Frattini theory for N-Lie algebras

Frattini theory for N-Lie algebras

We develop a Frattini Theory for n-Lie algebras by extending theorems of Barnes' to the n-Lie algebra setting. Specifically, we show some sufficient conditions for the Frattini subalgebra to be an ideal and find an example where the Frattini subalgebra fails to be an ideal.

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Дата:2009
Автор: Michael Peretzian Williams
Формат: Стаття
Мова:English
Опубліковано: Інститут прикладної математики і механіки НАН України 2009
Назва видання:Algebra and Discrete Mathematics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/154628
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Frattini theory for N-Lie algebras / Michael Peretzian Williams // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 108–115. — Бібліогр.: 7 назв. — англ.

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spelling irk-123456789-1546282019-06-16T01:29:28Z Frattini theory for N-Lie algebras Michael Peretzian Williams We develop a Frattini Theory for n-Lie algebras by extending theorems of Barnes' to the n-Lie algebra setting. Specifically, we show some sufficient conditions for the Frattini subalgebra to be an ideal and find an example where the Frattini subalgebra fails to be an ideal. 2009 Article Frattini theory for N-Lie algebras / Michael Peretzian Williams // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 108–115. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:15 Linear and multilinear algebra;matrix theory, 17 Nonassociative rings and algebras. http://dspace.nbuv.gov.ua/handle/123456789/154628 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We develop a Frattini Theory for n-Lie algebras by extending theorems of Barnes' to the n-Lie algebra setting. Specifically, we show some sufficient conditions for the Frattini subalgebra to be an ideal and find an example where the Frattini subalgebra fails to be an ideal.
format Article
author Michael Peretzian Williams
spellingShingle Michael Peretzian Williams
Frattini theory for N-Lie algebras
Algebra and Discrete Mathematics
author_facet Michael Peretzian Williams
author_sort Michael Peretzian Williams
title Frattini theory for N-Lie algebras
title_short Frattini theory for N-Lie algebras
title_full Frattini theory for N-Lie algebras
title_fullStr Frattini theory for N-Lie algebras
title_full_unstemmed Frattini theory for N-Lie algebras
title_sort frattini theory for n-lie algebras
publisher Інститут прикладної математики і механіки НАН України
publishDate 2009
url http://dspace.nbuv.gov.ua/handle/123456789/154628
citation_txt Frattini theory for N-Lie algebras / Michael Peretzian Williams // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 108–115. — Бібліогр.: 7 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT michaelperetzianwilliams frattinitheoryfornliealgebras
first_indexed 2025-07-14T06:40:18Z
last_indexed 2025-07-14T06:40:18Z
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2009). pp. 108 – 115 c⃝ Journal “Algebra and Discrete Mathematics” Frattini theory for N-Lie algebras Michael Peretzian Williams Communicated by V. M. Futorny Abstract. We develop a Frattini Theory for n-Lie alge- bras by extending theorems of Barnes’ to the n-Lie algebra setting. Specifically, we show some sufficient conditions for the Frattini sub- algebra to be an ideal and find an example where the Frattini sub- algebra fails to be an ideal. 1. Introduction Frattini theory has been studied extensively and has a rich history both in group theory and later in Lie algebras. The Frattini subalgebra is the intersection of all proper maximal subalgebras and we shall denote it �(A). In 1967, Barnes proved the following theorem for the Frattini subalgebra: if B,C ⊲ A where C ⊂ �(A) ∩ B and B/C is nilpotent, then B is nilpotent [4 p.348 Theorem 5]. A corollary to this is the following: if �(A) ⊲ A, then �(A) is nilpotent. This theorem raises an obvious question: when is �(A) ⊲ A? For groups it is known that �(A) ⊲ A always holds because all automorphisms permute maximal subgroups. Barnes and Chao [5 p.233 Theorem 3] proved that A is a nilpotent Lie algebra if and only if �(A) = A2. If A is nilpotent, clearly �(A) = A2 ⊲ A. In 1968, Barnes strengthened this statement and proved that if A is a solvable Lie algebra, then �(A) ⊲ A [2 p.348 Lemma 3.4]. One might believe �(A) ⊲ A is true in general for Lie algebras, but over F2 = {0, 1} if A is the cross product Lie algebra, then �(A)⋪A. This work is from my dissertation completed at North Carolina State University on February 17, 2004 under the direction of E. L. Stitzinger. 2000 Mathematics Subject Classification: 15 Linear and multilinear algebra; matrix theory, 17 Nonassociative rings and algebras. Key words and phrases: Lie algebras, non-associative algebras. M. P. Williams 109 The purpose of this paper is to develop a Frattini theory for n-Lie algebras. Where an n-Lie algebra, as introduced by Filipov [6], is an algebra equiped with an n-linear, skew symmetric n-ary bracket that satisfies the following Jacobi-like identity: [[x1, . . . , xn], y2, . . . , yn] = n∑ i=1 [x1, . . . , xi−1, [xi, y2, . . . yn], xi+1, . . . , xn] We also recall the following definitions given by Filipov for A an n-Lie algebra: Derivations If � is a linear transformation such that ([a1, . . . , an])� = n∑ i=1 [a1, . . . , (ai)�, . . . , an] for all aj ∈ A then � is a derivation of A. Right Multiplications R(y) = [__, y2, . . . , yn] where (y) will always denote the set y2, . . . , yn of n − 1 vectors right justified in the n-bracket. Example: xR(y) = [x, y2, . . . , yn]. R(A) R(A) is the vector space generated by all right multiplications of A. In [7 Theorem 2.2] we proved for A an n-Lie algebra, A is nilpotent if and only if �(A) = A2 which coresponds to Barnes’ and Chao’s work. In this paper we will prove the n-Lie algebra version of Barnes’ Frattini theorem and continue to examine when �(A) ⊲ A for n-Lie algebras. We will do so by proving the following three theorems: Theorem 1. Let A be an n-Lie algebra. If B,C ⊲ A where C ⊂ �(A) ∩ B and B/C is nilpotent, then B is nilpotent. Theorem 2. If A is a solvable n-Lie algebra, then �(A) ⊲ A. Theorem 3. If A is the n-Lie cross product over F2, then �(A)⋪A. 110 Frattini theory for N-Lie algebras 2. Proof of Theorem 1 We will prove the theorem by contradiction. Assume that the hypothesis holds but B is not nilpotent. By Engel’s theorem there exists an R ∈ R(B) such that BRs ∕= 0 for all s. We apply Fitting’s lemma. Let I be the final image and K be the Fitting null component. Recall that A = I ⊕ K and K is a subalgebra of A. Since B ⊲ A, we observe that I ⊂ B. Furthermore, since B/C is nilpotent, I ⊂ C ⊂ �(A). Now since A = I ⊕ K we see that A = �(A) + K. Since K is a subalgebra it is contained in some maximal subalgebra M and A = �(A)+M . Since M is a maximal subalgebra, �(A) ⊂ M and M = A but this is a contradiction as M is a maximal subalgebra. This proves Theorem 1. Corollary 1. If D ⊲ A and D ⊂ �(A), then D is nilpotent. Corollary 2. If �(A) ⊲ A, then A/�(A) is nilpotent if and only if A is nilpotent. The proof of these corollaries follow from standard Lie algebra argu- ments using Theorem 1. 3. Proof of Theorem 2 We begin the proof of the Theorem 2 with the following lemmas: Lemma 1. If I is a minimal ideal of A a solvable n-Lie algebra, then 1) [I, I, A . . . , A] = 0 2) I ∩ M = 0 or I ∩ M = I for all M maximal subalgebras of A. Proof. First we show 1. We observe [[I, I, A, . . . , A], A, . . . , A] = = [[I, A,A, . . . , A], I, . . . , A] + [I, [I, A, . . . , A]A, . . . , A]+ + [I, I, [A, . . . , A], A, . . . , A] ⊂ [[I, I, A, . . . , A]A, . . . , A] and [I, I, A, . . . , A] ⊲ A. Since A is solvable, [I, I, A, . . . , A] is prop- erly contained in I. Indeed, if I = [I, I, A, . . . , A], we can prove that I = [I, I, A, . . . , A] ⊂ A(n) for all n. We do so inductively. Obviously [I, I, A, . . . , A] ⊂ A(2) and taking the inductive step, we assume that I = [I, I, A, . . . , A] ⊂ A(n). Then I = [I, I, A, . . . , A] ⊂ [A(n), A(n), A, . . . , A] = A(n+1) and as a result I = [I, I, A, . . . , A] ⊂ A(n) for all n. Since A is solvable, I must be 0 contradicting the minimality condition of I. Hence, M. P. Williams 111 [I, I, A, . . . , A] is properly contained in I. But the only way this can happen is if [I, I, A, . . . , A] = 0 otherwise we will again contradict the minimality condition of I. This proves 1. Now we show 2. Assume that I ∩ M is properly contained in I. Since I is not contained in M we observe M is properly contained in I+M and I +M is a subalgebra. The only way this can happen is if I +M = A. Now using 1 we observe that [I ∩ M,A, . . . , A] = [I ∩ M,M + I, . . . ,M + I] = = [I ∩ M,M, . . . ,M ] + 0 + . . . ,+0 ∈ I ∩ M hence I ∩ M ⊲ A. As a result I ∩ M = 0, otherwise we contradict the minimality of I. This proves the lemma. Lemma 2. Let D be a nilpotent derivation of A, an n-Lie algebra over a field F and Dm+1 = 0. Then exp(D) = ∑m i=0 Di i! is an automorphism of A under the following field considerations: either cℎar(F) = 0 or, if cℎar(F) = p ∕= 0 and Dk = 0 for some minimal k, then k < p−1 2 . Proof. By Leibnitz’s rule for n-Lie algebras we see that (∗) [ x1, x2, . . . , xn ] Dk k! = = 1 k! ∑ i1+...+in=k ( k i1, i2, . . . , in ) [x1D i1 , x2D i2 , . . . , xnD in ] = = ∑ i1+...+in=k [ x1D i1 i1! , x2D i2 i2! , . . . , xnD in in! ] . Hence [expD(x1), expD(x2), . . . , expD(xn)] = = [ m∑ i1=1 x1D i1 i1! , m∑ i2=1 x2D i2 i2! , . . . , m∑ in=1 xnD in in! ] = = m∑ i1,...,in=0 [ x1D i1 i1! , x2D i2 i2! , . . . , xnD in in! ] = = nm∑ k=1 ∑ i1+...+in=k [ x1D i1 i1! , x2D i2 i2! , . . . , xnD in in! ] = = nm∑ k=1 [x1, x2, . . . , xn]D k k! = m∑ k=1 [x1, x2, . . . , xn]D k k! = 112 Frattini theory for N-Lie algebras = expD([x1, x2, . . . , xn]) Hence exp(D) is an n-Lie algebra homomorphism. Furthermore (exp(D))(exp(−D)) = (exp(D)) ( m∑ i=0 (−D)i i! ) = = m∑ i,j=0 Dj(−D)i j!i! = 2m∑ k=0 1 k! k∑ i=0 ( k i ) Di(−D)k−i = = 2m∑ k=0 1 k! (D −D)k = I Similarly, (exp(−D))(exp(D) = I. Hence exp(−D) = (exp(D))−1 and exp(D) is an automorphism. This proves Lemma 2. Proof of Theorem 2 The proof follows closely that of Barnes’ [2 p.348 Lemma 3.4] analogous proof for Lie Algebras. We induct on dim(A). Let I be a minimal ideal of A and let �I denote the intersection of all maximal subalgebras M such that I ⊂ M . Then �I/I = ( ∩ M ∣I⊂M M)/I = ∩ M ∣I⊂M M/I = �(A/I). By the induction hypothesis we have that �I/I ⊲ A/I and hence �I ⊲ A. If I ⊂ �(A), then �(A) = �I ⊲ A and we’re done. From here the proof can be broken up and conducted in two cases: 1) I is not a subset of �(A) and CA(I) ∕= I. 2) I is not a subset of �(A) and CA(I) = I. We define CA(I) = {x ∈ A∣[x, I, A, . . . , A] = 0} and call it the cen- tralizer of I in A. Note that since I ⊲ A, then CA(I) ⊲ A. Indeed, for x ∈ CI(A) we see by using the Jacobian property of n-Lie algebras that [[x,A, . . . , A], I, . . . , A] = [[x, I, A . . . , A], A, . . . , A]+ + [x, [A, I, . . . , A], A . . . , A] = 0 + [x, I, A . . . , A] = 0 + 0. Hence CA(I) ⊲ A. We now resume the proof. Case 1 Suppose I is not a subset of �(A) and CA(I) ∕= I. Due to this assumption M. P. Williams 113 there exists M , a maximal subalgebra of A such that I ∩ M is a proper subset of I. By our Lemma 1 I ∩ M = 0. Recall that CA(I) = {x ∈ A∣[x, I, A, . . . , A] = 0}. Recall also that if I ⊲ A, then CA(I) ⊲ A. Suppose I ⊂ CA(I), then CA(I) ∩ M ∕= 0 and [CA(I) ∩ M,A, . . . , A] = [CA(I) ∩ M,M + I, . . . ,M + I] = = [CA(I) ∩ M,M, . . . ,M ] + 0 + . . . ,+0 ⊂ CA(I) ∩ M because CA(I) ⊲ A and M is a subalgebra. Hence CA(I) ∩ M ⊲ A. As a result for each M a maximal subalgebra there exists J ⊂ M where J is a minimal ideal of A. We see that �(A) = ∩ M = ∩ J⊂M �J ⊲ A. Case 2 Suppose CA(I) = I and I is not contained in �(A). We will show that �(A) = 0. As in Case 1, due to this assumption and Lemma 1 there exists M , a maximal subalgebra of A such that I ∩ M = 0. For all m ∈ M we prove that m /∈ �(A). Since m /∈ I = CA(I) there exists i ∈ I and a′is ∈ A such that [m, i, a3, a4, . . . , an] = mRa ∕= 0. Note that Ra is a derivation on I. Since [I, I, A . . . , A] = 0 we see that R2 a = 0 and by Lemma 2, expRa = 1 + Ra is an automorphism of A . Set N = expRa(M) a maximal subalgebra. If m ∈ N then m = n(1 + Ra) = n + nRa for some n ∈ M . Since n,m ∈ M we see that m − n = nRa ∈ M . Hence nRa ∈ I ∩ M = 0 and m = n. But this means that mRa = nRa = 0 which contradicts the fact that mRa ∕= 0. This implies that M ∩ N = 0 and in turn that �(A) = 0 ⊲ A. This proves the theorem. 4. Proof of Theorem 3 As A is simple, it is enough to show that �(A) ∕= 0 and �(A) ∕= A. This is due to the following fact: a subspace S ⊂ A of codimension 1 is a subalgebra if and only if v = n+1∑ i=1 xi ∈ S. Let’s prove this fact. Note that S has a basis of the form {xi + �ixn+1∣�i ∈ F2} n i=1 114 Frattini theory for N-Lie algebras and x1, . . . , xn is the standard basis. This can be easily shown by induc- tion on n. Indeed if n = 2 and {v1, v2} = {v1, x1+x2+x3} is a basis for S, then {v1, v1 + x1 + x2 + x3} is as well and since 0 ∕= v1 ∕= x1 + x2 + x3, we see that v1 + v2 = ∑3 i=1 �ixi where at least one �i is zero. Now we induct on n. We consider An−1 the (n−1)-Lie algebra defined by [vi1 , . . . , v̂ij , . . . , vin−1 ]n−1 = [vi1 , . . . , v̂ij , . . . , vin−1 , vn] = vij where ik ∕= n for all k. By the induction hypothesis, {xi + �ixn+1} n+1 i=1,∕=n is a basis for An−1 and in turn, {xi + �ixn+1, vn} n+1 i=1,∕=n is a basis for A. We note that vn = n+1∑ i=1,∕=n tixi + xn, otherwise we do not have a basis. We observe vn − n+1∑ i=1,∕=n ti(xi + �ixn+1) = xn + n+1∑ i=1,∕=n ti�ixn+1. Replacing vn with the right hand side above, we obtain {xi+�ixn+1} n i=1 as a basis for A where �n = ∑n+1 i=1,∕=n ti�i. Using this basis, we note the only non-zero product is w = [x1 + �1xn+1, . . . , xn + �nxn+1] = xn+1 + n∑ i=1 �ixi. But w ∈ S if and only if w = ∑n i=1 ti(xi + �ixn+1) if and only if �i = ti, for all i and ∑n i=1 ti�i = 1. As a result n∑ i=1 ti�i = n∑ i=1 �2 i = ∑ �i = 1. Hence w ∈ S if and only if ∑ i �i = 1. On the other hand, v = ∑n+1 i=1 xi ∈ S if and only if v = ∑n i=1(xi + �ixn+1) which is equivalent to the fact that ∑ i �i = 1. This completes the proof and the paper. Acknowledgement The author would like to thank: Ernie Stitzinger for guidance and pa- tience, Ellison Anne Williams for love and support, Ellijah Williams M. P. Williams 115 for dancing, the referee for giving remarkable insight in simplifying the proofs, Vyacheslav Futorny for accepting this paper and God for either doing or making all of the above happen. References [1] Bela Bollobas, Graph Theory : An Introductory Course, Springer Verlag, New York (1979). [2] Donald W. Barnes; Humphrey M. Gastineau-Hills, On the Cohomology of Soluble Lie Algebras, Math. Zeitsch. 101 (1967) 343–349. [3] Donald W. Barnes; Humphrey M. Gastineau-Hills, On the Theory of Soluble Lie Algebras, Math. Zeitschr. 106 (1968) 343–354. [4] Donald W. Barnes; Martin L. Newell, Some Theorems on Saturated Homomorphs of Soluble Lie Algebras, Math. Zeitschr. 115 (1970) 179–187. [5] Chao, Chong-Yun, A nonimbedding theorem of nilpotent Lie algebras. Pacific J. Math. 22 (1967), 231–234. [6] V. T. Filippov, n-Lie Algebras, (Russian) Sibirskii Mathimaticheskii Zhurnal 26 (1985), no. 6, 126–140. [7] Michael Peretzian Williams, Nilpotent N-Lie Algebras, Pending Publication. Contact information M. P. Williams Department of Mathematics, Box 8205, NC State University, Raleigh, NC 27695-8205 E-Mail: skew1823@yahoo.com Received by the editors: 22.02.2005 and in final form 12.10.2009.

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