Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Irkutsk State University
2019-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
| Subjects: | |
| Online Access: | http://mathizv.isu.ru/en/article/file?id=1308 |
| Summary: | 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). |
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| ISSN: | 1997-7670 2541-8785 |