泛代数的Gröbner-Shirshov基方法

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.

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.