  We present a theorem which elucidates the connection between self-duality of
Markov processes and representation theory of Lie algebras. In particular, we
identify sufficient conditions such that the intertwining function between two
representations of a certain Lie algebra is the self-duality function of a
(Markov) operator. In concrete terms, the two representations are associated to
two operators in interwining relation. The self-dual operator, which arise from
an appropriate symmetric linear combination of them, is the generator of a
Markov process. The theorem is applied to a series of examples, including
Markov processes with a discrete state space (e.g. interacting particle
systems) and Markov processes with continuous state space (e.g. diffusion
processes). In the examples we use explicit representations of Lie algebras
that are unitary equivalent. As a consequence, in the discrete setting
self-duality functions are given by orthogonal polynomials whereas in the
continuous context they are Bessel functions.
