In this chapter we study the Brownian energy process, a diffusion process modelling heat conduction, where energy is diffusively exchanged among sites. We will introduce it as a many-particle limit of the symmetric inclusion process, in the spirit of population models where diffusion processes arise when limits of large population size are taken, and the variables become the ratios of the numbers individuals of different types with respect to the total population size. Naturally, the generator of the Brownian energy process, in its abstract form, reads exactly as the generator of the symmetric inclusion process, in terms of the generator of the \(\mathfrak {su}(1,1)\) Lie algebra. This algebraic structure is also at the origin of the duality relation between the two processes. We also introduce the Brownian momentum process, a diffusion process modelling the evolution of interacting momenta on a lattice, also sharing the same algebraic structure.