Indecomposable representation

An indecomposable representation is nonzero and cannot be expressed as a direct sum of two nonzero invariant subspaces. It is an indecomposable module over the associated group algebra or representing algebra.

A nonsplit extension contains an invariant subrepresentation with no invariant complementary subspace.

An extension's vertexwise split arrow matrices are $\left(\begin{array}{cc}{f}_X& \xi \\ 0& f_Z\end{array}\right)$. Scaling the subobject by $t$ multiplies $\xi$ by $t$. Nonzero $t$ preserves the middle isomorphism class; $t=0$ gives $X\oplus Z$. For a nonsplit extension, the split orbit is distinct and lies in the smaller-dimensional boundary of the middle orbit closure.

A unipotent operator has every eigenvalue equal to one and may create a nonsplit Jordan-block extension.