Gröbner-Shirshov基理论在非结合代数中的应用

我们将应用Gröbner-Shirshov基理论研究一系列重要的代数: (1)构造自由Lie-admissible代数的线性基底,并研究(Lie-admissible代数簇,李代数簇)是否构成一个PBW-pair; (2) 研究(assosymmetric代数簇,李代数簇)是否构成一个PBW-pair; (3)构造(没有幺元的)自由Novikov-Poisson代数的线性基底,研究是否每个Novikov-Poisson代数都能嵌入到它的泛包络Novikov-Poisson相容代数中;(4)建立双交换代数的Gröbner-Shirshov基理论，并尝试证明每个有限生成的双交换代数的Gelfand-Kirillov维数都是一个自然数.

We shall apply Gröbner-Shirshov bases theory to study a series of important algebras: (1) We would construct linear bases for free Lie-admissible algebras, and see whether the pair (Lie-admissible, Lie) of the varieties of Lie-admissible algebras and Lie algebras is a PBW-pair; (2) We would answer the question on whether the pair (Assosymmetric, Lie) of the varieties of assosymmetric algebras and Lie algebras is a PBW-pair; (3) We would construct linear bases for free Novikov-Poisson algebras (without unit), and check whether every Novikov-Poisson algebra can be embedded into its universal enveloping Novikov-Poisson admissible algebra; (4) We shall establish the Gröbner-Shirshov bases theory for bicommutative algebras and try to prove that the Gelfand-Kirillov dimension of an arbitrary finitely generated bicommutative algebra is a non-negative integer.