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 (13 of classical type. For type A(n-1) it is associated to algebra NT (n, K) of (lower) nil -triangular n x n- matrices over K. The algebra R = NT (F, K) of all nil -triangular F -matrices alpha = parallel to a(ij)parallel to(ij is an element of T) Er over K with indices from chain F 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 Tz,z_i of all F -matrices with zeros above i-throw and in columns with numbers > 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(F, K)). As corollary we obtain that the automorphism groups Aut R and Aut R" coincide. Partially automorphisms studied earlier, in particulary, for Aut UT (F, K) when K is a field. Well-known (1990) special matrix representation of Lie algebras N.:13(K) allows to construct non-finitary generalizations NG(K) of type G = Br, Cr and Dr. Be research automorphisms by transfer to factors of Lie ring NG(K) which is isomorphic to NT (F, K). On the other hand, for any chain F finitary nil -triangular F -matrices forms finitary Lie algebra F NG(F, K) of type G = Ar ( i.e., F NG(F, K)), Br, Cr and Dr. Earlier automorphisms was studied (V. M. Levchuk and G. S. Sulejmanova, 1987 and 2009) for Lie ring F NT (F, 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).