= Representation variety of an associative algebra
{title2=$\operatorname{Rep}_A(\mathbf n)$}
For a finitely generated <associative algebra>, choose matrices for its generators and impose its polynomial relations. This defines an <affine variety>. With prescribed <orthogonal idempotents> $1=\sum_ie_i$, require their images to be the standard projections on $\bigoplus_i k^{n_i}$. The resulting variety is $\operatorname{Rep}_A(\mathbf n)$, acted on by $\prod_i\operatorname{GL}_{n_i}(k)$. Without a chosen idempotent decomposition, the usual variety is $\operatorname{Rep}_A(r)=\operatorname{Hom}_{k\text{-alg}}(A,M_r(k))$. Its orbits are module isomorphism classes, and their stabilizers are automorphism groups.
Back to article page
