On Nilpotent Groups of Algebra Automorphisms (original) (raw)

Automorphism Groups of Nilpotent Lie Algebras

Journal of the London Mathematical Society, 1987

In [2], the author and J. R. J. Groves showed that every linear algebraic group over a field of characteristic 0 arises from some nilpotent Lie algebra as the group of linear transformations induced on the commutator quotient by the automorphism group of the algebra. The purpose of the present paper is to extend this result to apply to arbitrary fields. Let L be a finitely generated nilpotent Lie algebra over a field k, and let m be the dimension of L. Choosing a basis of L, we may regard the automorphism group Aut (L) as a subgroup of GL (m, k). Let K be an algebraically closed extension field of k. Then K® L (tensor product taken over k) has the structure of a Lie algebra over K. Taking the basis of K® L corresponding to the chosen basis of L we may regard Aut (K ® L) as a subgroup of GL (m, K), and we have Aut (L) = Aut (K ® L) n GL (m, k). As remarked in [2], Aut(K® L) is a fc-closed subgroup of GL(m, K), that is, a subgroup closed in the Zariski fc-topology. Thus (see [4, p. 217]) Aut (K®L) is defined over some field K, where K is a finite purely inseparable extension of k and k £ K £ K. Let L denote the derived algebra of L, that is, L = [L, L], (We shall use commutator notation for Lie products.) Then Aut(L) induces on L/L' a group of linear transformations which we denote by Aut* (L). Choosing a basis of L/L, we may regard Aut*(L) as a subgroup of GL in, k), where n = dim (L/L'). Similarly, with the corresponding basis of (K® L)/(K® L)', Aut* (K (g) L) is a subgroup of The natural map from L to L/L' gives rise to a homomorphism : Aut OK L) > GL (n, K) with image Aut* (K ® L). It is easy to verify that 0 is a morphism of algebraic groups defined over K. Therefore (see [4, p. 218]) Aut* (K® L) is an algebraic subgroup of GL(n, K) defined over K. In particular, Aut* (K® L) is /c-closed. If char(/:) = 0, then K = k, and so Aut* (A^ ® L) is A:-closed; while if char (k) = p # 0, then Aut* (K ® L) is jc'-closed for every power q of p (by an easy argument), and so again Aut*(AT® L) is ^-closed. If char (k) = 0 then, as noted in [2], Aut* (L) = Aut* (K® L) D GL in, k), and consequently Aut*(L) is a Zariski-closed subgroup of Gh(n,k). The latter statement also holds if k is algebraically closed (by the discussion above with K=k)

Automorphisms of Non-Singular Nilpotent Lie Algebras

For a real, non-singular, 2-step nilpotent Líe algebra n, the group Aut(n)/ Auto(n), where Auto(n) is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the l-dimensional group of dilations. Maximality of some automorphísms groups of n follows and is related to how close is n to being of Heisenberg type. For example, at least when the dimension ofthe center is two, dimAut(n) is maximal if and only if n is of Heisenberg type. The connection with fat distributions is discussed. Matbematics Subject Classification 2010: 17B30, 16W25.

Group algebras and Lie nilpotence

Journal of Algebra, 2013

Let * be an involution of a group algebra F G induced by an involution of the group G. For char F = 2, we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of *-skew elements is nilpotent.

Automorphisms of the Category of the Free Nilpotent Groups of the Fixed Class of Nilpotency

International Journal of Algebra and Computation, 2007

This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0 of all automorphisms of the category Θ0 is from the group Inn Θ0 of inner automorphisms of Θ0 (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎. Let Θ = 𝔑d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.

Group algebras of torsion groups and Lie nilpotence

Journal of Group Theory, 2000

Let à be an involution of a group algebra FG induced by an involution of the group G. For char F 0 2, we classify the torsion groups G with no elements of order 2 whose Lie algebra of Ã-skew elements is nilpotent.

The geometric classification of nilpotent algebras

2021

We give a geometric classification of n-dimensional nilpotent, commutative nilpotent and anticommutative nilpotent algebras. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit generic families of algebras which define each of these varieties. We show some applications of these results in the study of the length of anticommutative algebras.

A method to obtain the lie group associated with a nilpotent lie algebra

Computers & Mathematics With Applications, 2006

According to Ado and Cm'tan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform. (~)