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This article is cited in 2 scientific papers (total in 2 papers)
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.
Received: 10.05.2019
Citation:
J. V. Bekker, V. M. Levchuk, E. A. Sotnikova, “Non-finitary generalizations of nil-triangular subalgebras of Chevalley algebras”, Bulletin of Irkutsk State University. Series Mathematics, 29 (2019), 39–51
Linking options:
https://www.mathnet.ru/eng/iigum383 https://www.mathnet.ru/eng/iigum/v29/p39
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Abstract page: | 159 | Full-text PDF : | 58 | References: | 12 |
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