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.

Lie nilpotence in group algebras

2014

Let * be the natural involution on a group algebra FG induced by setting 9-+ g-1, for all 9 E G. Here we survey on the results recently obtained on the Lie nilpotence of the symmetric and skew elements of FG under *.

The algebraic classification of nilpotent algebras

Journal of Algebra and Its Applications, 2021

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

An overview of free nilpotent Lie algebras

2014

Any nilpotent Lie algebra is a quotient of a free nilpotent Lie algebra of the same nilindex and type. In this paper we review some nice features of the class of free nilpotent Lie algebras. We will focus on the survey of Lie algebras of derivations and groups of automorphisms of this class of algebras. Three research projects on nilpotent Lie algebras will be mentioned.

Anosov automorphisms of nilpotent Lie algebras

Journal of Modern Dynamics, 2009

Each matrix A in GL n (Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra f n,r . We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give applications to the study of Anosov automorphisms.