Bernoulli-type Relations in Some Noncommutative Polynomial Ring

TL;DR

This paper discovers Bernoulli-type relations within a specific noncommutative polynomial ring, linking algebraic structures to Bernoulli numbers and providing a new representation framework.

Contribution

It introduces Bernoulli-type relations in a noncommutative polynomial ring isomorphic to a universal enveloping algebra, revealing new algebraic structures and representations.

Findings

Bernoulli-type relations are established in the ring.

A representation involving Bernoulli numbers is constructed.

The relations connect to the structure constants of the algebra.

Abstract

We find particular relations which we call "Bernoulli-type" in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional non-abelian Lie algebra. From these Bernoulli-type relations in our ring, we can obtain a representation on a certain left ideal with the Bernoulli numbers as structure constants.