Exchange Study Of The Typical Group On The Ring And Typical Lie Algebra Structure

In this paper, we have studied construction of classical groups and that of the derivation algebras of classical lie algebras over rings.In chapter 1, one type of maximal subgroups in unitary groups over local rings is obtained. Let R be a local ring with a automorphism φ of order 2: a(?)(a|ˉ) and CharR≠2, m be a positive integer, S be the unique maximal ideal of R, We define:then G(S) is a maximal subgroups of U(2m,R),In chapter 2, one type of maximal subgroups in unitary groups over polynomialrings is obtained. Let F be a field with a automorphism φ of order 2: a(?)(a|ˉ), R=F [λ] is a ring of polynomials in one variable over F, then a_mλ^m+…+a₁λ +a₀ (?)(a|ˉ)_mλ^m+…+(a|ˉ)₁,λ+(a|ˉ)₀ be an automorphism of order 2 induced by φ. Let m be a positive integer, (f|ˉ) = f∈F and S be the ideal of R generated by λ - f. Wedefine G(5) =Then G(5) is a subgroup of U(2m,R).In chapter 3, we have studied construction of the derivation algebras of classicallie algebras over a commutative ring. Let R be an arbitrary commutative ring with identity, gl(n, R) the general linear lie algebra over R consisting of all n × n matricesover R and with the bracket operation [x,y] = xy -yx, t ( resp., u) the lie subalgebraof gl(n, R) consisting of all n×n upper triangular (resp., strictly upper triangular ) matrices over R and d the lie subalgebra of gl(n,R) consisting of all n×n diagonal matrices over R. We will describe the intermediate lie algebras between d and t, then give an explicit description on the derivations of the intermediate lie algebras between d and t.