A Class of Polynomial Modules over Map Lie Algebras

Abstract

For any finitely generated unital commutative associative algebra $\mathcal {R}$ over $\mathbb {C}$ and any complex finite-dimensional simple Lie algebra $\mathfrak {g}$ with a fixed Cartan subalgebra $\mathfrak {h}$, we classify all $\mathfrak {g}\otimes \mathcal {R}$-modules on $U(\mathfrak {h})$ such that $\mathfrak {h}$ as a subalgebra of $\mathfrak {g}\otimes \mathcal {R}$, acts on $U(\mathfrak {h})$ by the multiplication. We construct these modules explicitly and study their module structures.

Keywords

Map Lie algebras / $U(\mathfrak {h})$-free modules')">$U(\mathfrak {h})$-free modules / Simple Lie algebras / 17B10 / 17B20 / 17B35 / 17B65

Funding

National Natural Science Foundation of China(11931009)

Anhui Initiative in Quantum Information Technologies(AHY150000)

National Natural Science Foundation of China(11771410)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature