Abstract
We introduce the boundary-driven multispecies harmonic process, a colored-particle generalization of the model proposed by Frassek, Giardinà, and Kurchan in 2019 [1] and generalized in [2] . After defining the Markov generator, we prove that the process is integrable in the Bethe-ansatz sense.
To this end, we solve the Yang-Baxter and boundary Yang-Baxter equations, deriving the corre- sponding R- and K-matrices in terms a Verma module of the Lie algebra \(gl(M + 1) (M\) is the number of different species). The associated transfer matrix is then constructed, generating an infnite family of commuting conserved quantities. We show that the logarithmic derivative of the transfer matrix yields the Hamiltonian, which is in one-to-one correspondence with the Markov generator.
Finally, exploiting the underlying \(gl(M + 1)\) symmetry, we defne an absorbing dual process that allows for the characterization of the non-equilibrium steady state.
This talk is based on the work in progress (soon on the ArXiv) with Rouven Frassek and Cristian Giardinà :

ˆ Francesco Casini, Rouven Frassek and Cristian Giardinà: « Integrability for the multi-species harmonic process », 2025+.

References
[1] R. Frassek, C. Giardinà, and J. Kurchan, Non-compact quantum spin chains as integrable stochastic particle processes, J. Stat. Phys. 180, 135-171 (2020).
[2] R. Frassek and C. Giardinà, Exact solution of an integrable non-equilibrium particle system, J. Math. Phys. 63, 103301 (2022).