Decomposition of Automorphisms of Certain Solvable Subalgebra of Symplectic Lie Algebra over Commutative Rings

Xing Tao Wang,Lei Zhang

doi:10.1155/2012/242736

Abstract

LetCl+1(R)be the2(l+1)×2(l+1)matrix symplectic Lie algebra over a commutative ringRwith 2 invertible. Thentl+1CR = {m-1m-20-m-1T ∣ m̅1is anl+1upper triangular matrix,m̅2T=m̅2, over R}is the solvable subalgebra ofCl+1(R). In this paper, we give an explicit description of the automorphism group oftl+1(C)(R).

Highlights

We give an explicit description of the automorphism group of tl Classical Lie algebras occupy an important place in matrix algebras
R-algebra of n by n matrices over R that has a structure of a Lie algebra over R with bracket operation x, y xy − yx for any x, y ∈ Mn R
Encouraged by Dokovic 9 and Cao’s 10 papers which described the automorphism groups of Lie algebra consisting of all upper triangular n × n matrices of trace 0 over a connected commutative ring and a commutative ring with n invertible, respectively, in this paper we use similar techniques to those in to prove that any automorphism ψ of tl can be uniquely expressed as ψ θλD, where θ and λD are inner and diagonal automorphisms, respectively, for l ≥ 1 and R is a commutative ring with 2 invertible

Summary

Introduction

Classical Lie algebras occupy an important place in matrix algebras. 1.1 is one of classical Lie algebras, where T denotes the matrix transpose. Encouraged by Dokovic 9 and Cao’s 10 papers which described the automorphism groups of Lie algebra consisting of all upper triangular n × n matrices of trace 0 over a connected commutative ring and a commutative ring with n invertible, respectively, in this paper we use similar techniques to those in to prove that any automorphism ψ of tl can be uniquely expressed as ψ θλD, where θ and λD are inner and diagonal automorphisms, respectively, for l ≥ 1 and R is a commutative ring with 2 invertible. Let I and D be the inner and diagonal automorphism groups, respectively.

Preliminaries

Then we obtain that a111

Proofs of Main Results