In this chapter we illustrate how the Lie algebraic approach reproduces the basic self-duality relation for independent random walkers. After providing the description of the process generator in terms of the generators of the Heisenberg algebra, we show how the triangular self-duality function (related to multi-variate factorial moments) arises in two ways. The first method is by means of a symmetry acting on the diagonal self-duality function associated to reversibility. The symmetry is the total annihilation operator and is easily deduced from the algebraic description. The second method starts instead from basic dualities relating the generators of the Heisenberg algebra and then promotes them to self-duality of the Markov process by composing dualities. We close the chapter by showing how self-duality is used in the ergodic theory of the process on the infinite lattice \(\mathbb Z^d\) . In particular the infinite-volume process has products of Poisson distributions as reversible and ergodic measures. Under an appropriate moment growth condition, we show that these product Poisson distributions are the only ergodic distributions.