Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras

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