Gröbner-Shirshov基及其计算

本项目利用已经建立的和将建立的相关代数的Gröbner-Shirshov基理论，研究以下相关课题的前沿和热点问题：建立三代数、Leibniz超代数、generic Poisson代数、Novikov-Poisson代数、严格的monoidal范畴的Gröbner-Shirshov基理论并给出它们的一些重要的应用；利用我们建立的二代数的Gröbner-Shirshov理论，解决以下公开问题：构造自由幂等二半群、自由二群和二群的自由积；证明广义的Hecke-Grothendieck多项式的Kirillov猜想；研究一关系generic Poisson代数的字问题；研究Novikov-Poisson代数的Freiheitssatz定理是否成立；给出Leibniz超代数扩张的完全刻画.

It is known that Gröbner-Shirshov bases theories in universal algebras is a powerful tool to solve some classical problems in algebras. It is not easy to establish Gröbner-Shirshov bases method in some algebras. The program is mainly concerned in the following hot problems in algebras: establish Gröbner-Shirshov bases theories for trialgebras, Leibniz superalgebras, Novikov-Poisson algebras, generic Poisson algebras, and strict monoidal categories. Some important applications will be given on such algebras. By using our Gröbner-Shirshov bases theory for dialgeras obtained before, solve the open problems: construct free idempotent disemigroups, free digroups and free product of digroups; show that the Kirillov conjecture in 2016; research the word problem for one-relator generic Poisson algebras; show that Freiheitssatz theorem of Novikov-Poisson algebras; and give a complete characterization of extensions of Leibniz superalgebras.