Gröbner–Shirshov Bases for Irreducible sln+1-Modules

In [10], inspired by an idea of Gröbner, Buchberger discovered an effective algorithm for solving the reduction problem for commutative algebras, which is now called the Gröbner Basis Theory. It was generalized to associative algebras through Bergman’s Diamond Lemma [2], and the parallel theory for Lie algebras was developed by Shirshov [21]. The key ingredient of Shirshov’s theory is the Composition Lemma, which turned out to be valid for associative algebras as well (see [3]). For this reason, Shirshov’s theory for Lie algebras and their universal enveloping algebras is called Gröbner–Shirshov Basis Theory. For finite-dimensional simple Lie Algebras, Bokut and Klein constructed the Gröbner–Shirshov bases explicitly [5–7]. In [4], Bokut, et al. unified the Gröbner–Shirshov basis theory for Lie superalgebras and their universal