泛代数的Gröbner-Shirshov基方法

We will establish Gröbner-Shirshov bases theory for the following algebras over a commutative algebra: modules over an associative conformal algebra, difference-differential associative algebras, differential modules over a differential algebra, Poisson algebras, Novikov algebras, strict monoidal categories, dialgebras, pre-Lie algebras, Rota-Baxter algebras, etc. By using the known Gröbner-Shirshov bases theory, the following applications will be done: automatic structures for some one-relator semigroups (monoids, groups, resp.); proofs of Kac theorem for simple finite dimensional Lie superalgebras and Gabber-Kac theorem for simple infinite dimensional Kac-Moody Lie algebras by Gröbner-Shirshov bases method; extensions of some algebras; normal forms, word problem, etc. of braid groups with B, C, D type, general braid groups (virtual braid groups, ...) and some Coxter groups; the Dehn functions of some groups and semigroups; and so on.

本项目试图建立交换代数上以下代数的Gröbner-Shirshov基理论: 结合的conformal代数上的模, 差分-微分结合代数, 微分代数上的微分模,Poisson代数,Novikov代数,严格monoidal范畴, di-代数, pre-李代数, Rota-Baxter代数等. 利用已经建立的Gröbner-Shirshov基方法给出其应用：研究一关系半群（幺半群，群）的automatic性质；寻找单有限维李superalgebra上的Kac定理、单无限维Kac-Moody李代数上Gabber-Kac定理的Gröbner-Shirshov基证明；某些代数的扩张；B、C、D类型的辫子群、一般的辫子群（virtual braid群，…）和某些Coxter群的normal form、字问题等；寻找某些群和半群的Dehn函数等等.

建立了以下代数簇的Grobner-Shirshov基理论：Gelfand-Dorfman-Novikov-Poisson 代数；Gelfand-Dorfman-Novikov超代数；结合的共形代数；微分Rota-Baxter李代数；带算子Rota-Baxter李代数；二代数；交换二代数。利用已经建立的相关代数的Grobner-Shirshov基理论，构造了自由Gelfand-Dorfman-Novikov超代数的一组线性基底，并证明了任何一个Gelfand-Dorfman-Novikov超代数都能嵌入它的泛包络微分结合超交换代数中; 给出了群、结合代数和李代数等的扩张问题的刻画，这是Grobner-Shirshov基理论的新的应用；给出了二代数的Gelfand-Kirillov维数的刻画；给出两类李代数的Grobner-Shirshov基，刻画其中一类李代数的不可解字问题；给出Schubert多项式的若干组合性质并给出两个新的算法；给出了长度小于四的一关系自动半群的的刻画；证明了任何GDN-Poisson代数可嵌入它的泛包络特殊GDN-Poisson相容代数；构造了由任意集合生成的Rota-Baxter系统的一组线性基底，并在该系统上定义了左Hopf代数的结构；证明了每个有限生成的交换二代数有一个有限的Grobner-Shirshov基，此重要结论将应用于计算机代数。