Integral Bases for the Universal Enveloping Algebras of Map Algebras

TL;DR

This paper constructs an explicit integral basis for the universal enveloping algebra of map algebras derived from simple Lie algebras, providing new tools for algebraic and representation-theoretic analysis.

Contribution

It introduces an explicit integral form and basis for the universal enveloping algebra of map algebras, including detailed commutation formulas for sl_2.

Findings

Explicit integral basis for the universal enveloping algebra of map algebras.

Commutation formulas enabling Poincare-Birkhoff-Witt order expressions.

Framework applicable to complex simple Lie algebras.

Abstract

Given a finite-dimensional, complex simple Lie algebra we exhibit an integral form for the universal enveloping algebra of its map algebra, and an explicit integral basis for this integral form. We also produce explicit commutation formulas in the universal enveloping algebras of the map algebras of sl_2 that allow us to write certain elements in Poincare-Birkhoff-Witt order.