This article is cited in
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Algebraic and logical methods in computer science and artificial intelligence
Non-finitary generalizations of nil-triangular subalgebras of Chevalley algebras
J. V. Bekker,
V. M. Levchuk,
E. A. Sotnikova Siberian Federal University, Krasnoyarsk, Russian Federation
Abstract:
Let
$N\Phi(K)$ be a niltriangular subalgebra of Chevalley algebra
over a field or ring
$K$ associated with root system
$\Phi$ of
classical type. For type
$A_{n-1}$ it is associated to algebra
$NT(n,K)$ of (lower) nil-triangular
$n \times n$- matrices over
$K$. The algebra
$R=NT(\Gamma,K)$ of all nil-triangular
$\Gamma$-matrices
$\alpha =||a_{ij}||_{i,j\in \Gamma}$ over
$K$
with indices from chain
$\Gamma$ of natural numbers gives its
non-finitary generalization. It is proved, (together with
radicalness of ring
$R$) that if
$K$ is a ring without zero
divizors, then ideals
$T_{i,i-1}$ of all
$\Gamma$-matrices with
zeros above
$i$-th row and in columns with numbers
$\geq i$
exhausts all maximal commutative ideals of the ring
$R$ and associated
Lie rings
$R^{(-)}$, and also maximal normal subgroups
of adjoint group (it is isomorphic to the generalize unitriangular
group
$UT(\Gamma,K)$). As corollary we obtain that the
automorphism groups
$Aut\ R$ and
$Aut\ R^{(-)}$ coincide.
Partially automorphisms studied earlier, in particulary, for
$Aut\ UT(\Gamma,K)$ when
$K$ is a field.
Well-known (1990) special matrix representation of Lie algebras
$N\Phi(K)$ allows to construct non-finitary generalizations
$NG(K)$ of type
$G=B_\Gamma,C_\Gamma$ and
$D_\Gamma$. Be research
automorphisms by transfer to factors of Lie ring
$NG(K)$ which is
isomorphic to
$NT(\Gamma,K)$.
On the other hand, for any chain
$\Gamma$ finitary nil-triangular
$\Gamma$-matrices forms finitary Lie algebra
$FNG(\Gamma,K)$ of
type
$G=A_{\Gamma}$ ( i.e.,
$FNG(\Gamma,K)$),
$B_{\Gamma},C_{\Gamma }$ and
$D_{\Gamma}$. Earlier automorphisms
was studied (V. M. Levchuk and G. S. Sulejmanova, 1987 and 2009)
for Lie ring
$FNT(\Gamma,K)$ over ring
$K$ without zero divizors
and, also, for finitary generalizations of unipotent subgroups of
Chevalley group of classical type over the field (including
twisted types).
Keywords:
Chevalley algebra, nil-triangular subalgebra, unitriangular group, finitary and nonfinitary generalizations, radical ring.
UDC:
512.5
MSC: 22E05 Received: 10.05.2019
DOI:
10.26516/1997-7670.2019.29.39