= Central character of an associative-algebra module
{title2=$\theta:Z(A)\to\mathbb C$}
= Central character
{synonym}
A module over a complex associative algebra $A$ has central character $\theta$ if every element $z$ of the centre acts as the scalar $\theta(z)$, where $\theta$ is a unital algebra homomorphism. A <Verma module> has a central character: the centre preserves its one-dimensional highest-weight space, and centrality extends that scalar action from the cyclic generator to every vector. A nonzero homomorphism between two modules with scalar central actions forces their central characters to coincide. A generalized central character allows each $z-\theta(z)$ to act locally nilpotently instead of as zero.
Back to article page
