Shuang Algebraic Structure On A Lie Algebra Class

In many branches of mathematics and physics, affine Kac-Moody algebra andits representation play an important role, which are based on the single variableLaurent polynomial ring as its coordinate algebra. As a non-commutative general-ization of the Laurent polynomial algebra, the quantum torus algebra has alreadybeen studied by many scholars. In this paper, we begin the quantum torus C withtwo variables, then we discuss the Lie bialgebra structures on the derivation algebraof C.Lie bialgebra is a vector space with both the structures of Lie algebra and Liecoalgebra. In the first chapter, we mainly introduce the current situations of Liebialgebra, the purpose and the significance of this investigate.In the second chapter, we first recall the concept of Lie bialgebra and somebasic knowledge, then we introduce the derivation algebra of the quantum torusand some related results of derivation.In the third chapter, we show the main results of this paper, which is thatevery Lie bialgebra structures on the derivation algebra of the quantum torus is atriangular coboundary Lie bialgebra.