非零权的Rota-Baxter代数的模

【作者】 刘娜；

【导师】 唐孝敏；

【作者基本信息】 黑龙江大学 ， 数学， 2022， 硕士

【摘要】 Rota-Baxter代数由一个结合代数和一个线性算子组成,它起源于上世纪60年代对随机理论的研究.一直以来,Rota-Baxter代数引起了许多数学家和物理学家的广泛关注,并将其用于代数学及物理学等诸多领域.本文分别对非零权Rota-Baxter代数(k[x],P)的模以及非零权单变量无常数项的Rota-Baxter代数(k*[x],P)的模进行了研究.主要结果表明每个非零权Rota-Baxter代数(k[x],P)的模及(k-[x],P)的模都等价于具有确定生成关系的两个变元所生成的非交换代数的模.进而,我们通过解矩阵方程,分别刻画了非零权Rota-Baxter代数(k[x],P)的模及(k*[x],P)的模的分类.此外,我们研究了(k*[x],P)的不可约模和不可分解模.最后,给出了(k*[x],P)模的一些例子.更多还原

【Abstract】 In the 1960s,Rota-Baxter algebra originated from random theory,which consists of an associative algebra and a linear operator.It has been widely concerned by many mathematicians and physicists,and has been used in many fields such as algebra and physics.In this paper,we study the modules of polynomial RotaBaxter algebras(k[x],P)of weight nonzero and the free commutative non-unital Rota-Baxter algebra(k*[x],P)which is the algebra of polynomials in one variable without constant term with Rota-Baxter operators of nonzero weight.The main result shows that every module over a non-zero-weighted Rota-Baxter algebra(k[x],P)and(k*[x],P)is equivalent to the modules over a noncommutative algebra generated by two indeterminates with a generation relationship.Furthermore,we provide the classification of modules of non-zero-weighted Rota-Baxter algebras(k[x],P)and(k*[x],P)through solution to some matrix equations.In addition,we study the irreducible and indecomposable modules of(k*[x],P).Finally,we give some examples of modules of(k*[x],P).更多还原