Quantum group structure in the unitary minimal model

Vincent Pasquier

Alan L. Carey

From Pauli matrices to r-matrices, A.J. Bracken statistical mechanics - from Ising models to unitary minimal models, P.A. Pearce symmetry breaking and nucleon structure, A.W. Thomas electron-atom collisions, I.E. McCarthy exact s-functions and quantum-scattering potentials, K. Amos dimensional odd-ities, R. Delbourgo mathematical remarks on the cohomology of gauge groups and anomalies, A.L. Carey and M.K. Murray the Ising model and its descendants, R.J. Baxter integrable SU(2)-invariant spin chains and the haldane conjecture, M.T. Batchelor and C.M. Yung generalized characteristic identities and applications, P.D. Jarvis and D.S. McAnally quantum group and group link polynomials, M.D. Gould discrete space-time, P. Szekeres.

Lingaraj Sahu, Preetinder Singh

We consider the GNS Hilbert space H of a uniformly hyper-finite C * -algebra and study a class of unbounded Lindbladian arises from commutators.Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution.The vacuum expectation of homomorphic cocycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian.We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.

Zhe Chang

B. Basu Mallick, A. Kundu

Ge Li

We prove that the saturation condition (g) = ^2 for the quantum geometric tensor---which characterizes ideal flat bands and lowest Landau levels---is equivalent to a Cauchy--Riemann condition dJ_M = J d, where J is the complex structure governing unitary quantum dynamics. This gives a dynamical interpretation of quantum band geometry: a state family saturates the metric--curvature inequality if and only if it inherits the phase-ordering structure of the ambient Hilbert space. When saturation holds, the quantum metric g_{} and Berry curvature F_{} carry identical geometric information, related by F_{} = 2g_{}J^{}{}_{}. The proof proceeds through a structural analysis of quantum dynamics under minimal assumptions. Starting from linearity, strong continuity, group composition, and norm preservation, Stone's theorem forces time evolution to be a one-parameter unitary group. We identify the imaginary unit i as the minimal complex structure J satisfying J^2 = -I---the unique algebraic object confining evolution to constant-norm orbits. This leads to the Phase-Ordering Principle: under unitary evolution, time is exclusively a phase-ordering parameter with no amplitude modulation. The same J descends from the Hilbert space to the projective state space ^{N-1} as the K\"{a}hler structure, unifying the quantum metric and Berry curvature into the quantum geometric tensor Q_{} = g_{} + i_{}. Extending to position-dependent phase redefinitions, local ordering consistency uniquely introduces a U(1)$ connection field, providing a geometric interface to gauge structure. The boundary of the Phase-Ordering Principle is made precise: it holds if and only if evolution is unitary; Lindblad dynamics breaks it by introducing amplitude modulation alongside phase ordering.