Abstract
The first Weyl algebra \(A_1\) arises naturally in quantum mechanics as the algebra generated by position and momentum operators, satisfying the canonical commutation relation \([∂,x]=1\). In characteristic zero, this algebra admits no nontrivial finite-dimensional representations. In contrast, over a field of characteristic \(p>0\), \(A_1\) has a much richer structure: it admits a large center and possesses many finite-dimensional irreducible representations. In this talk, we describe how these representations can be classified by central characters, and how their dimensions are governed by algebraic properties of the center.