Lie algebras with S3 or S4-action, and generalized Malcev algebras

0.35 MB·English

by  Alberto Elduque

LIE ALGEBRAS WITH S OR S -ACTION, AND
3 4
GENERALIZED MALCEV ALGEBRAS
ALBERTO ELDUQUE⋆ AND SUSUMUOKUBO∗
Abstract. Lie algebras endowed with an action by automorphisms of
8 any of thesymmetric groups S3 or S4 are considered, and their decom-
0 position into adirect sum of irreduciblemodules for thegiven action is
0 studied.
2 In case of S3-symmetry, the Lie algebras are coordinatized by some
n nonassociativesystems,whicharetermedgeneralizedMalcevalgebras,as
a they extend the classical Malcev algebras. These systems are endowed
J with a binary and a ternary products, and include both the Malcev

6 algebras and the Jordan triple systems.
1

]
A
R Introduction
.
h The Tetrahedron algebra is an infinite dimensional Lie algebra which is
t
a endowedwithanaturalactionbyautomorphismsofS4,thesymmetricgroup
m of degree 4 (see [Eld07]). Lie algebras with suchan action have been investi-
[ gated bytheauthorsin[EO07], whereitis shownthattheseLiealgebras are

coordinatizedbyaclassofnonassociativealgebrasthatinclude,asaveryim-
1
v portant case, the structurable algebras introduced by Allison [All78]. These
1 algebras are very useful in providing models of simple Lie algebras, models
8
which reflect a S -symmetry (see [EO07] and the references there in).
4 4
2 In[IT07, Problem20.10] theauthorsposedthequestionof howtheTetra-
. hedron algebra decomposes into a direct sum of irreducible modules for S .
1 4
0 The first aim of this paper is to study the decomposition into direct sums
8 of irreducible S -modules of any Lie algebra endowed with an action of S
4 4
0
by automorphisms. This will be done in Section 1. The answer for the
:
v Tetrahedron algebra is particularly simple, as only the two irreducible three
i
X dimensional modules for S live inside the Lie algebra. This will be checked
4
r in Section 2.
a
On the other hand, given any Lie algebra endowed with an action of S
4
by automorphisms, Klein’s 4-group induces a grading of the Lie algebra
over Z Z . The (¯0,¯0)-component of this grading, that is, the subspace
2 2
×
of elements fixed by any element in Klein’s 4-group, is naturally endowed
with an action of the symmetric group S by automorphisms. Lie algebras
3
with such an action are studied in Section 3, where the generalized Malcev
algebras are introduced.
Date: January 15, 2008.
⋆ Supported by the Spanish Ministerio de Educaci´on y Ciencia and FEDER (MTM
2007-67884-C04-02) and by the Diputaci´on General de Arago´n (Grupo de Investigaci´on
deA´lgebra).
∗ Supported in part byU.S. Department of Energy Grant No. DE-FG02-91ER40685.
1
2 ALBERTOELDUQUEANDSUSUMUOKUBO
As already noticed by Mikheev [Mik92] and Grishkov [Gri03], Malcev
algebrasappearascoordinatealgebrasofLiealgebraswithanactionofS by
3
automorphisms satisfying an extra condition, which in our terms translates
into the fact that, as a module for S , there is no submodule isomorphic to
3
the alternating one dimensional module for S . This will be the subject of
3
Section 4, where the examples of Malcev algebras and Jordan triple systems
are discussed. Different examples of Lie algebras with an action of S by
3
automorphisms and of their coordinate algebras will be given in the last
Section 5.
Throughout the paper, all the vector spaces considered will be defined
over a ground field k of characteristic = 2,3. Unadorned tensor products
6
will be defined over k.
Recall that the symmetric group S is the semidirect product of Klein’s
4
4-group V = τ ,τ and the symmetric group S = ϕ,τ . Here
4 1 2 3
h i h i
τ = (12)(34),
1
τ = (23)(14),
2
ϕ = (123) : 1 2 3 1
7→ 7→ 7→
τ = (12).
The symmetric group S appears both as a subgroup and as a quotient
3
S /V of the symmetric group S .
4 4 4
There are exactly five non-isomorphic irreducible modules for the sym-
metric group S (see, for instance, [FH91]):
4
U = ku, the trivial module: σ(u) = u for any σ S .
4
• ∈
U′ = ku′, the alternating module: σ(u′) = ( 1)σu′ for any σ S ,
4
• − ∈
where ( 1)σ denotes the signature of the permutation σ.
−
W = (α ,α ,α ) k3 : α +α +α = 0 , thetwodimensionalirre-
1 2 3 1 2 3
• { ∈ }
ducible module. This is a natural module for S (σ (α ,α ,α ) =
3 1 2 3
(ασ−1(1),ασ−1(2),ασ−1(3)) and hence a module for S4 in which the
(cid:0) (cid:1)
elements of V act trivially.
4
V = (α ,α ,α ,α ) k4 : α +α +α +α = 0 , the standard
1 2 3 4 1 2 3 4
• { ∈ }
module.
V′ = U′ V.
• ⊗
Themodules U, U′ and W form a family of representatives of the isomor-
phism classes of the irreducible modules for S .
3
1. Lie algebras with S -action
4
Let g be a Lie algebra over our ground field k endowed with a group
homomorphism
S Aut(g).
4
→
As in [EO07], the action of Kleins’s 4-group V gives a Z Z -grading
4 2 2
×
on g:
g = t g g g , (1.1)
0 1 2
⊕ ⊕ ⊕
LIE ALGEBRAS WITH S3 OR S4-ACTION 3
where
t = x g : τ (x) = x, τ (x) = x (= g ),
1 2 (¯0,¯0)
{ ∈ }
g = x g : τ (x) = x, τ (x) = x (= g ),
0 1 2 (¯1,¯0)
{ ∈ − } (1.2)
g = x g : τ (x) = x, τ (x) = x (= g ),
1 1 2 (¯0,¯1)
{ ∈ − }
g = x g : τ (x) = x, τ (x) = x (= g ).
2 1 2 (¯1,¯1)
{ ∈ − − }
(Here, the subindices 0,1,2 must be considered modulo 3.)
Assume that the action of V is not trivial, as otherwise the S -action is
4 4
just an action of S S /V , and let then A denote the subspace g . For
3 4 4 0
≃
x g , define
0
∈
ι (x) = x g , ι (x) = ϕ(ι (x)) g , ι (x) = ϕ2(ι (x)) g . (1.3)
0 0 1 0 1 2 0 2
∈ ∈ ∈
Thus
g = t 2 ι (A) .
⊕ ⊕i=0 i
Asshownin[EO07],Abecomesanalgebrawithinvolution(A, , )where:
(cid:0) (cid:1)
· −
The involution (involutive antiautomorphism) is given by
•
ι (x¯)= τ(ι (x))
0 0
−
foranyx A. Note thatτ(g ) =g andthatthisgives immediately,
0 0
∈
since ϕτ = τϕ2, the following actions:
τ(ι (x)) = ι (x¯), τ(ι (x)) = ι (x¯), τ(ι (x)) = ι (x¯). (1.4)
0 0 1 2 2 1
− − −
The multiplication is given by
•
ι (x y) = [ι (x),ι (y)], (1.5)
0 1 2
·
for any x,y A.
∈
Moreover, consider the space of Lie related triples (see [AF93]):
lrt(A, , ) = (d ,d ,d ) gl(A)3 :
0 1 2
· − { ∈
d¯(x y) =d (x) y+x d (y), x,y A, i= 0,1,2 ,
i i+1 i+2
· · · ∀ ∈ ∀ }
where
d¯(x) = d(x¯).
This space lrt(A, , ) is a Lie algebra under componentwise bracket, and it
· −
comes endowed with the action of S by automorphisms given by:
3
ϕ (d ,d ,d ) = (d ,d ,d ),
0 1 2 2 0 1
(1.6)
τ(cid:0)(d0,d1,d2)(cid:1) = (d¯0,d¯2,d¯1),
which is compatible with(cid:0)the action (cid:1)of S3 on t, that is,
ρ σ(d) = σ ρ(d) ,
for any σ S3 and d t (see(cid:0)[EO0(cid:1)7, eq(cid:0)s. (1.(cid:1)8) and (2.3)]), where ρ is the
∈ ∈
linear map:
ρ:t gl(A)3
−→
d ρ (d),ρ (d),ρ (d) ,
0 1 2
7→
with (cid:0) (cid:1)
ι ρ (d)(x) = [d,ι (x)]
i i i
(cid:0) (cid:1)
4 ALBERTOELDUQUEANDSUSUMUOKUBO
for any d t and x A. It turns out that ρ is a Lie algebra homomorphism
∈ ∈
and that ρ(t) is contained in lrt(A, , ). Moreover, there appears a skew-
· −
symmetric bilinear map
δ :A A lrt(A, , )
× −→ · −
(x,y) ρ [ι (x),ι (y)] = δ (x,y),δ (x,y),δ (x,y) .
0 0 0 1 2
7→
Theorem 1.7. ([EO07(cid:0), Theorem 2(cid:1).4])(cid:0)For any a,b,x,y,z A(cid:1)and i,j =
∈
0,1,2:
(i) δ (a,b),δ (x,y) = δ δ (a,b)(x),y +δ (x,δ (a,b)(y) ,
i j j i−j j i−j
(ii) δ (x¯,y z)+δ (y¯,z x)+δ (z¯,x y)= 0,
(cid:2)0 1 (cid:3) (cid:0) 2 (cid:1) (cid:1)
· · ·
(iii) δ (x,y)(z)+δ (y,z)(x)+δ (z,x)(y) = 0,
0 0 0
(iv) δ (x,y) = L L L L ,
1 y¯ x x¯ y
−
(v) δ (x,y) = R R R R ,
2 y¯ x x¯ y
−
(vi) δ (x,y) = δ (x¯,y¯) (or τ δ(x,y) = δ(x¯,y¯)),
i −i
where L and R denote, respectively, the left and right multiplication by x
x x (cid:0) (cid:1)
in the algebra (A, ).
·
The 4-tuple (A, , ,δ) is then called a normal Lie related triple algebra
· −
(see [Oku05]), or normal LRTA for short.
Conversely, given a normal LRTA (A, , ,δ), consider three copies of A:
· −
ι (A), i= 0,1,2, and the Lie subalgebra of inner Lie related triples:
i
2
inlrt(A, , ,δ) = ϕi δ(A,A) .
· −
i=0
X (cid:0) (cid:1)
Then the vector space
g(A, , ,δ) = inlrt(A, , ,δ) 2 ι (A)
· − · − ⊕ ⊕i=0 i
is a Lie algebra [EO07, Theorem 2.6] with bracket determined by:
(cid:0) (cid:1)
inlrt(A, , ,δ) is a Lie subalgebra,
• · −
[(d ,d ,d ),ι (x)] = ι d (x) , (d ,d ,d ) inlrt(A, , ,δ), x A,
0 1 2 i i i 0 1 2
• ∀ ∈ · − ∀ ∈
[ι (x),ι (y)] = ι (x y) x,y A, i= 0,1,2,
i i+1 i+2 (cid:0) (cid:1)
• · ∀ ∈ ∀
[ι (x),ι (y)] = ϕi δ(x,y) , x,y A, i= 0,1,2.
i i
• ∀ ∈ ∀
(1.8)
(cid:0) (cid:1)
Moreover, the symmetric group S acts naturally by automorphisms on
4
g(A, , ,δ) by means of:
· −
V acts trivially on inlrt(A, , ,δ) and ϕ and τ act by (1.6),
4
• · −
ϕ ι (x) = ι (x), i= 0,1,2,
i i+1
• ∀
τ((cid:0)ι0(x))(cid:1)= ι0(x¯), τ(ι1(x)) = ι2(x¯), τ(ι2(x)) = ι1(x¯), (1.9)
• − − −
τ ι (x) = ι (x), τ ι (x) = ι (x) for i = 1,2,
1 0 0 1 i i
• −
τ2(cid:0)ι1(x)(cid:1) = ι1(x), τ2(cid:0)ιi(x)(cid:1) = ιi(x) for i = 0,2,
• −
for any x A.
(cid:0) (cid:1) (cid:0) (cid:1)
∈
One can also consider the larger Lie algebra
˜g(A, , ,δ) = lrt(A, , ,δ) 2 ι (A) (1.10)
· − · − ⊕ ⊕i=0 i
(cid:0) (cid:1)
LIE ALGEBRAS WITH S3 OR S4-ACTION 5
with the same bracket given in (1.8), which again is endowed with an action
of S by automorphisms given by (1.9). Besides, if g is a Lie algebra with
4
an action of S and (A, , ,δ) is the associated normal LRTA, then the
4
· −
Lie algebra homomorphism ρ : t lrt(A, , ) extends to a Lie algebra
→ · −
homomorphism
ρ˜:g ˜g(A, , ,δ)
−→ · −
compatible with the action of S , and such that ρ˜ ι (x) = ι (x) for any
4 i i
x A and i = 0,1,2.
∈ (cid:0) (cid:1)
Proposition 1.11. Let g be a Lie algebra with S -action and let g = t
4
g g g be the associated Z Z -grading as in (1.1). Assume tha⊕t
0 1 2 2 2
⊕ ⊕ ×
g = 0 and that g has no proper ideals invariant under the action of S
0 4
6
(this happens, in particular, if g is simple). Then the homomorphism ρ˜ is
one-to-one and its image is g(A, , ,δ).
· −
Proof. Since ρ˜is a homomorphism of Lie algebras with S -action, its kernel
4
kerρ˜is an ideal invariant under the action of S , so it is trivial. Hence ρ˜is
4
one-to-one. Ontheother hand,thesubalgebraofg generated byg g g
0 1 2
⊕ ⊕
is an ideal invariant under S , so it is the whole g, and hence ρ˜(g) is the
4
subalgebra generated by 2 ι (A), which is precisely g(A, , ,δ). (cid:3)
⊕i=0 i · −
Now we are ready to decompose the subspace g g g in (1.1) as a
0 1 2
⊕ ⊕
direct sum of irreducible modules for the action of S .
4
Theorem 1.12. Let g be a Lie algebra endowed with an action of S by
4
automorphisms, and let 0 = x A= g . Then:
0
6 ∈
(i) If x¯ = x, then 2 kι (x) is a S -module isomorphic to the stan-
− i=0 i 4
dard module V.
P
(ii) Ifx¯ = x, then 2 kι (x)isaS -module isomorphic toV′ = U′ V.
i=0 i 4 ⊗
Proof. Equations (1.2), (1.3) and (1.4) show that in both cases the sub-
P
space 2 kι (x) is invariant under the action of S . In the first case the
i=0 i 4
assignment
P
ι (x) (1,1, 1, 1), ι (x) ( 1,1,1, 1), ι (x) (1, 1,1, 1),
0 1 2
7→ − − 7→ − − 7→ − −
provides the required isomorphism, while in the second case, the assignment
ι (x) u′ (1,1, 1, 1), ..., gives the result. (cid:3)
0
7→ ⊗ − −
Corollary 1.13. Let g be a Lie algebra endowed with an action of S by
4
automorphisms. Then 2 ι (A) = g g g is a direct sum of copies of
⊕i=0 i 0⊕ 1⊕ 2
the two irreducible three-dimensional modules for S .
4
Letusturnnow ourattention totheaction ofS (actually of S S /V )
4 3 4 4
≃
on the Lie algebra lrt(A, , ,δ).
· −
On the two one-dimensional irreducible modules (the trivial one U and
the alternating one U′ for S (or S )), the cycle ϕ acts trivially, while ϕ acts
4 3
with minimal polynomial X2 + X + 1 on the two-dimensional irreducible
module. But for (A, , ) an algebra with involution:
· −
(d ,d ,d ) lrt(A, , ) : ϕ (d ,d ,d ) = (d ,d ,d )
0 1 2 0 1 2 0 1 2
{ ∈ · − }
= (d,d,d) gl((cid:0)A)3 : d¯(x y(cid:1))= d(x) y+x d(y) x,y A
{ ∈ · · · ∀ ∈ }
= (d,d,d) gl(A)3 : d(x y) = d(x) y+x d(y) x,y A
{ ∈ ∗ ∗ ∗ ∀ ∈ }
6 ALBERTOELDUQUEANDSUSUMUOKUBO
where x y = x y for any x,y A. Hence, the subspace of fixed elements
∗ · ∈
by ϕ in lrt(A, , ) is naturally isomorphic to the Lie algebra of derivations
· −
of the algebra (A, ):
∗
Fixlrt(A,·,−)(ϕ) der(A, ).
≃ ∗
Given an algebra with involution (A, , ), der(A, , ) will denote the Lie
· − · −
algebra:
der(A, , ) = d der(A, ) : d¯= d .
· − { ∈ · }
Proposition 1.14. Let (A, , ) be an algebra with involution. Then:
· −
(i) The direct sum of the trivial S -submodules of lrt(A, , ) is the sub-
4
· −
space (d,d,d) :d der(A, , ) .
{ ∈ · − }
(ii) The direct sum of the S -submodules of lrt(A, , ) isomorphic to the
4
· −
alternating moduleis (d,d,d) :d sder(A, , ) , wheresder(A, , )
{ ∈ · − } · −
is the subspace of skew-derivations of (A, , ):
· −
sder(A, , ) = d gl(A) :d(x y)= d(x) y x d(y) and d¯= d .
· − { ∈ · − · − · − }
(iii) The direct sum of the irreducible S -submodules of lrt(A, , ) iso-
4
· −
morphic to the two-dimensional irreducible module W is
(d ,d ,d ) lrt(A, , ) : d +d +d =0 .
0 1 2 0 1 2
{ ∈ · − }
Proof. The direct sum of the trivial submodules is
(d ,d ,d ) lrt(A, , ) : ϕ (d ,d ,d ) = (d ,d ,d )= τ (d ,d ,d )
0 1 2 0 1 2 0 1 2 0 1 2
{ ∈ · − }
= (d ,d ,d ) lrt(A, , ) : d = d = d = d¯
(cid:0) 0 1 2 (cid:1) 0 (cid:0) 1 2 (cid:1)0
{ ∈ · − }
because of (1.6), and this gives (i).
Also, the direct sum of the alternating submodules is
(d ,d ,d ) lrt(A, , ) : ϕ (d ,d ,d ) = (d ,d ,d ) = τ (d ,d ,d )
0 1 2 0 1 2 0 1 2 0 1 2
{ ∈ · − − }
= (d ,d ,d ) lrt(A, , ) :d = d = d = d¯ ,
(cid:0) 0 1 2 (cid:1) 0 (cid:0)1 2 (cid:1)0
{ ∈ · − − }
and this proves (ii).
Finally, the direct sum of the S -submodules of lrt(A, , ) which are
4
· −
isomorphic to the two-dimensional irreducible module W is the kernel of
1+ϕ+ϕ2, and this gives the result in (iii). (cid:3)
Let us pause to give an example of a normal LRTA (A, , ,δ) where
· −
der(A, , ) is strictly contained in the Lie algebra of derivations der(A, ).
· − ·
Example 1.15. Let g be any Z Z -graded Lie algebra:
2 2
×
g = g g g g ,
(¯0,¯0) (¯1,¯0) (¯0,¯1) (¯1,¯1)
⊕ ⊕ ⊕
with g = 0 = g , and let ν and µ the order two automorphisms of g
(¯0,¯1) (¯1,¯1)
6 6
given by:
id on g g ,
ν = (¯0,¯0)⊕ (¯1,¯0)
( id on g(¯0,¯1) g(¯1,¯1),
− ⊕
id on g g ,
µ = (¯0,¯0)⊕ (¯0,¯1)
( id on g(¯1,¯0) g(¯1,¯1).
− ⊕
LIE ALGEBRAS WITH S3 OR S4-ACTION 7
The symmetric group S acts by automorphisms on g3 as follows:
4
τ (x,y,z) = (x,ν(y),ν(z)),
1
τ2(cid:0)(x,y,z)(cid:1) = (ν(x),y,ν(z)),
ϕ(cid:0)(x,y,z)(cid:1) = (z,x,y),
τ(cid:0)(x,y,z)(cid:1) = (µ(x),µ(z),µ(y)).
(Compare to [EO07, E(cid:0)xample 3(cid:1).5].) Then the decomposition in (1.1) is
given by:
3
t= g g ,
(¯0,¯0) (¯1,¯0)
⊕
g = (x,0,0) : x g g ,
0 (cid:0) (cid:1) (¯0,¯1) (¯1,¯1)
{ ∈ ⊕ }
g = (0,x,0) : x g g ,
1 (¯0,¯1) (¯1,¯1)
{ ∈ ⊕ }
g = (0,0,x) : x g g ,
2 (¯0,¯1) (¯1,¯1)
{ ∈ ⊕ }
so that A= g g is a normal LRTA with
(¯0,¯1) (¯1,¯1)
⊕
x y = 0, x¯ = µ(x),
· −
for any x A.
∈
The Lie algebra of derivations of (A, ) is then the whole general linear
·
algebra:
der(A, ) = gl(A),
·
while
der(A, , ) = f gl(A) :f(x¯)= f(x) x
· − { ∈ ∀ }
= f gl(A) :fµ= µf
{ ∈ }
= f gl(A) :f g g , f g g ,
(¯0,¯1) (¯0,¯1) (¯1,¯1) (¯1,¯1)
{ ∈ ⊆ ⊆ }
and (cid:0) (cid:1) (cid:0) (cid:1)
sder(A, , ) = f gl(A) : fµ= µf
· − { ∈ − }
= f gl(A) : f g g , f g g . (cid:3)
(¯0,¯1) (¯1,¯1) (¯1,¯1) (¯0,¯1)
{ ∈ ⊆ ⊆ }
(cid:0) (cid:1) (cid:0) (cid:1)
In [EO07, Theorem 2.4] it is proved that the unital normal LRTA’s are
precisely the structurable algebras. In this case, the previous Proposition
becomes simpler.
Corollary 1.16. Let (A, , ) be a unital algebra with involution, then
· −
sder(A, , ) = 0.
· −
In particular, the alternating module does not appear in the decomposition
of lrt(A, , ) as a direct sum of irreducible S -modules.
3
· −
Proof. Any d sder(A, , ) satisfies d¯= d and
∈ · − −
d(x y)= d(x) y x d(y)
· − · − ·
for any x,y A. With x = y = 1 it follows that d(1) = 0, and then with
y = 1 that d(∈x) = d(x) for any x, so d= 0. (cid:3)
−
8 ALBERTOELDUQUEANDSUSUMUOKUBO
Remark 1.17. Given an algebra with involution (A, , ), the Lie algebra
· −
l = lrt(A, , ) decomposes as
· −
l = l l l ,
+ − 2
⊕ ⊕
where l is the subspace of fixed elements by the action of any σ S (that
+ 3
∈
is, the sum of the trivial S -submodules of l), l is the direct sum of the
3 −
alternating S -submodules of l and l is the sum of the S -submodules of l
3 2 3
which are isomorphic to the unique two-dimensional irreducible module for
S . Then any (d ,d ,d ) l decomposes as
3 0 1 2
∈
(d ,d ,d ) = (d,d,d)+(f,f,f)+(f ,f ,f ),
0 1 2 0 1 2
where (d,d,d) l , (f,f,f) l and (f ,f ,f ) l are given by:
+ − 0 1 2 2
∈ ∈ ∈
1
d = (d +d +d )+(d¯ +d¯ +d¯ ) ,
0 1 2 0 1 2
6
f = 1(cid:0)(d +d +d ) (d¯ +d¯ +d¯ )(cid:1),
0 1 2 0 1 2
6 −
1(cid:0) (cid:1)
f = (2d d d ) (indices modulo 3).
i i i+1 i+2
3 − −
Notethatindeed(1+ϕ+ϕ2)(f ,f ,f ) = 0,so(f ,f ,f ) l and,similarly,
0 1 2 0 1 2 2
(d,d,d) l and (f,f,f) l . ∈ (cid:3)
+ −
∈ ∈
2. The Tetrahedron algebra
The results in the previous section, together with [Eld07], give an imme-
diate answer to [IT07, Problem 2.10], which asks for the decomposition of
the Tetrahedron algebra into a direct sum of irreducible S -modules.
4
Recall that the Tetrahedron algebra g⊠ has been defined in [HT07] in
connection with the so called Onsanger algebra introduced in [Ons44], in
which the free energy of the two dimensional Ising model was computed.
The Tetrahedron algebra g⊠ is the Lie algebra over k with generators
X :i,j 0,1,2,3 , i= j
ij
∈ { } 6
and relations (cid:8) (cid:9)
X +X = 0 for i = j,
ij ji
6
[X ,X ] = 2(X +X ) for mutually distinct i,j,k,
ij jk ij jk
[X ,[X ,[X ,X ]]] = 4[X ,X ] for mutually distinct h,i,j,k.
hi hi hi jk hi jk
Consider the basis x = −1 2 , y = −10 , z = 1 0 of the three
0 1 −21 0 −1
dimensional simple Lie algebra sl . Then [HT07, Proposition 6.5 and The-
2
(cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9)
orem 11.5] the Tetrahedron algebra is isomorphic to the three-point loop
algebra sl , =k[t,t−1,(t 1)−1], by means of the isomorphism
2
⊗A A −
Ψ :g⊠ sl2 , (2.1)
→ ⊗A
LIE ALGEBRAS WITH S3 OR S4-ACTION 9
determined by:
Ψ(X ) =x 1, Ψ(X ) = y 1, Ψ(X )= z 1,
12 23 31
⊗ ⊗ ⊗
Ψ(X )= y t+z (t 1),
03
⊗ ⊗ −
Ψ(X )= z t′+x (t′ 1),
01
⊗ ⊗ −
Ψ(X )= x t′′+y (t′′ 1),
02
⊗ ⊗ −
where
t′ = 1 t−1 and t′′ = (1 t)−1. (2.2)
− −
ThesymmetricgroupS embedsnaturallyinthegroupofautomorphisms
4
Aut(g⊠) by means of
σ(X ) = X ,
ij σ(i)σ(j)
for any σ S and 0 i < j 3. Here we identify 0 and 4. On the other
4
∈ ≤ ≤
hand, S embeds as a group of automorphisms of g = sl as follows
4 2
⊗ A
([Eld07, Theorem 1.4]):
(i) ϕ= ϕs ϕA, where ϕs is the order 3 automorphism of sl2 given by
⊗
ϕs(x) = y, ϕs(y)= z, ϕs(z) = x,
and ϕ is the order 3 automorphism of the k-algebra determined
A
A
by
ϕ (t) = 1 t−1 =t′.
A
−
(ii) τ = τs τA, where τs is the order 2 automorphism of sl2 given by
⊗
τs(x) = x, τs(y) = z, τs(z) = y,
− − −
andτ istheorder2automorphismof determinedbyτ (t)= 1 t.
A A
A −
(iii) τ is the automorphism of g, as a Lie algebra over , given by
1
A
τ (x 1) = x 1,
1
⊗ − ⊗
τ (y 1) = z t′+x (t′ 1) , (2.3)
1
⊗ − ⊗ ⊗ −
τ (z 1) = x t′′+y (t′′ 1).
1 (cid:0) (cid:1)
⊗ ⊗ ⊗ −
(iv) τ is the automorphism of g, as a Lie algebra over , given by
2
A
τ (x 1) = y t+z (t 1),
2
⊗ ⊗ ⊗ −
τ2(y 1) = y 1, (2.4)
⊗ − ⊗
τ (z 1) = x t′′+y (t′′ 1) .
2
⊗ − ⊗ ⊗ −
With this action of S4, the iso(cid:0)morphism Ψ in (2.1)(cid:1)becomes an isomor-
phism of Lie algebras with S -action.
4
A particular basis u ,u ,u of g = sl , as a module for , plays a
0 1 2 2
{ } ⊗A A
key role in [Eld07]. It is defined by:
1
u = Ψ(X +X ),
0 02 31
4
1
u = Ψ(X +X ), (2.5)
1 03 12
4
1
u = Ψ(X +X ),
2 01 23
4
and satisfies [Eld07, Theorem 1.9]:
[u ,u ]= u t, [u ,u ] = u t′, [u ,u ] = u t′′. (2.6)
0 1 2 1 2 0 2 0 1
− − −
10 ALBERTOELDUQUEANDSUSUMUOKUBO
Moreover, theseelements also generate g as a Liealgebra over k. Theaction
of S on this -basis is given by:
4
A
τ (u ) = u = τ (u ),
1 0 0 2 0
−
τ (u ) = u = τ (u ),
1 1 1 2 1
− −
τ (u ) = u = τ (u ), (see [Eld07, Theorem 2.2])
1 2 2 2 2
−
ϕ(u ) = u , (indices modulo 3, see [Eld07, (1.8)])
i i+1
1
τ(u ) = Ψ(X X ) = [u ,u ] = u t, (see [Eld07, Theorem 1.9])
1 03 12 0 1 2
4 − −
τ(u ) = τϕ(u ) = ϕ2τ(u ) = u t′′,
2 1 1 1
τ(u ) = τϕ2(u )= ϕτ(u ) = u t′.
0 1 1 0
Besides, the subspaces t, g , g and g in (1.2) become:
0 1 2
t = 0, g = u , g = u , g = u ,
0 0 1 1 2 2
A A A
and the associated normal LRTA can be identified to ( , , ,δ) with (see
A · −
[Eld07, Proposition 3.1]):
a b = τ ϕ (a) (τ ϕ2 (b) ,
· A A A A
a¯ = t′τ (a).
(cid:0) A (cid:1) (cid:1)
−
(Here, the usual multiplication of = k[t,t−1,(1 t)−1] is denoted by
A −
juxtaposition.)
Then Corollary 1.13, together with t = 0, gives the answer to [IT07,
Problem 2.10]:
Proposition 2.7. The Tetrahedron algebra is a direct sum of copies of the
two irreducible three-dimensional modules for S .
4
But a more precise statement can be given, by providing a concrete de-
composition into a direct sum of irreducible S -modules. To do so, and
4
because of Theorem 1.12, let us first compute the elements a with
∈ A
a¯ = a. Note that for a ,
± ∈ A
a¯ = a a = t′τ (a)
A
⇔ −
ta= τ (ta), as tt′t′′ = 1 and τ (t) = (t′′)−1.
A A
⇔ −
But, as = k[t,t−1,(1 t)−1] k(t), any b is written uniquely as
A − ⊆ ∈ A
p(t)
b = , with p(t) a polynomial in k[t] with either p(0) = 0 or p(1) = 0.
(t(1−t))s 6 6
Since τ (t)= 1 t, τ (b) = b if and only if τ p(t) = p(t) or, equivalently,
A A A
−
p(t) = p(1 t). This is equivalent to the condition p(t) k[t(1 t)] =
− (cid:0) (cid:1) ∈ −
k[ t 1 2]. Note that the set (t(1 t))r : r N 0 is a k-basis of the
− 2 { − ∈ ∪{ }}
polynomial ring k[t(1 t)].
(cid:0) (cid:1) −
In a similar vein, τ (b) = b if and only if p(t) (2t 1)k[t(1 t)].
A
− ∈ − −
Hence, for any a :
∈A
a¯ = a a t−1k[t(1 t),(t(1 t))−1],
⇔ ∈ − −
a¯ = a a t−1(2t 1)k[t(1 t),(t(1 t))−1].
− ⇔ ∈ − − −
Therefore, Theorem 1.12 gives:
Theorem 2.8. For any integer s,