2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67833Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Let $A,A^*$ denote a Leonard pair on $V$. There exists a decomposition of $V$ into a direct sum of 1-dimensional subspaces, with respect to which $A$ is lower bidiagonal and $A^*$ is upper bidiagonal. This is known as the {\it split decomposition}. We use the split decomposition to obtain several characterizations of Leonard pairs.18 pagesRings and AlgebrasQuantum AlgebraRepresentation Theory05E35, 05E30, 33C45, 33D45text