Depth Generated Simple Lie Algebras

Algebr Represent Theor (2009) 12:53–61DOI 10.1007/s10468-008-9101-x

Depth Generated Simple Lie Algebras

Christopher Kennedy · David J. Winter

Received: 25 September 2006 / Accepted: 25 September 2007 / Published online: 3 June 2008© Springer Science + Business Media B.V. 2008

Abstract A Lie module algebra for a Lie algebra L is an algebra and L-module Asuch that L acts on A by derivations. The depth Lie algebra LD of a Lie algebra Lwith Lie module algebra A acts on a corresponding depth Lie module algebra AD .This determines a depth functor D : (A, L) �→ (AD , LD) from the category of Liemodule algebra pairs to itself. Remarkably, this functor preserves central simplicity.It follows that the Lie algebras AD LD corresponding to faithful central simple Liemodule algebra pairs (A, L) with A commutative are simple. Upon iteration atsuch (A, L), the Lie algebras AD i LD i are simple for all i ∈ ω. In particular, theAD i LD i (i ∈ ω) corresponding to central simple Jordan Lie algops (A, L) are simpleLie algebras.

Keywords Lie algop · Lie module algebra · Depth Lie algebra

Mathematics Subject Classifications (2000) 17B20 · 17B40 · 17B99

1 Introduction

The key structures in this paper are depth Lie algebras and Lie module algebras.Depth Lie algebras were originally introduced and used in [2] for the purpose ofstudying deep matrix algebras.

Presented by Don Passman.

C. Kennedy (B)Department of Mathematics, Christopher Newport University,1 University Place, Newport News, VA 23606-2998, USAe-mail: [email protected]

D. J. WinterDepartment of Mathematics, University of Michigan,2074 East Hall 530 Church Street, Ann Arbor, MI 48109-1043, USAe-mail: [email protected]

54 C. Kennedy, D.J. Winter

Definition 1 [2, Definition VI.15] Let L be a Lie algebra and let αi : L → Li (i ∈ ω)

be Lie algebra isomorphisms. Then the depth Lie algebra on L is the Lie alge-bra LD ≡ ⊕i∈ω Li with bracket determined by [αi(x), α j(y)] ≡ αmax(i, j )([x, y]) (x, y ∈L; i, j ∈ ω).

Lie module algebras are modules of a Lie algebra satisfying certain conditions(Definition 2). Whenever L possesses a Lie module algebra A, the depth Liealgebra LD acts naturally on a corresponding depth Lie module algebra AD —thereby determining a depth functor D : (A, L) �→ (AD , LD) from the category ofLie module algebra pairs to itself.

Depth Lie algebras are never simple (in particular, ⊕i≥n Li is an ideal for anyn). Remarkably, however, central simplicity of Lie module algebras is preservedunder passage from a Lie module algebra pair (A, L) to the corresponding depthLie module algebra pair (AD , LD). Consequently, for a faithful central simple Liemodule algebra pair (A, L) with A commutative, simple Lie algebras AD i LD i aredepth generated for all i ∈ ω. In particular, the AD i LD i (i ∈ ω) corresponding to thecentral simple Jordan Lie algops (A, L) studied in [6, 8, 9] are simple Lie algebras.This is noted in Theorems 4, 5.

1.1 Related Earlier and Continuing Work

1.1.1 McCrimmon’s Deep Matrix Algebras; Kennedy’s Depth Lie Algebras

Deep matrix algebras were first introduced by Kevin McCrimmon, together with thecorresponding associative structure and representation theories [4].

Building on the work of McCrimmon [4], Christopher Kennedy further investi-gates deep matrix algebras in [2, 3] — with an emphasis on the corresponding Liestructure and representation theories. In this work, depth Lie algebras LD of a Liealgebra L occur naturally and are used accordingly.

1.1.2 The Simplicity Theorem of Jordan

The following theorem of David Jordan is needed for Theorem 5.

Theorem 1 [1, Theorem 5] Let L be a Lie algebra of derivations of a unital commu-tative algebra A and suppose that A is L-simple. Then AL is a simple Lie algebraexcept when the characteristic is 2 and AL is of the form Ax for some x in L. Whenthe characteristic is 2 and AL is of the form Ax for some x in L, AL is simple if andonly if x is surjective.

1.1.3 The Jordan Lie Algop Program of Winter

Motivated by [1, 5, 6], David J. Winter introduced Lie algops and launched a programto investigate and determine the simple Jordan Lie algops (A, L), their absolutelyirreducible modules, and their corresponding simple Lie algebras AL over arbitraryfields [8, 9].

Depth generated simple Lie algebras 55

The Lie algops of [8, 9] are the Lie module algebra pairs (A, L) of this paperfor which A is commutative. These are of particular interest for the theory of Liealgebras since their corresponding AL are Lie algebras.

The simple Lie algops (A, L) determined by Su et al. [6] in their classificationof simple Lie algebras of Witt type are encompassed within this Jordan Lie algopprogram in the sense that their closures are the simple Jordan Lie algops of finitetype with L abelian over an algebraically closed field of characteristic 0.

By Theorem 4 of this paper, all central simple Lie algops (A, L), such as thosediscussed in [5, 6, 8, 9], determine corresponding central simple Lie algops (AD i , LD i)

and simple Lie algebras AD i LD i — those generated by the depth functor of Section 5.A key unpublished theorem that underlies the theory of Lie algops and [8, 9] is the

central simplicity of the tensor product of central simple Lie algops. This theorem isa special case of a theorem for more general algops of work in progress. It followsfrom Theorem 2 below — another special case of the theorem for algops.

1.2 Conventions

The set of natural numbers is ω. For any set S, Sω consists of all ω-tuples x = (xi)i∈ω

with xi ∈ S for i ∈ ω. The identity map on a set S is IS.An algebra A is a nonzero unital associative algebra over a field k. Its identity is

1A, and k is identified with k1A. Homomorphisms and subalgebras of algebras areunital. The center of A is denoted C(A).

2 Lie Module Algebras

2.1 Lie Module Algebras and Central Simplicity

Definition 2 A Lie module algebra for a Lie algebra L over a field k is an associativealgebra and L-module A over k such that L acts on A by derivations; that is, x(ab) =(xa)b + a(xb) (x ∈ L; a, b ∈ A). And a Lie module algebra pair is a pair (A, L)

where L is a Lie algebra and A is a Lie module algebra for L.

LieModAlgs denotes the category of Lie module algebra pairs — wherein amorphism α : (A, L) → (B, M) of Lie module algebra pairs (A, L), (B, M) is amorphism αA : A → B of algebras together with a morphism αL : L → M of Liealgebras such that αA(xa) = αL(x)αA(a) (x ∈ L, a ∈ A).

Definition 3 For a Lie module algebra A for a Lie algebra L:

1. A is faithful if A is faithful as L-module.2. A is central if C(A, L) ≡ {a ∈ C(A) | La = 0} is k1A.3. A sub Lie module algebra (resp. ideal) of A as Lie module algebra is a subalgebra

(resp. ideal) B of A as algebra and L-submodule of A as L-module. A sub Liemodule algebra or ideal B of A is proper if B �= 0 and B �= A.

4. When A has no proper ideals, that is, when A is L-simple, the Lie module algebraA is simple.


The Lie module algebra pair (A, L) is faithful (resp. central, resp. simple) if the Liemodule algebra A for L is faithful (resp. central, resp. simple).

2.2 Finite Tensor Products

Definition 4 The tensor product of Lie module algebra pairs (Ai, Li) (0 ≤ i ≤ r)over a field k is the Lie module algebra pair ⊗r

i=0(Ai, Li) ≡ (⊗ri=0 Ai,⊕r

i=0 Li) whosemodule action is

(⊕ri=0xi

) (⊗ri=0ai

) = x0a0 ⊗ a1 ⊗ . . . ⊗ ar + . . . + a0 ⊗ a1 ⊗ . . . ⊗ xrar

for x0 ∈ L0, . . . , xr ∈ Lr and a0 ∈ A0, . . . , ar ∈ Ar.

Theorem 2 The tensor product ⊗ri=0(Ai, Li) of central simple Lie module algebra

pairs (Ai, Li) (0 ≤ i ≤ r) is central simple for r > 1.

Proof Evidently, it suffices to show (A, L) ⊗k (B, M) = (A ⊗k B, L ⊕ M) is centralsimple for central simple Lie module algebra pairs (A, L), (B, M). This reducesestablishing central simplicity of the r-fold product to showing it for a similar (r − 1)-fold product and applying induction.

For this, let Lλ, Mλ denote the Lie algebras of left translations xλ, yλ on A, Bby elements of x ∈ L, y ∈ M respectively, let J be a nonzero ideal of A ⊗k B stableunder L ◦ M ≡ Lλ ⊗k IB + IA ⊗k Mλ, and let c = ∑n

i=1 ai ⊗ bi be a nonzero elementof J with n minimal. By the minimality of n, the ai are nonzero and the bi are linearlyindependent. Since (A, L) is simple, the L-stable ideal of A generated by an is A, sosuch a c can be chosen with an = 1A. For such a c, the commutator (a ⊗ 1A)c − c(a ⊗1A) = ∑n−1

i=1 (aai − aia) ⊗ bi is in J with a smaller n. So, by the minimality of n, it is 0.Consequently, the commutators aai − aia are 0 and aai = aia for all a ∈ A, 1 ≤ i < n.So, the ai are in C(A).

Since J is an L ◦ M-stable ideal that contains c = ∑ni=1 ai ⊗ bi, J contains

(xλ ⊗ IB)

(n∑

i=1

ai ⊗ bi

)

=n∑

i=1

xai ⊗ bi

for all x ∈ L. Since xan = x1A = 0, J then contains the sum

n∑

i=1

xai ⊗ bi =n−1∑

i=1

xai ⊗ bi

of n − 1 terms. By the choice of c with n minimal, it then follows that

n−1∑

i=1

xai ⊗ bi = 0.

But then, by the independence of the bi, it follows that xai = 0 (x ∈ L; 1 ≤ i ≤ n).This establishes that ai ∈ C(A, L) = k (1 ≤ i ≤ n).


It follows that c = 1A ⊗ b with b = ∑ni=1 aibi nonzero in B. Since B is M-simple,

the L ◦ M-stable ideal of A ⊗k B generated by c = 1A ⊗ b is then A ⊗k B, that is,J = A ⊗k B and (A, L) ⊗ (B, M) is simple.

It remains to show that the Lie module algebra pair (A ⊗k B, L ⊕ M) is central;that is, the c in C(A ⊗k B, L ⊕ M) are in k. Suppose, to the contrary, that somec = ∑n

i=1 ai ⊗ bi in C(A ⊗k B, L ⊕ M) is not in k and choose such a c with n minimal.Again, by the minimality of n, the bi are linearly independent.

Since c ∈ C(A ⊗k B), 0 = (a ⊗ 1B)c − c(a ⊗ 1B) = ∑ni=1(aai − aia) ⊗ bi. It follows

that aai − aia = 0 and aai = aia for all a ∈ A and all i — by the independence of thebi. So, the ai are in C(A).

Since c ∈ C(A ⊗k B, L ⊕ M), 0 = (xλ ⊗ IB)(c) = ∑xaibi. So, x(ai) = 0 (x ∈ L) by

the independence of the bi. Since the ai are in C(A), this implies that they are inC(A, L) = k. It follows that c = 1A ⊗ b with b = ∑n

i=1 aibi nonzero in B.Taking n = 1, reversing the roles of the ai and bi, and arguing as before but with

c = 1A ⊗ b, b is in C(B, M) = k. But then c ∈ k — contrary to the choice of c. So,there is no such c and C(A ⊗k B, L ⊕ M) = k. �

2.3 Infinite Tensor Products

The direct limit conventions of [7, E.0.43, p20] are employed in the followingdefinition.

Definition 5 Let Ai (i ∈ ω) be a collection of algebras over k. Then the tensor productof the Ai (i ∈ ω) is the direct limit ⊗i∈ω Ai ≡ limr∈ω ⊗r

i=0 Ai of the ⊗ri=0 Ai (r ∈ ω)

with respect to the morphisms βsr(a0 ⊗ . . . ⊗ ar) ≡ a0 ⊗ . . . ⊗ ar ⊗ 1Ar+1 ⊗ . . . ⊗ 1As

defined for 0 ≤ r < s ∈ ω and a0 ∈ A0, . . . , ar ∈ Ar.

Each element ⊗i∈ωai of the tensor algebra ⊗i∈ω Ai of Definition 5 can be repre-sented as ⊗i∈ωai = ⊗r

i=0ai ⊗i>r 1Ai for some r ∈ ω.

Definition 6 ⊗ri=0 Ai ⊗ 1 is the subalgebra consisting of the ⊗r

i=0ai⊗1 (a0 ∈ A0,

· · · , ar ∈ Ar) — wherein ⊗1 ≡ ⊗i>r1Ai .

Evidently, then, ⊗ri=0 Ai ⊗ 1 (r ∈ ω) is a filtration and ⊗i∈ω Ai = ⋃

r∈ω ⊗ri=0 Ai ⊗ 1

(ascending union) — where ⊗ri=0 Ai ⊗ 1 is canonically isomorphic to ⊗r

i=0 Ai for r ∈ ω.

Definition 7 Let Ai be a Lie module algebra for a Lie algebra Li for i ∈ ω and letL ≡ ∏

i∈ω Li be the Lie algebra consisting of the ω-tuples (xi)i∈ω (xi ∈ Li) equippedwith componentwise operations. Then the corresponding tensor Lie module algebrafor L is the tensor algebra ⊗i∈ω Ai as an L -module relative to the module actions

(xi)i∈ω

(⊗ri=0ai ⊗ 1

) = x0a0 ⊗ a1 ⊗ . . . ⊗ ar ⊗ 1 + . . . + a0 ⊗ a1 ⊗ . . . ⊗ xrar ⊗ 1

of L on the ⊗ri=0 Ai ⊗ 1 (r ∈ ω) with union ⊗i∈ω Ai. The tensor product ⊗i∈ω(Ai, Li)

of the (Ai, Li) (i ∈ ω) is the Lie module algebra pair (⊗i∈ω Ai,L ).


Evidently, the sub Lie module algebra ⊗ri=0 Ai ⊗ 1 of the Lie module algebra

⊗i∈ω Ai for L of Definition 6 can be regarded as a Lie module algebra for ⊕ri=0 Li that

is canonically isomorphic to the Lie module algebra ⊗ri=0 Ai for ⊕r

i=0 Li of Definition4. This plays a role in the proof of the following theorem.

Theorem 3 The tensor product ⊗i∈ω(Ai, Li) = (⊗i∈ω Ai,L ) of central simple Liemodule algebra pairs (Ai, Li) (i ∈ ω) is central simple.

Proof Suppose that J is a nonzero L -stable ideal of ⊗i∈ω Ai. Since ⊗i∈ω Ai is theascending union of the ⊗r

i=0 Ai ⊗ 1 for r ∈ ω, there exists r such that J ∩ ⊗ri=0 Ai ⊗ 1 is

nonzero. Then J ∩ ⊗ri=0 Ai ⊗ 1 is a nonzero ideal of the Lie module algebra ⊗r

i=0 Ai ⊗1 for ⊕r

i=0 Li— which is simple by Theorem 2. But then J ∩ ⊗ri=0 Ai ⊗ 1 is ⊗r

i=0 Ai ⊗ 1.It follows that J contains ⊗r

i=0 Ai ⊗ 1 for all r, hence J = ⊗i∈ω Ai. Therefore,⊗i∈ω Ai is a simple Lie module algebra for L .

Finally, the ⊗ri=0 Ai ⊗ 1 are central for ⊕r

i=0 Li by Theorem 2, hence their union iscentral for L .

3 Depth Lie Module Algebras

Each Lie module algebra A of L determines a depth Lie module algebra AD for LD .Moreover, each morphism α : (A, L) → (B, M) of Lie module algebra pairs extendsuniquely to a morphism αD : (AD , LD) → (BD , MD) of the corresponding depthLie module algebras that preserves their depth structure. This determines the depthfunctor.

3.1 Construction

The depth Lie algebra LD can be constructed more concretely as a subalgebra ofLω equipped with componentwise operations. Specifically, LD can be thought of asall (xi)i∈ω where xi = xi+1 for large enough i. The following (evident) propositionestablishes this formally and is used in Definition 8.

Proposition 1 For a Lie algebra L and LD ≡ {(xi)i∈ω|(∃r > 0)xi = xi+1 for i ≥ r},define αi : L → LD by αi(x) ≡ (x j ) j∈ω where x j = 0 for j < i and x j = x for j ≥i, x ∈ L. Then the αi : L → αi(L) (i ∈ ω) are isomorphisms, LD = ⊕i∈ωαi(L), and[αi(x), α j(y)] = αmax(i, j )([x, y]) for i, j ∈ ω and x, y ∈ L.

Definition 8 For an algebra A, the depth algebra AD of A is AD ≡ ⊗i∈ω A. For a Liemodule algebra A of L, the depth Lie module algebra AD for LD is the depth algebraAD as LD -module with respect to the restriction to LD of the action of Lω on ⊗i∈ω Aof Definition 7. The depth Lie module algebra pair of (A, L) is then (AD , LD).

3.2 Structure

Definition 9 A sub Lie module algebra pair of a Lie module algebra pair (A′, L′) isa Lie module algebra pair (A, L) where L is a Lie subalgebra of L′ and A is a subLie module algebra of A′ as Lie module algebra for L.


Definition 10 For a Lie module algebra pair (A, L), a depth Lie module algebra pairover (A, L) is a Lie module algebra pair (A′, L′) containing (A, L) as a sub Liemodule algebra pair together with a morphism δ : (A′, L′) → (A′, L′) such that

1. δA : A′ → A′ is an injective homomorphism.2. δ0

A(A) · · · δrA(A) is the internal tensor product of δ0

A(A), · · · , δrA(A) for r ∈ ω.

3. A′ = ⋃r∈ω δ0

A(A) · · · δrA(A).

4. δL : L′ → L′ is an injective Lie algebra homomorphism.5. L′ = ⊕i∈ωδi

L(L) (internal direct sum).6. [δi

L(x), δjL(y)] = δ

max(i, j )L ([x, y]) (x, y ∈ L; i, j ∈ ω).

7. The module action of L′ by derivations on A′ is uniquely determined by theconditions δi

L(x)δjA(a) = δ

jA(xa) (i ≤ j ), δi

L(x)δjA(a) = 0 (i > j ).

The δA, δL, δ are the depth maps of A′, L′, (A′, L′).

Identify an algebra A with the sub algebra {⊗i∈ωai | a0 ∈ A, ai = 1A (i > 0)} of⊗i∈ω A. Similarly, identify a Lie algebra L with the diagonal Lie subalgebra {(xi)i∈ω |x0 ∈ L, xi = xi+1 (i ≥ 0)} of Lω. Per these identifications, a Lie module algebra pair(A, L) can be regarded as a sub Lie module algebra pair of (AD , LD).

The following proposition is evident from the foregoing considerations.

Proposition 2 For a Lie module algebra pair (A, L), (AD , LD) together with themorphism δ : (AD , LD) → AD , LD) defined by the maps

δA(⊗i∈ωai) = ⊗i∈ωbi where b 0 = 1A, bi+1 = ai (i ≥ 0),

δL((xi)i∈ω) = (yi)i∈ω where y0 = 0, yi+1 = xi (i ≥ 0)

is a depth Lie module algebra pair over (A, L).

Corollary 1 For any two Lie module algebra pairs (A′, L′), (A′′, L′′) over a Liemodule algebra pair (A, L), there is a unique morphism ρ : (A′, L′) → (A′′, L′′)whose restriction to (A, L) is I(A,L) such that ρδA′ = δA′′ρ and ρδL′ = δL′′ρ.

Proof By Proposition 2, it may be assumed without loss of generality that (A′, L′)is (AD , LD) together with its canonical depth maps — in which case it is evident byvirtue of the universality of the tensor product. �

3.3 The Depth Functor

Each morphism α : (A, L) → (B, M) of Lie module algebra pairs extends uniquelyto a morphism αD : (AD , LD) → (BD , MD) of the corresponding depth Lie modulealgebras that preserves the depth structure described above. Specifically, the algebramorphism αA : A → B extends to the algebra morphism αAD : AD → BD definedby the conditions αAD (δi

A(a)) = δiA(αA(a)) (a ∈ A; i ∈ ω). Similarly, the Lie algebra

morphism αL : L → M extends to the Lie algebra morphism αLD : LD → MD

defined by the conditions αLD (δiL(x)) = δi

L(αL(x)) (x ∈ L; i ∈ ω). Verification thatαD : (AD , LD) → (BD , MD) is a morphism of Lie module algebra pairs is routine.It uniquely preserves the depth structure by construction.


The depth functor D : LieModAlgs → LieModAlgs is defined by D(A, L) ≡(AD , LD) and D(α : (A, L) → (B, M)) = αD : (AD , LD) → (BD , MD).

The depth functor preserves central simplicity by the following theorem.

Theorem 4 The depth Lie algebra pair (AD , LD) of a central simple Lie modulealgebra pair (A, L) is central simple.

Proof LD is a sufficiently large part of Lω that the proof, sketched below, is almostidentical to that of Theorem 3.

Suppose that J is a nonzero LD -stable ideal of AD = ⊗i∈ω A. Since AD is theascending union of the ⊗r

i=0 A ⊗ 1 for r ∈ ω, there exists r such that J ∩ ⊗ri=0 Ai ⊗ 1 is

nonzero. Since ⊗ri=0 Ai ⊗ 1 is simple, J contains ⊗r

i=0 Ai ⊗ 1 for all r. Thus, J = AD

and AD is a simple Lie module algebra for LD .The ⊗r

i=0 Ai ⊗ 1 are central for ⊕ri=0 Li by Theorem 2. Thus, their union AD is

central for LD . �

3.4 Depth Generated Simple Lie Algebras

For a commutative Lie module algebra A for a Lie algebra L, AD LD is a Lie algebra.Its structure is determined by the structure of (AD , LD) described in Section 3.1.

Theorem 5 The Lie algebras AD i LD i of the depth Lie module algebra pairs(AD i , LD i) corresponding to a central simple Lie module algebra (A, L) with Acommutative are simple.

Proof The (AD i , LD i) are central simple by Theorem 4. But then the AD i LD i aresimple by Jordan’s Theorem 1. �

Example 1 When (A, L) is the central simple Lie module algebra with polynomialalgebra A = k[t], Lie algebra L = k∂ , and action ∂(t) ≡ 1, then D(A, L) is centralsimple and AD LD is a simple Lie algebra. In fact, the depth Lie module algebra(AD , LD) with its depth maps δA, δL has as its depth algebra the polynomial algebraAD = k[δi

A(t) (i ∈ ω)] in the variables δiA(t) (i ∈ ω) and as its depth Lie algebra LD =

⊕i∈ωkδiL(∂). It is taken together with the depth maps δA on AD and δL on LD , and

the module action

δiL(∂)δ

jA(t) = δ

jA(∂t) (i ≤ j ), δi

L(∂)δjA(t) = 0 (i > j ).

or δiL(∂)δ

jA(t) = 1 for i ≤ j and δi

L(∂)δjA(t) = 0 for i > j. It follows that δi

L(∂) =⊕ j≥i∂ j (i ∈ ω) where the ∂ j are the partial derivatives with respect to the variablesδ

jA(t) ( j ∈ ω).

Example 2 The Lie module algebra pair (A2, sl2) = (k[x], kf + kh + ke) with f ≡∂x, h ≡ x∂x, and e ≡ x2∂x is central simple — as is the corresponding depth Liemodule algebra pair (A2D , sl2D). The Lie algebra A2Dsl2D is simple.

Acknowledgements This paper has benefited greatly from the helpful and incisive comments of thereviewer. It is a pleasure to take this opportunity to thank the editor and reviewer for their effortsand contributions toward completion of this project.


References

1. Jordan, D.A.: On the simplicity of Lie algebras of derivations of commutative algebras. J.Algebra 228, 580–585 (2000)

2. Kennedy, C.: An exploration of deep matrix algebras, p. 109. Dissertation, University ofMichigan (2004)

3. Kennedy, C.: Deep matrix algebras of finite type. Algebr. Represent. Theory 9, 525–537 (2006)4. McCrimmon, K.: Deep matrix algebras and their Frankenstein actions. In: Sabinen, L., Sbitneva,

L., Shestakov, I. (eds.) Non-Associative Algebra and its Applications. Lecture Notes in Pure andApplied Mathematics, vol. 246, pp. 261–274. CRC, Boca Raton (2006)

5. Osborn, J.M., Winter, D.J.: Simple Lie algebras. Comm. Algebra 31(11), 5405–5420 (2003)6. Su, Y., Xu, X., Zhang, H.: Derivation-simple algebras and the structures of Lie algebras of Witt

type. J. Algebra 233, 642–662 (2000)7. Winter, D.J.: The Structure of Fields. Grad. Texts in Math., vol. 16, p. 205. Springer, Berlin (1974)8. Winter, D.J.: Lie algops and simple Lie algebras. Comm. Algebra 33, 3157–3178 (2005)9. Winter, D.J.: Simple nil Lie algops. Comm. Algebra 35, 1093–1126 (2007)

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