Diagonal locally finite Lie algebras and a version of Ado's theorem

33 Pages·2010·0.31 MB·English

Diagonal locally finite Lie algebras and a
version of Ado’s theorem
A.A. Baranov ∗
Institute of Mathematics, Academy of Sciences of Belarus,
Surganova 11, Minsk, 220072, Belarus.
1 Introduction
A.E. Zalesskii developed in [14] a new approach to studying the (two-sided) ideal
lattices of group algebras of locally finite groups via representation theory of finite
groups. He established a 1-1 correspondence between ideals of a group algebra and so-
calledinductivesystems(ofmodulesoverfinitegroups). Thisobservation,inparticular,
enabled him to describe in [13] the ideal lattices of complex group algebras for locally
finite groups that are unions of finite alternating groups. It is now known that this
method can be applied to locally finite Lie algebras and their universal enveloping
algebras. This approach is used in studying the existence of an embedding of a locally
finite Lie algebra L into a locally finite associative algebra. When L is assumed to be
finite-dimensional, this result follows from Ado’s theorem. However, in general, locally
finite Lie algebras do not have this embedding property even if simplicity is assumed.
Therefore, it is natural to seek a description of the simple locally finite Lie algebras
for which the locally finite analog of Ado’s theorem holds. The machinery developed
below allows one to solve this problem as indicated in Corollary 5.11 which is one of
the main results of this paper. We observe that, according to a discussion in [15],
this problem is a natural analog of the following problem (going back to I. Kaplansky
(1965)) for locally finite simple groups: describe the groups G such that the only
nontrivial proper ideal of the complex group algebra CG is the augmentation ideal.
LetLbealocallyfiniteLiealgebra. ThismeansthateveryfinitesetofelementsofL
iscontainedinafinite-dimensionalsubalgebra. Ifthelattercanbechosen(semi)simple,
then L is called locally (semi)simple. Observe that locally finite Lie algebras can be
regarded as an asymptotic version of finite-dimensional ones, and appear in various
applications as the Lie algebras of direct limits of Lie groups. Suppose that L has
countable dimension. (This is in fact a purely technical assumption which allows us
to make the exposition more transparent.) Then L can be expressed in the form
∗
This paper is prepared in the framework of an INTAS project “Noncommutative algebra and
geometrywiththefocusonrepresentationtheory”andsupportedbyINTASandbytheFundamental
Research Foundation of Belarus (Grant No F94-011)
1
L = ∪i∈NLi (or L = −li→mLi) where Li are finite-dimensional Lie algebras, and Li ⊂ Li+1
for all i ∈ N. Since L ⊂ L , each L -module ϕ may be considered to be an L -
i i+1 i+1 i
module. We use the notation ϕ↓L to denote this L -module. Let hψi denote the set
i i
of inequivalent composition factors of a module ψ, and IrrL be the set of inequivalent
i
irreducible Li-modules, Let Φi be a finite subset of IrrLi. We say that Φ = {Φi}i∈N is
an inductive system for L if
∪ hϕ↓L i = Φ
ϕ∈Φi+1 i i
for all i ∈ N (cf. Definition 2.9). It can be shown (see, for instance, [15]) that there is a
1-1 correspondence between the inductive systems for a locally semisimple Lie algebra
L and the ideals X of its universal enveloping algebra U(L) with U(L)/X locally finite.
This translates the problem into the language of representation theory. For instance,
if L is simple, then the analog of Ado’s theorem for L holds if and only if there exists
a nontrivial inductive system for L.
One can ask whether the class of locally semisimple Lie algebras contains all simple
locally finite Lie algebras. Recently Yu. Bahturin and H. Strade [3] have constructed
examples of simple locally finite Lie algebras that are not locally simple. Actually,
their arguments in [3] can be used to prove that these Lie algebras are not locally
semisimple for the zero characteristic case. Thus, for studying the problem above for
simple Lie algebras one cannot restrict oneself to locally semisimple Lie algebras. In
particular, it is necessary to look for a reasonable extension of the result above on the
1-1 correspondence to a wider class of locally finite Lie algebras. Our experience shows
that it should be the class of locally perfect Lie algebras. This class is a radical one
(Theorem 2.6), and the quotient of a locally finite Lie algebra by its locally perfect
radical is locally solvable. Observe that simple locally finite Lie algebras are locally
perfect ([2, Theorem 3.2] and our Theorem 2.8). We establish (Theorem 3.9, see also
[4]) a 1-1 correspondence between the set of inductive systems for a locally perfect
Lie algebra L and the set of semiprimitive ideals X of U(L) with U(L)/X locally
finite. This result is similar to Theorem 1.25 in [15] for group algebras over a field of
positivecharacteristic. Theproofisbasedonthefollowinginterestingfact(Lemma3.3)
which seems to be unknown. Let L be a finite-dimensional perfect Lie algebra (i.e.
[L,L] = L), and Φ a finite subset of IrrL. Then codimensions of the annihilators in
U(L) of all finite-dimensional L-modules ϕ such that hϕi = Φ are bounded by some
constant depending on L and Φ.
The correspondence established explains our interest to the inductive systems for
locally perfect Lie algebras. The problem of an explicit description of inductive sys-
tems for a locally simple Lie algebras was investigated by A.G. Zhilinskii [17, 18]. In
Section 5 we extend the notion of diagonality introduced by A.G. Zhilinskii [17] to
locally perfect Lie algebras and prove (Theorem 5.7) that a locally perfect Lie algebra
has a nondegenerate (see Definition 5.4) inductive system if and only if it is diagonal.
In particular, a simple locally finite Lie algebra has a nontrivial inductive system if
and only if it is diagonal. This generalizes the result of A.G. Zhilinskii [17] proved
for locally simple Lie algebras. It follows from the 1-1 correspondence above that a
simple locally finite Lie algebra is diagonal if and only if it can be embedded into a
locally finite associative algebra (Corollary 5.11). Observe that diagonal locally finite
Lie algebras (and only they) have nice faithful representations such that their images
2
generate locally finite associative algebras. One can consider these representations as a
natural analog of finite-dimensional representations of finite-dimensional Lie algebras.
Another, somewhat different approach to representation theory of locally finite Lie
algebras is given in [1] where highest weight modules are studied.
Let L be a perfect finite-dimensional Lie algebra, S = S ⊕ ... ⊕ S be its Levi
1 n
subalgebra where S ,...,S are simple components of S, and V denote the standard
1 n i
S -module. Since L is perfect, for each i there exists the unique irreducible L-module
i
V such that V ↓S = V . The modules V ,...,V are called the standard L-modules.
i i i i 1 n
Let Q be another perfect finite-dimensional Lie algebra. An embedding L ⊂ Q is called
diagonal if hW↓Li ⊆ {V ,...,V ,V∗,...,V∗,T} for all standard Q-modules W, where
1 n 1 n
T is the trivial one-dimensional L-module, V∗ is the dual module for V . For example,
i i
an embedding f : sl(V) → sl(W) is diagonal if and only if there exist bases of V
and W such that f(A) = diag(A,...,A,−At,...,−At) for all matrices A ∈ sl(V).
Assume that L = limL where all L are perfect and all embeddings L ⊂ L are
−→ i i i i+1
diagonal. Then one can check that L is diagonal. Observe that the simple locally finite
Lie algebras constructed in [3] are inductive limits of diagonal embeddings, so they
are diagonal. One can show that all simple diagonal locally finite Lie algebras can be
constructed in such way [5].
Section6containssomeauxiliarylemmasaboutbranchingrulesforrepresentations.
In Section 7 the notion of a Bratteli diagram for a locally perfect Lie algebra is intro-
duced. In Section 8 (Corollary 8.5) we give a diagonality criterion for locally perfect
Lie algebras (on the language of Bratteli diagrams). In Section 9 we prove a general
“Ado’s theorem” for locally perfect Lie algebras with the trivial centers (Theorem 9.4),
i.e. we describe all such algebras that can be embedded into locally finite associative
algebras.
A motivation for investigating locally finite Lie algebras can also be found in [2] and
[16]. We only want to notice here that there exists some parallelism between the theory
ofdiagonal locally finiteLiealgebras and thatof locally semisimpleassociative algebras
(see [8, 10]) via the notion of Bratteli diagrams. On the other hand, the questions
consideredarecloselylinkedwiththerepresentationtheoryofgroupsthatareinductive
limits of Lie or algebraic groups. Representations of the most natural examples of such
groups were studied by G.I. Olshanskii [11], S. Stratila and D. Voiculescu [12], and
others.
I am sincerely grateful to A.E. Zalesskii for attracting me to this circle of problems
and useful discussions, I.D. Suprunenko for her attention and helpful conversations
on representation theory, and the referee for correcting the proof of Lemma 3.3 and
numerous comments which helped to improve the text.
Notation.
The ground field F is an algebraically closed field of zero characteristic. N is the
set of natural numbers. All Lie algebras considered are of finite dimension or locally
finite. U(L) denotes the universal enveloping algebra of a Lie algebra L. If V is an
L-module, then Ann V denotes the annihilator of V in U(L).
U(L)
Let L be a finite-dimensional Lie algebra. Denote by IrrL the set of inequivalent
irreducible finite-dimensional L-modules. For an L-module V let hVi denote the set
3
of inequivalent composition factors of V. If Φ = {V } is a set of L-modules, then
i i∈I
hΦi = ∪ hV i. If S is a subalgebra of L, then V↓S denotes the restriction of the L-
i∈I i
module V to S. More general, let θ : S → L be a homomorphism of Lie algebras, and
V be an L-module. Then θ gives an action of S on V. The corresponding module is
denoted by V↓θ. Sometimes the symbol θ is omitted if it is clear which homomorphism
S
is considered. If Φ = {V } is a set of L-modules, then Φ↓θ = {V ↓θ} . Denote by
i i∈I S i S i∈I
RadL the solvable radical of L. Let S be a Levi subalgebra of L, W be an S-module.
We can consider W as an L-module, setting (RadL)W = 0. Denote this module by
W↑L. If Ψ = {W } is a set of S-modules, then Ψ↑L = {W ↑L} . L is called perfect
i i∈I i i∈I
if [L,L] = L.
Let A be an associative algebra. An ideal X of A is called primitive if it is the
annihilator of an irreducible A-module and semiprimitive if it is the intersection of
primitive ideals. Equivalently, X is semiprimitive if and only if the Jacobson radical
Rad(A/X) is trivial. Let A(−) be the Lie algebra with the basic set A and the multi-
plication [a,b] = ab−ba. We say that ε : L → A is an embedding of a Lie algebra L
into A if ε is an injective homomorphism of L into A(−).
2 Locally perfect Lie algebras
Thebasictoolforinvestigatinglocally finiteLiealgebrasisthenotionofalocal system.
Definition 2.1 LetLbealocallyfiniteLiealgebra. Aset{L } offinite-dimensional
i i∈I
subalgebras of L is called a local system of L if L = ∪ L and for each pair i,j ∈ I
i∈I i
there exists k ∈ I such that L ,L ⊆ L .
i j k
Set i ≤ j if L ⊆ L . Then I is a directed set, i.e. for each pair i,j ∈ I there exists
i j
k ∈ I such that i,j ≤ k. It is clear that L is an inductive limit of the algebras L , that
i
is L = limL .
−→ i
Definition 2.2 A local system {L } of L is called perfect if all L are perfect.
i i∈I i
Definition 2.3 A locally finite Lie algebra is called locally solvable if it has a local
system of solvable algebras and it is called locally perfect if no nontrivial quotient is
locally solvable.
The following lemma explains this terminology.
Lemma 2.4 A locally finite Lie algebra is locally perfect if and only if it has a perfect
local system.
(∞)
Proof. Let L = limL be locally perfect. For every i ∈ I denote by L the smallest
−→ i i
(∞) (∞)
member of the derived series of L . It is clear that L is perfect, L /L is solvable,
i i i i
and L ⊆ L implies L(∞) ⊆ L(∞). Hence L(∞) = limL(∞) is an ideal of L and L/L(∞) is
i j i j −→ i
locally solvable. This implies L = L(∞). Therefore, {L(∞)} is a perfect local system
i i∈I
of L.
Conversely, let {L } be a perfect local system of L, and M be an ideal of L with
i i∈I
L/M locally solvable. Then M = M ∩L is an ideal of L and the quotient L /M is
i i i i i
solvable. Since all L are perfect, M = L for all i ∈ I, forcing M = L.
i i i
4
(∞)
Remark 2.5 By the proof of Lemma 2.4, if L = limL , then {L } is a perfect
−→ i i i∈I
local system of the ideal L(∞) = limL(∞) of L where L(∞) is the smallest member of
−→ i i
the derived series of L . If L is locally perfect, then L = L(∞).
i
In what follows, when we write L = limL for locally perfect L, we mean that
−→ i
{L } is a perfect local system of L. The significance of locally perfect Lie algebras
i i∈I
is shown by the following
Theorem 2.6 Let L be a locally finite Lie algebra. Then there exists a locally perfect
ideal P(L) of L which contains all other locally perfect ideals of L. P(L) is called the
locally perfect radical of L. The quotient algebra L/P(L) is locally solvable.
(∞)
Proof. Let L = limL . Set P(L) = limL (see Remark 2.5). Then P(L) is a locally
−→ i −→ i
perfect ideal of L, and L/P(L) is locally solvable. Let Q be a locally perfect ideal of L.
Then Q/(Q∩P(L)) is locally solvable. This implies Q = Q∩P(L), forcing Q ⊆ P(L).
Definition 2.7 Let L be a locally finite Lie algebra. The largest locally solvable ideal
R(L) of L is called the locally solvable radical. If R(L) = 0, then L is called semisimple.
It is well known that the solvable radical of a perfect finite-dimensional Lie algebra
is nilpotent. Consequently, the locally solvable radical of a locally perfect Lie algebra
is locally nilpotent (R(L)∩L ⊂ RadL ).
i i
Proposition 2.8 Let L be a simple locally finite Lie algebra. Then L is semisimple
and locally perfect.
Proof. We have to show that R(L) = 0 and P(L) = L (see Theorem 2.6 and Defini-
tion 2.7). If R(L) = 0, then P(L) 6= 0. Since L is simple, P(L) = L. Therefore, it
suffices to prove that R(L) 6= L. Assume that R(L) = L. Then L = limL where all
−→ i
L are solvable. Since [L,L] 6= 0, there exist x,y ∈ L such that z = [x,y] 6= 0. Let M
i
be the ideal of L generated by z. Since L is simple, M = L, so x,y ∈ M. Therefore,
there exists i ∈ I such that z ∈ L and x,y ∈ N where N is the ideal of L generated
i i
(n−1) (n) (n−1)
by z. Take n ∈ N such that z ∈ L and z 6∈ L . Then x,y ∈ L . Therefore,
i i i
(n−1) (n−1) (n)
z = [x,y] ∈ [L ,L ] = L . This contradiction establishes the proposition.
i i i
Note that a result similar to Proposition 2.8 has been obtained earlier by Yu. Bah-
turin and H. Strade ([2], Corollary 3.2, Theorem 3.2).
The following notion introduced by A.E. Zalesskii is crucial for what follows.
Definition 2.9 Let L = limL be a locally finite Lie algebra, Φ a finite subset of
−→ i i
IrrL . The set Φ = {Φ } is called an inductive system if hΦ ↓L i = Φ for each pair
i i i∈I j i i
i < j.
5
3 Inductive systems and ideals
It is not difficult to show that there is a 1-1 correspondence between the inductive sys-
tems for a locally semisimple Lie algebra L and the ideals X of its universal enveloping
algebra U(L) with U(L)/X locally finite (see [15]). In this section we extend this result
to arbitrary locally perfect Lie algebras. Recall that the solvable radical of a perfect
finite-dimensional Lie algebra L coincides with the nilpotent radical of L, i.e. RadL
annihilates all irreducible L-modules (see, for instance, [6]). Therefore, we can state
Lemma 3.1 Let L be a perfect finite-dimensional Lie algebra, S a Levi subalgebra of L.
Then the map V 7→ V↓S is a 1-1 correspondence between irreducible L- and S-modules,
respectively, (the inverse map is given by W 7→ W↑L). In particular, V↓S↑L = V and
W↑L↓S = W.
Sometimes we shall identify irreducible L- and S-modules.
Lemma 3.2 Let L be a finite-dimensional Lie algebra, V , V be finite-dimensional
1 2
L-modules such that Ann V = Ann V . Then hV i = hV i.
U(L) 1 U(L) 2 1 2
Proof. Set A = U(L)/Ann V = U(L)/Ann V . Then A is finite-dimensional,
U(L) 1 U(L) 2
and V , V are faithful A-modules. But for each faithful A-module V the set hVi
1 2
consists of all irreducible A-modules. Therefore, hV i = hV i as A-modules. Hence
1 2
hV i = hV i as L-modules.
1 2
Lemma 3.3 Let L be a finite-dimensional perfect Lie algebra, Φ a finite subset of
IrrL. Then codimensions of annihilators in U(L) of all finite-dimensional L-modules
V such that hVi = Φ are bounded by some constant depending on L and Φ.
Proof. Let L = S ⊕ R, where S is a Levi subalgebra, R is the radical. Let {r | i =
i
1,...,m} be a basis of R. Then, by PBW theorem, U(S) is a subalgebra of U(L), and
U(L) = ⊕ U(S)rk1...rkm.
k1,...,km 1 m
Denote by M(Φ) the set of all finite-dimensional L-modules V such that hVi = Φ.
Let V ∈ M(Φ). It is clear that the image of the ideal U(L)R in the quotient algebra
U = U(L)/Ann V is nilpotent. Since the ideal
V U(L)
U(S)∩Ann V = Ann V = ∩ Ann ϕ
U(L) U(S) ϕ∈Φ U(S)
of U(S) is semiprimitive and does not depend on the choice of V, the subalgebra
P = U(S)/U(S) ∩ Ann V of U is semisimple and the same for all V ∈ M(Φ).
U(L) V
Hence R = U(L)R/U(L)R∩Ann V is the radical of U . We have U = P ⊕R .
V U(L) V V V
Let {p | j = 1,...,l} be a basis of P, r¯ the image of r in U . Then the elements
j i i V
p r¯, r¯ generate R , so R is a finitely generated algebra. Suppose that there exists
j i i V V
n = n(L,Φ) such that Rn = 0 (or equivalently, RnV = 0) for all V ∈ M(Φ). Then it is
V
easy to see that the dimensions of all R ,V ∈ M(Φ), are bounded by some constant.
V
Therefore, the same is true for all algebras U ,V ∈ M(Φ), as required.
V
6
So we have to show that there exists n = n(L,Φ) such that RnV = 0 for all
V ∈ M(Φ). We proceed by induction on dimR, the case R = 0 being clear. Assume
that dimR > 0. Since R is a nilpotent Lie algebra, it has the nontrivial center Z(R).
As S is semisimple, Z(R) is a completely reducible S-module (with respect to the
adjoint action). Consider the following two cases.
Case 1. Z(R) contains an irreducible submodule Z such that dimZ > 1.
We have hV↓Si = hVi↓S = Φ↓S. Therefore, the set Λ of weights of the module
V↓S does not depend on the choice of V ∈ M(Φ) and coincides with the set of all
weights of the modules from Φ↓S. Denote by Ω the set of weights of the S-module Z.
We have the decomposition of V and Z in the sum of weight spaces: V = ⊕ V ,
λ∈Λ λ
Z = ⊕ Z . Since Λ is finite, there exists k ∈ N such that (kω + λ) 6∈ Λ for all
ω∈Ω ω
nonzero ω ∈ Ω and all λ ∈ Λ. Let ω 6= 0, z ∈ Z , v ∈ V . Consider the element
ω ω λ λ
zkv ∈ V. Let h be an element from the Cartan subalgebra of S. Then
ω λ
hzkv = [h,z ]zk−1v +z [h,z ]zk−2v +...+zk−1[h,z ]v +zkhv =
ω λ ω ω λ ω ω ω λ ω ω λ ω λ
(kω(h)+λ(h))zkv = (kω +λ)(h)zkv .
ω λ ω λ
Since Λ does not contain the weight kω + λ, we have zkv = 0 for all v . Hence
ω λ λ
zk ∈ Ann V. Now consider elements of Z (if Z 6= 0). Denote by E the linear
ω U(L) 0 0 0
subspace of Z generated by all elements of type [s ,z ] where α is a root of S,
0 α −α
s ∈ S , z ∈ Z . It is clear that ⊕ Z ⊕ E is an S-submodule of Z. The
α α −α −α ω6=0 ω 0
irreducibility of Z forces E = Z . Thus, the elements of type [s ,z ] generate Z as
0 0 α −α 0
a linear space. Since [Z,Z] = 0 (i.e. U(Z) is commutative), and zk ∈ Ann V (as
−α U(L)
the weight α is nonzero), we have
(ads )kzk = k![s ,z ]k +z u ∈ Ann V
α −α α −α −α U(L)
where u ∈ Zk−1. This implies
[s ,z ]k2 ≡ (−1/k!)k(z u)k ≡ (−1/k!)kzk uk ≡ 0 (mod Ann V).
α −α −α −α U(L)
Therefore, there exists a basis {z | i = 1,...,p} of the module Z such that zk2 ∈
i i
Ann V for i = 1,...,p and all V ∈ M(Φ). In view of commutativity of U(Z),
U(L)
there exists m ∈ N such that Zm ∈ Ann V for all V ∈ M(Φ). Consider a chain of
U(L)
L-modules
V ⊇ ZV ⊇ ... ⊇ Zm−1V ⊇ ZmV = 0
Set L′ = L/Z, W = ⊕ ϕ, Q = Zj−1V/ZjV, Q¯ = Q ⊕W, j = 1,...,m. Since Z
ϕ∈Φ j j j
annihilates W and all Q , the modules Q¯ can be considered as L′-modules. The Lie
j j
algebra L′ is perfect and contains S as a Levi subalgebra. We have
hQ¯ ↓L′i↓S = hQ¯ ↓Si = hQ ↓Si∪hΦ↓Si = hΦ↓Si.
j j j
In view of Lemma 3.1, one can assume hQ¯ ↓L′i = Φ, j = 1,...,m. It is valid because
j
we use only properties of weights of Φ↓S. Note that R′ = R/Z is the radical of L′.
Since dimR′ < dimR, by inductive hypothesis, there exists n = n(L′,Φ), such that
7
R′nQ¯ = 0 for all j. In particular, R′nQ = 0, so Rn(Zj−1V/ZjV) = 0, j = 1,...,m.
j j
This implies RnmV = 0 for all V ∈ M(Φ), as required.
Case 2. Z(R) does not contain nontrivial S-submodules.
Since L = S ⊕ R is perfect, R = [L,L] ∩ R = [R,R] + [S,R]. Therefore, R 6= 0
implies [S,R] 6= 0, so the S-module R contains an element r of a nonzero weight ω.
ω
Recall that R is nilpotent. Therefore, there exist a ,...,a ∈ R,k ,...,k ∈ N, such
1 t 1 t
that
z = (ada )k1...(ada )ktr
1 t ω
is a nonzero element of Z(R), but
(ada )ki+1(ada )ki+1...(ada )ktr = 0
i i+1 t ω
for each i = 1,...,t. As in Case 1, take m ∈ N such that (mω +λ) 6∈ Λ for all λ ∈ Λ.
Then rm ∈ Ann V. Since
ω U(L)
(ada )mk1...(ada )mktrm = c (ada )mk1...(ada )mkt−1((ada )ktr )m = ... =
1 t ω t 1 t−1 t ω
c((ada )k1...(ada )ktr )m = czm
1 t ω
where c ,c ∈ F, we conclude that zm ∈ Ann V. Denote by Z the one-dimensional
t U(L)
subspace generated by z. Then Z is an ideal of L and ZmV = 0 for all V ∈ M(Φ).
Arguments analogous to those of Case 1 (the inductive hypothesis is applied to L′ =
L/Z) show that there exists n such that RnmV = 0 for all V ∈ M(Φ), as required.
Theorem 3.4 Let L be a perfect finite-dimensional Lie algebra, Φ a finite subset of
IrrL, F(Φ) the set of all ideals X of U(L) such that U(L)/X is finite-dimensional and
hU(L)/Xi = Φ. Then F(Φ) is nonempty and has the smallest element N(Φ) and the
largest element M(Φ) such that N(Φ) ⊆ X ⊆ M(Φ) for all X ∈ F(Φ). The algebra
U(L)/M(Φ) is semisimple, the algebra M(Φ)/N(Φ) is nilpotent.
Proof. Assume that X ∈ F(Φ). Then X is the annihilator of the U(L)-module
V = U(L)/X. Since the set of composition factors of V is Φ, we have X ⊆ M(Φ)
where M(Φ) is the annihilator of the completely reducible U(L)-module ⊕ ϕ. It is
ϕ∈Φ
clear that U(L)/M(Φ) is semisimple, and (M(Φ)/X)k = 0 where k is the number of
composition factors of V. Therefore, M(Φ) is the largest element of F(Φ). By Lem-
ma 3.3, codimensions of all ideals from F(Φ) are bounded by some constant. Therefore,
every descending chain of ideals from F(Φ) stabilizes. On the other hand, the inter-
section of any two ideals from F(Φ) belongs to F(Φ) again (the annihilator of the sum
of modules is the intersection of their annihilators). Therefore, F(Φ) has the smallest
element N(Φ). Since N(Φ) ∈ F(Φ), we get (M(Φ)/N(Φ))k = 0 for some k.
It is convenient to assume that F(∅) = {U(L)}. Observe that F(Φ) is the set of all
ideals X of U(L) such that N(Φ) ⊆ X ⊆ M(Φ). So we obtain
Corollary 3.5 The set F(Φ) is a finite sublattice of the lattice of all ideals of U(L).
8
If L is semisimple, then for each X ∈ F(Φ) the L-module U(L)/X is completely
reducible. Therefore, U(L)/X-module U(L)/X is completely reducible. Hence the
algebra U(L)/X is semisimple. But F(Φ) contains only one semiprimitive ideal M(Φ).
Therefore, we get
Corollary 3.6 If L is semisimple, then |F(Φ)| = 1 (i.e. N(Φ) = M(Φ)).
Lemma 3.7 Let L ⊆ L be perfect finite-dimensional Lie algebras, U(L ) ⊆ U(L );
1 2 1 2
Φ , Φ finite subsets of IrrL , IrrL , respectively, such that hΦ ↓L i = Φ . Then
1 2 1 2 2 1 1
X ∈ F(Φ ) implies X ∩U(L ) ∈ F(Φ ).
2 1 1
Proof. Since Ann (U(L )/X) = X ∩U(L ), by Lemma 3.2,
U(L1) 2 1
hU(L )/X ∩U(L )i = hU(L )/X↓L i = hΦ ↓L i = Φ .
1 1 2 1 2 1 1
Hence X ∩U(L ) ∈ F(Φ ).
1 1
Lemma 3.8 Let L = limL be a locally perfect Lie algebra, X be an ideal of U(L)
−→ i
with U(L)/X locally finite. Then the set
Φ(X) = {hU(L )/X ∩U(L )i}
i i i∈I
is an inductive system for L.
Proof. Since U(L)/X is locally finite, U(L )/X ∩ U(L ) is finite-dimensional, so the
i i
set Φ (X) = hU(L )/X ∩U(L )i is finite. Further, if i ≤ j then Ann (U(L )/X ∩
i i i U(Li) j
U(L )) = X ∩U(L ). Therefore, by Lemma 3.2,
j i
hΦ (X)↓L i = hhU(L )/X ∩U(L )i↓L i = hU(L )/X ∩U(L )↓L i =
j i j j i j j i
hU(L )/X ∩U(L )i = Φ (X),
i i i
so Φ(X) is an inductive system for L.
Denote by IS, LF the sets of inductive systems for a locally perfect Lie algebra
L, and ideals X of its universal enveloping algebra U(L) with U(L)/X locally finite,
respectively. Define the map f : LF → IS, setting f(X) = Φ(X) where X ∈ LF,
Φ(X) as in Lemma 3.8. Denote by LF(Φ) the inverse image of an inductive system Φ.
Theorem 3.9 Let L = limL be a locally perfect Lie algebra, and the map f : LF →
−→ i
IS be as above. Then for each inductive system Φ the set LF(Φ) is nonempty and has
the smallest element N(Φ) and the largest element M(Φ) such that N(Φ) ⊆ X ⊆ M(Φ)
for all X ∈ LF(Φ). The algebra U(L)/M(Φ) is semisimple, the algebra M(Φ)/N(Φ)
is locally nilpotent. Moreover, the map f produces a 1-1 correspondence between the
semiprimitive ideals from LF and the inductive systems for L (the inverse map is given
by Φ 7→ M(Φ)).
9
Proof. Let Φ = {Φ } be an inductive system. For every i ∈ I denote by F(Φ ) the
i i∈I i
set of ideals X of U(L ) such that U(L )/X is finite-dimensional and hU(L )/Xi = Φ .
i i i i
Then by Theorem 3.4, F(Φ ) is nonempty and has the smallest element N(Φ ). Let i ≤
i i
j. Then L ⊆ L , U(L ) ⊆ U(L ) and by definition of an inductive system, hΦ ↓L i =
i j i j j i
Φ . By Lemma 3.7, N(Φ )∩U(L ) ∈ F(Φ ), so N(Φ ) ⊆ N(Φ ). Since for each i N(Φ )
i j i i i j i
is an ideal of U(L ), we conclude that N(Φ) = limN(Φ ) is an ideal of U(L). Observe
i −→ i
that U(L)/N(Φ) is locally finite. It is clear that for each i ∈ I there exists j ≥ i such
that N(Φ)∩U(L ) = N(Φ )∩U(L ) ∈ F(Φ ). Therefore, hU(L )/N(Φ)∩U(L )i = Φ ,
i j i i i i i
forcing N(Φ) ∈ LF(Φ). Denote by M(Φ) the inverse image of the Jacobson radical
of U(L)/N(Φ) in U(L). Recall that the Jacobson radical of a locally finite algebra is
locally nilpotent. Therefore, the algebra (M(Φ)∩U(L ))/(N(Φ)∩U(L )) is nilpotent
i i
for each i ∈ I. Since N(Φ) ∩ U(L ) ∈ F(Φ ), we have M(Φ) ∩ U(L ) ∈ F(Φ ), so
i i i i
M(Φ) ∈ LF(Φ). Let X ∈ LF(Φ). By definition, hU(L )/X ∩ U(L )i = Φ . Hence
i i i
X ∩ U(L ) ∈ F(Φ ). By Theorem 3.4, N(Φ ) ⊆ X ∩ U(L ), and X ∩ U(L )/N(Φ )
i i i i i i
is nilpotent. Consequently, N(Φ) ⊆ X, and X/N(Φ) is locally nilpotent, forcing
N(Φ) ⊆ X ⊆ M(Φ). This completes the proof.
Note that LF(Φ) is the set of all ideals X of U(L) such that N(Φ) ⊆ X ⊆ M(Φ).
So the set LF(Φ) is a sublattice of the lattice of all ideals of U(L). By Lemma 3.7,
X ∩U(L ) ∈ F(Φ ) for i ≤ j and X ∈ F(Φ ), so we have a morphism of finite lattices
i i j
f : F(Φ ) → F(Φ ). Therefore we have
ji j i
Corollary 3.10 The lattice LF(Φ) is a projective limit of the finite lattices F(Φ ).
i
If all L are semisimple, then by Corollary 3.6, all F(Φ ) are one-element sets. So
i i
we obtain already known result:
Corollary 3.11 If L = limL where all L are semisimple (i.e. L is locally semisim-
−→ i i
ple), then |LF(Φ)| = 1 (i.e. N(Φ) = M(Φ)) for all inductive systems Φ.
4 Abstract Levi subalgebras
Definition 4.1 A subalgebra S of a locally finite Lie algebra L = limL is called a
−→ i
Levi subalgebra associated with the local system {L } , if S = limS where S is a
i i∈I −→ i i
Levi subalgebra of L and S ⊆ S for each pair i ≤ j.
i i j
Note that a Levi subalgebra is locally semisimple.
Lemma 4.2 If a locally perfect Lie algebra L has a Levi subalgebra S, then S is asso-
ciated with a perfect local system of L.
Proof. Assume that S is associated with a local system {L } . Since L is locally
i i∈I
(∞)
perfect, by Remark 2.5, {L } is a perfect local system of L. Since S = S ∩ L
i i∈I i i
(∞) (∞)
is semisimple, we obtain S ⊆ L . Hence S is a Levi subalgebra of L , so S is
i i i i
(∞)
associated with {L } .
i i∈I
Itisnotclear,whetheralllocallyfiniteLiealgebrashaveLevisubalgebras. However,
we have
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