Bases for local Weyl modules for the hyper and truncated current ${\mathfrak{sl}}_2$-algebras

Abstract

We use the theory of Gröbner-Shirshov bases for ideals to construct linear bases for graded local Weyl modules for the (hyper) current and the truncated current algebras associated to the finite-dimensional complex simple Lie algebra ${\mathfrak{sl}}_2$. The main result is a characteristic-free construction of bases for this important family of modules for the hyper current ${\mathfrak{sl}}_2$-algebra. In the positive characteristic setting this work represents the first construction in the literature. In the characteristic zero setting, the method provides a different construction of the Chari-Pressley (also Kus-Littelmann) bases and the Chari-Venkatesh bases for local Weyl modules for the current ${\mathfrak{sl}}_2$-algebra. Our construction allows us to obtain new bases for the local Weyl modules for truncated current ${\mathfrak{sl}}_2$-algebras with very particular properties.

22 pages. Comments are welcome!