# Finite gradings of special linear Lie algebras

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- FINITE PRESENTATION OF ABELIAN-BY-FINITE-DIMENSIONAL LIE ALGEBRAS. BRYANT, R. M.; GROVES, J. R. J. // Journal of the London Mathematical Society;08/01/1999, Vol. 60 Issue 1, p45
In this paper we shall prove the main part ((3) implies (1)) of the following theorem. The proof of the theorem will be completed in a forthcoming paper [6].

- N-commutators for simple Lie algebras. Dzhumadil'daev, A. S. // AIP Conference Proceedings;11/14/2007, Vol. 956 Issue 1, p159
The N-commutator sN(X1,...,XN) of N vector fields X1,...,XN (differential operators of order 1) is defined as the skew-symmetric sum of the N! compositions XÏƒ(1)...XÏƒ(N) for all permutations ÏƒâˆˆSymN. In general it is a differential operator of order more than 1, but for some special...

- Enumeration of maximal subalgebras in free restricted lie algebras. Petrogradskiĭ, V. M.; Smirnov, A. A. // Siberian Mathematical Journal;Nov2008, Vol. 49 Issue 6, p1101
Given a finitely generated restricted Lie algebra L over the finite field $$ \mathbb{F}_q $$ , and n â‰¥ 0, denote by a n ( L) the number of restricted subalgebras H âŠ† L with $$ \dim _{\mathbb{F} _q} $$ L/H = n. Denote by Ã£ n ( L) the number of the subalgebras satisfying the...

- The hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra. Kabanov, A. N. // Siberian Mathematical Journal;Mar2009, Vol. 50 Issue 2, p261
We describe the hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra over an arbitrary field. Using it, we prove that this group admits no faithful representation by matrices over a field provided that the algebra rank is at least four.

- Test rank of an abelian product of a free Lie algebra and a free abelian Lie algebra. EKICI, NAIME; ÖĞÜŞLÜ, NAZAR // Proceedings of the Indian Academy of Sciences: Mathematical Scie;Aug2011, Vol. 121 Issue 3, p291
Let F be a free Lie algebra of rank n â‰¥ 2 and A be a free abelian Lie algebra of rank m â‰¥ 2. We prove that the test rank of the abelian product F Ã— A is m. Morever we compute the test rank of the algebra $F/\gamma _{k}\left( F\right) ^{^{\prime }}$.

- On generalized Witt algebras in one variable. Ki-Bong Nam; Pakianathan, Jonathan // Turkish Journal of Mathematics;2011, Vol. 35 Issue 3, p405
We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples. We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra...

- Automorphisms, definable relations, and covers of elements of the computability potential scale for all finite algebras. Pinus, A. G. // Algebra & Logic;Jul2008, Vol. 47 Issue 4, p263
Definable subsets are distinguished, based on which automorphisms of the scale of all finite algebras are studied.

- Embeddings of Sz(32) in E8(5). Saxl, Jan; Wales, David B.; Wilson, Robert A. // Bulletin of the London Mathematical Society;2000, Vol. 32 Issue 2, p196
We show that the Suzuki group Sz(32) is a subgroup of E8(5), and so is its automorphism group. Both are unique up to conjugacy in E8(F) for any field F of characteristic 5, and the automorphism group Sz(32):5 is maximal in E8(5). 1991 Mathematics Subject Classification 20E28.

- Endomorphisms of Free Metabelian Lie Algebras Which Preserve Orbits. Chirkov, I. V.; Shevelin, M. A. // Siberian Mathematical Journal;Nov/Dec2004, Vol. 45 Issue 6, p1135
For the free rank 2 metabelian Lie algebra over an infinite field we prove that an endomorphism of the algebra which preserves the automorphic orbit of a nonzero element is an automorphism. We construct some counterexamples over finite fields.