A Lie-Jordan geometric formulation of Lindblad dynamics

We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors \(Cμν{}^λ\) and \(Bμν{}^λ\). Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is an explicit Lie--Jordan tensor decomposition of the standard elementary GKLS block, \( D(X,Y)Z=XZY-12\{YX,Z\}\). Its operator space components are determined entirely by the Lie--Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. The combinations \(B+C\) and \(B-C\) encode twice the left and right regular actions, whose mutual compatibility follows from associativity. Their affine extensions define two linear connections, and the elementary dissipative map is expressed as a left-right bimodule composition corrected by an ordered Jordan contribution. The curvature tensors of these connections reconstruct the universal elementary GKLS map through an explicit four-term identity. The map obeys inherent identities related to trace annihilation and conjugation. In the case of an autonomous physical contraction with a Hermitian Hamiltonian and a positive-semidefinite Kossakowski matrix, the resulting evolution is CPTP (Completely Positive, Trace-Preserving semigroup), and its generator has real entries when expressed in a Hermitian Hilbert--Schmidt basis. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. A correlated multichannel case likewise produces invariant population and coherence sectors, along with a stationary operator-space dimension determined by the rank of the Kossakowski matrix. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.