Bases for local Weyl modules for the hyper and truncated current [formula omitted]-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 sl2. The main result is a characteristic-free construction of bases for this important family of modules for the hyper current sl2-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 sl2-algebra. Our construction allows us to obtain new bases for the local Weyl modules for truncated current sl2-algebras with very particular properties.