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22
Lie methods in growth of groups and groups of finite width
, 2000
"... In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an Nseries of subgroups. The asymptotics of the Poincaré series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial ..."
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Cited by 20 (14 self)
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In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an Nseries of subgroups. The asymptotics of the Poincaré series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial growth and growth of type e √ n in the class of residually–p groups, and gives examples of finitely generated p–groups of uniformly exponential growth. In the second part, we produce two examples of groups of finite width and describe their Lie algebras, introducing a notion of Cayley graph for graded Lie algebras. We compute explicitly their lower central and dimensional series, and outline a general method applicable to some other groups from the class of branch groups. These examples produce counterexamples to a conjecture on the structure of justinfinite groups of finite width.
RESTRICTED SIMPLE LIE ALGEBRAS AND THEIR INFINITESIMAL DEFORMATIONS
, 2007
"... Abstract. In the first two sections, we review the BlockWilsonPremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Rest ..."
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Cited by 4 (2 self)
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Abstract. In the first two sections, we review the BlockWilsonPremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Restricted Lie algebras We fix a field F of characteristic p> 0 and we denote with Fp the prime field with p elements. All the Lie algebras that we will consider are of finite dimension over F. We are interested in particular class of Lie algebras, called restricted (or pLie algebras). Definition 1.1 (Jacobson [JAC37]). A Lie algebra L over F is said to be restricted (or a pLie algebra) if there exits a map (called pmap), [p] : L → L, x ↦ → x [p], which verifies the following conditions: (1) ad(x [p]) = ad(x) [p] for every x ∈ L. (2) (αx)[p] = αpx [p] for every x ∈ L and every α ∈ F. (3) (x0 + x1) [p] = x [p] 0 + x[p] 1 + ∑ p−1 i=1 si(x0, x1) for every x, y ∈ L, where the element si(x0, x1) ∈ L is defined by si(x0, x1) = − 1 ∑ adxu(1) ◦ adxu(2) ◦ · · · ◦ adxu(p−1)(x1), r u the summation being over all the maps u: [1, · · · , p − 1] → {0, 1} taking rtimes the value 0. Example. (1) Let A an associative Falgebra. Then the Lie algebra DerFA of Fderivations of A is a restricted Lie algebra with respect to the pmap D ↦ → Dp: = D ◦ · · · ◦ D. (2) Let G a group scheme over F. Then the Lie algebra Lie(G) associated to G is a restricted Lie algebra with respect to the pmap given by the differential of the homomorphism G → G, x ↦ → xp: = x ◦ · · · ◦ x. One can naturally ask when a FLie algebra can acquire the structure of a restricted Lie algebra and how many such structures there can be. The following criterion of Jacobson answers to that question. Proposition 1.2 (Jacobson). Let L be a Lie algebra over F. Then (1) It is possible to define a pmap on L if and only if, for every element x ∈ L, the pth iterate of ad(x) is still an inner derivation. (2) Two such pmaps differ by a semilinear map from L to the center Z(L) of L, that is a map f: L → Z(L) such that f(αx) = α p f(x) for every x ∈ L and α ∈ F.
On The Distribution Of ARComponents Of Restricted Lie Algebras
 Contemp. Math
"... In this paper we study the distribution of indecomposable modules of the reduced enveloping algebra u(L; ) associated to a finite dimensional restricted Lie algebra (L; [p]). Each component of the stable AuslanderReiten quiver possesses only finitely modules of a given dimension. We combine this fa ..."
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In this paper we study the distribution of indecomposable modules of the reduced enveloping algebra u(L; ) associated to a finite dimensional restricted Lie algebra (L; [p]). Each component of the stable AuslanderReiten quiver possesses only finitely modules of a given dimension. We combine this fact with geometric techniques in order to produce families of components of type ZZ[A 1 ]. 0. Introduction and Preliminaries Let F be an algebraically closed field, a finite dimensional F algebra of infinite representation type. If is tame, it follows from Brauer Thrall II and the work of CrawleyBoevey [5, Corollary E] that there exists a number d 2 IN, such that for each multiple `d there are infinitely many nonisomorphic indecomposablemodules of dimension `d. In case is wild, this conclusion continues to hold (with d even) since there is a representation embedding from the module category of the Kronecker algebra F [X; Y ]=(X 2 ; Y 2 ) into the module category of . A related questi...
Computing Resolutions over Finite pGroups
, 2000
"... . A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any nite pgroup is detailed. The resolutions we construct ([32]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian pgroup of the same order as t ..."
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. A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any nite pgroup is detailed. The resolutions we construct ([32]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian pgroup of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented. 1 Introduction In this paper, we present a uniform constructive approach to calculating relatively small resolutions over nite pgroups. The algorithm we use comes from [32, 8.1.8 and the penultimate paragraph of 9.4]. There has been a massive amount of work done on the structure of pgroups since the beginning of group theory. A good introduction is [22]. We combine mathematical and computer methods to construct the uniform resolut...
Simple finite group schemes and their infinitesimal deformations, preprint available at arXiv:0811.2668
"... Abstract. We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these simple objects have been classified. We review this ..."
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Abstract. We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes. 1.
On the Structure of Cohomology of Hamiltonian pAlgebras
, 2004
"... Abstract. We demonstrate advantages of nonstandard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theor ..."
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Abstract. We demonstrate advantages of nonstandard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theorem on the structure of cohomology of Lie algebra with inner grading element, we show that all nontrivial cohomology classes are located in the grades which are the multiples of the characteristic p. Besides, this grading implies another symmetries in the structure of cohomology. These symmetries are based on the Poincaré duality and symmetry with respect to transpositions of conjugate variables of the symplectic space. Some results obtained by computer program utilizing these peculiarities in the cohomology structure are presented. 1
Some problems in the theory of rings that are nearly associative
 IN [SSS05]. MR0102532 34 M. VAUGHANLEE, SUPERALGEBRAS AND DIMENSIONS OF ALGEBRAS, INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
, 1958
"... The words “some problems ” in the title of this article mean primarily that the article considers absolutely no results about algebras of finite dimension. Among other questions that remain outside the scope of the article, we mention, for example, various theorems about decomposition of algebras (s ..."
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The words “some problems ” in the title of this article mean primarily that the article considers absolutely no results about algebras of finite dimension. Among other questions that remain outside the scope of the article, we mention, for example, various theorems about decomposition of algebras (see for example [70, 47]) which are closely related to the theory of algebras of finite dimension. The author is grateful to A. G. Kurosh and L. A. Skornyakov who got acquainted with the first draft of the manuscript and made a series of very valuable comments.
Simple finite group schemes and their infinitesimal deformations
, 811
"... We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classi ..."
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We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.