Decomposition Of Automorphisms Of Subalgebras Of Matrix Algebra Over Commutative Rings

It is theoretically important to solve the problems of decomposition of automorphisms of solvable subalgebra and nilpotent subalgebra of matrix algebra over commutative rings.Let t_n+1 (R) be the algebra of all upper triangular square matrices of order n + 1 over a 2-torsionfree commutative ring R with the identity. For n ≥ 1, we prove that any Jordan automorphism of t_n+1(R) can be decomposed uniquely as a product of graph, inner and diagonal automorphisms.Over a 2-torsionfree commutative ring R with identity, the algebra of all strictly upper triangular n + 1 by n + 1 matrices is denoted by N_n+1(R). We prove that any Jordan automorphism of N_n+1(R) can be uniquely decomposed as a product of a graph automorphism, a diagonal automorphism, a central automorphism and an inner automorphism for n ≥ 3. We also give a decomposition for any Jordan automorphism of N₂(R) and N₃(R).Let t_n+1 (R) be the algebra of all upper triangular square matrices of order n + 1 over a commutative ring R with the identity 1 and unit 2. For n ≥ 2, we prove that any Lie automorphism of t_n+1(R) can be uniquely written as a product of graph, central, inner and diagonal automorphisms.Let D_l+1(R) be the orthogonal Lie algebra over a commutative ring R with 2 invertible and m₁ an l+ 1 upper triangular matrix.is the solvable subalgebra of D_l+1(R). For l≥ 1,l≠3, we prove that any automorphism of t_n+1^D(R) can be uniquely written as a product of graph, inner, diagonal and generalized diagonal automorphisms.Let C_l+1(R) be the symplectic Lie algebra over a commutative ring R with 2 invertible and m₁ an l + 1 upper triangular matrix.