Nonfinitary algebras and their automorphism groups
I. N. Zotov,
V. M. Levchuk Siberian Federal University, Krasnoyarsk
Abstract:
Let
$\Gamma$ be a linearly ordered set (chain), and let
$K$ be an associative commutative ring with a unity. We study the module of all matrices over
$K$ with indices in
$\Gamma$ and the submodule
$NT({\Gamma},K)$ of all matrices with zeros on and above the main diagonal. All finitary matrices in
$NT({\Gamma},K)$ form a nil-ring. The automorphisms of the adjoint group (in particular, Ado's and McLain's groups) were already described for a ring
$K$ with no zero divisors. They depend on the group
$\mathcal{A} (\Gamma)$ of all automorphisms and antiautomorphisms of
$\Gamma$. We show that
$NT({\Gamma}, K)$ is an algebra with the usual matrix product iff either (a)
$\Gamma$ is isometric or anti-isometric to the chain of naturals and
$\mathcal{A} (\Gamma)=1$ or (b)
$\Gamma$ is isometric to the chain of integers and
$\mathcal{A} (\Gamma)$ is the infinite dihedral group. Any of these algebras is radical but not a nil-ring. When
$K$ is a domain, we find the automorphism groups of the ring
$\mathcal{R}=NT({\Gamma}, K)$ of the associated Lie ring
$L(\mathcal{R})$ and the adjoint group
$G(\mathcal{R})$ (Theorem 3). All three automorphism groups coincide in case {(a)}. In the main case (b) the group
$\operatorname{Aut} \mathcal{R}$ has more complicated structure, and the index of each of the groups
$\operatorname{Aut} L(\mathcal{R})$ and
$\operatorname{Aut} G(\mathcal{R})$ is equal to
$2$. As a consequence, we prove that every local automorphism of the algebras
$\mathcal{R}$ and
$L(\mathcal{R})$ is a fixed automorphism modulo
$\mathcal{R}^2$.
Keywords:
nil-triangular subalgebra, nonfinitary generalizations, radical ring, associated Lie ring, adjoint group, automorphism group, local automorphism.
UDC:
512.54+
512.55 Received: 12.05.2021
Revised: 13.09.2021
Accepted: 11.10.2021
DOI:
10.33048/smzh.2022.63.107