FREE AMALGAMATION AND AUTOMORPHISM GROUPS

AbstractWe show that the class of graded c-nilpotent Lie algebras over a fixed field K is closed under free amalgamation. In [1] this result was applied, but its proof was incorrect. In case of a finite field K we obtain a Fraïssé limit of all finite graded c-nilpotent Lie algebras over K. This gives an example for the following more general considerations. The existence of free amalgamation for the age of a Fraïssé limit implies the universality of its automorphism group for all automorphism groups of substructures of that Fraïssé limit. We use [6] and [5].

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On the automorphism groups of certain Lie algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100067980 ◽

1989 ◽

Vol 106 (1) ◽

pp. 67-76 ◽

Cited By ~ 3

Author(s):

Dan Segal

Keyword(s):

Automorphism Group ◽

Lie Algebra ◽

Lie Algebras ◽

Automorphism Groups ◽

Simple Condition ◽

Separable Extension ◽

Ground Field ◽

Nilpotent Lie Algebras ◽

Nilpotent Matrices

We fix a ground field k and a finite separable extension K of k. To a Lie algebra L over k is associated the Lie algebra KL = K ⊗kL over K. If we forget the action of K, we can think of KL as a larger Lie algebra over k; in particular we can ask what is the automorphism group Autk KL of KL as a k-algebra. There does not seem to be any simple answer to this question in general; the purpose of this note is to give a simple condition on L which makes Autk KL quite easy to determine. Examples of algebras which satisfy this condition include the free nilpotent Lie algebras and the algebras of all n × n triangular nilpotent matrices.

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AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

International Journal of Algebra and Computation ◽

10.1142/s021819670700372x ◽

2007 ◽

Vol 17 (03) ◽

pp. 527-555 ◽

Cited By ~ 5

Author(s):

YOU'AN CAO ◽

DEZHI JIANG ◽

JUNYING WANG

Keyword(s):

Automorphism Group ◽

Commutative Ring ◽

Lie Algebra ◽

Lie Algebras ◽

Commutative Rings ◽

Nilpotent Lie Algebras ◽

Simple Lie Algebra ◽

Finite Dimensional ◽

Positive Roots ◽

Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

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Strong map-symmetry of SL(3,K) and PSL(3,K) for every finite field K

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500481 ◽

2020 ◽

pp. 2150048

Author(s):

Antonio Breda d’Azevedo ◽

Domenico A. Catalano

Keyword(s):

Automorphism Group ◽

Finite Field ◽

Automorphism Groups ◽

Group Automorphism ◽

Regular Maps ◽

Strong Map

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

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Finite Groups Which are Automorphism Groups of Infinite Groups Only

Canadian Mathematical Bulletin ◽

10.4153/cmb-1985-008-4 ◽

1985 ◽

Vol 28 (1) ◽

pp. 84-90

Author(s):

Jay Zimmerman

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Finite Field ◽

Finite Groups ◽

Finite Simple Group ◽

Automorphism Groups ◽

Outer Automorphism ◽

Schur Multiplier ◽

Infinite Groups ◽

Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

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Automorphisms of free metabelian Lie algebras

International Journal of Algebra and Computation ◽

10.1142/s0218196716500326 ◽

2016 ◽

Vol 26 (04) ◽

pp. 751-762 ◽

Cited By ~ 2

Author(s):

C. E. Kofinas ◽

A. I. Papistas

Keyword(s):

Automorphism Group ◽

Lie Algebra ◽

Lie Algebras ◽

Automorphism Groups ◽

Free Algebras ◽

Free Metabelian Lie Algebra ◽

Dense Subgroups

We give a sharpening of a result of Bryant and Drensky [R. M. Bryant and V. Drensky, Dense subgroups of the automorphism groups of free algebras, Canad. J. Math. 45(6) (1993) 1135–1154] for the automorphism group [Formula: see text] of a free metabelian Lie algebra [Formula: see text], with [Formula: see text]. In particular, we prove that the subgroup of [Formula: see text] generated by [Formula: see text] and two more IA-automorphisms is dense in [Formula: see text] and, for [Formula: see text], the subgroup generated by [Formula: see text] and one more IA-automorphism is dense in [Formula: see text].

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EXTENSIONS OF JOHNSON'S AND MORITA'S HOMOMORPHISMS THAT MAP TO FINITELY GENERATED ABELIAN GROUPS

Journal of Topology and Analysis ◽

10.1142/s1793525313500027 ◽

2013 ◽

Vol 05 (01) ◽

pp. 57-85 ◽

Cited By ~ 2

Author(s):

MATTHEW B. DAY

Keyword(s):

Abelian Group ◽

Automorphism Group ◽

Lie Algebras ◽

Mapping Class Group ◽

Finite Rank ◽

Class Group ◽

Homogeneous Spaces ◽

Group Cohomology ◽

Mapping Class ◽

Nilpotent Lie Algebras

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [5]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.

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Automorphism Groups of Nilpotent Lie Algebras

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-36.2.257 ◽

1987 ◽

Vol s2-36 (2) ◽

pp. 257-274 ◽

Cited By ~ 1

Author(s):

Roger M. Bryant

Keyword(s):

Lie Algebras ◽

Automorphism Groups ◽

Nilpotent Lie Algebras

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Automorphisms of Free Nilpotent Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-015-1 ◽

1990 ◽

Vol 42 (2) ◽

pp. 259-279 ◽

Cited By ~ 17

Author(s):

Vesselin Drensky ◽

C. K. Gupta

Keyword(s):

Automorphism Group ◽

Lie Algebra ◽

Lie Algebras ◽

Linear Group ◽

General Linear Group ◽

General Linear ◽

Free Lie Algebra ◽

Nilpotent Lie Algebras

Let Fm be the free Lie algebra of rank m over a field K of characteristic 0 freely generated by the set ﹛x1,… ,xm﹜, m ≧ 2. Cohn [7] proved that the automorphism group Aut Fm of the K-algebra Fm is generated by the following automorphisms: (i) automorphisms which are induced by the action of the general linear group GLm (= GLm(K)) on the subspace of Fm spanned by ﹛x1, … ,xm﹜; (ii) automorphisms of the form x1 → x1 +f(x2,… ,xm),Xk → xk, k ≠ 1, where the polynomial f(x2,…,xm) does not depend on x1.

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