algebra E.
For X ∈ E denote by X t the transposed with respect to the base ex, x ∈ β. For a M-set
Γ with corresponding reflection it holds that M (Γ)t = M (Γσ ). Hence, the map X 7→ X t
is an automorphism of E.
Definition 1.3.
b denote tg the automorphisim of E sending X ∈ E to (X g )t = (X t)g . In
For g ∈ H
particular, the fixed space CE(tg ) is a sub-Lie-algebra of E.
b give rise to sub-Lie-algebras of type F4. Let L be a totally
Central involution γ of H
singular line of V, i.e. L is a 2-dimensional subspace, such that Q(x) = 0 for all x ∈ L.
b with CV (γ) = L⊥.
Let γ = γL the uniquely determined element of H
Proposition 1.6.
(1) γ is an involution and [V, γ] = L < L⊥ = CV (γ).
(2) (x|xγ ) = 0 for all x in V.
(3) γ fixes exactly 12 vectors s with Q(s) = 1, the elements of L⊥ \ L.
Proof :
Let x ∈ V and v ∈ L⊥. Then (x|v) = (xγ |v γ ) = (xγ |v). Hence, x − xγ ∈ L⊥⊥ = L.
This proves (1). Moreover, xγ = x + u for some u ∈ L.
As Q(x) = Q(xγ ) = Q(x) + (x|u) + Q(u) = Q(x) + (x|u), it follows that
0 = (x|u) = (x|x + u) = (x|xγ ). Hence (2). As L is totally singular and Q has
Witt-index 2, all vectors s ∈ L⊥ \ L are non-singular. Hence (3).
Corollary 1.2.
Let γ = γL as above. tγ permutes the roots M (Γ), Γ a M-set. tγ normalizes the
Cartan-algebra H, defined in Proposition 1.3.
(1) tγ fixes exactly 24 roots M (Γ) and has 24 orbits of length 2.
(2) If M (Γ)tγ 6= M (Γ), then [M (Γ), M (Γ)tγ ] = 0.
(3) dim CH (tγ ) = 4.
(4) dim CE(tγ ) = 52.
R´
ef´
erences
[1] M. Aschbacher, The 27-dimensional module for E6. I, Invent. math. 89, 159-195(1987).
[2] J. E. Humphreys. Introduction to Lie-algebras and representation Theory. SpringerVerlag, Berlin, Heidelberg, New York, 1970.
[3] J. P. Serre, Lie-algebras and Lie-groups, New York, W. A, Benjamin, 1965.
[4] D. J. Winter, Abstract Lie-algebras, Cambridge, Mass, M. I. T-Press, 1972.

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