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...
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Irkutsk State University
2019-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | http://mathizv.isu.ru/en/article/file?id=1308 |
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| author | J. V. Bekker V. M. Levchuk E. A. Sotnikova |
| author_facet | J. V. Bekker V. M. Levchuk E. A. Sotnikova |
| author_sort | J. V. Bekker |
| collection | DOAJ |
| description | 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). |
| first_indexed | 2024-12-11T12:02:49Z |
| format | Article |
| id | doaj.art-1f49a3e28e264483b732e7fdfc9047e5 |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-12-11T12:02:49Z |
| publishDate | 2019-09-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-1f49a3e28e264483b732e7fdfc9047e52022-12-22T01:08:02ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852019-09-012913951https://doi.org/10.26516/1997-7670.2019.29.39Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley AlgebrasJ. V. BekkerV. M. LevchukE. A. SotnikovaLet $N\Phi(K)$ 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).http://mathizv.isu.ru/en/article/file?id=1308Chevalley algebranil-triangular subalgebraunitriangular groupfinitary and nonfinitary generalizationsradical ring |
| spellingShingle | J. V. Bekker V. M. Levchuk E. A. Sotnikova Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras Известия Иркутского государственного университета: Серия "Математика" Chevalley algebra nil-triangular subalgebra unitriangular group finitary and nonfinitary generalizations radical ring |
| title | Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras |
| title_full | Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras |
| title_fullStr | Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras |
| title_full_unstemmed | Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras |
| title_short | Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras |
| title_sort | non finitary generalizations of nil triangular subalgebras of chevalley algebras |
| topic | Chevalley algebra nil-triangular subalgebra unitriangular group finitary and nonfinitary generalizations radical ring |
| url | http://mathizv.isu.ru/en/article/file?id=1308 |
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