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A generalization of the Lorenz equations is proposed where the variables takevalues in a Lie algebra. The finite dimensionality of the representationencodes the quantum fluctuations, while the non-linear nature of the equationscan describe chaotic fluctuations. We identify a criterion, for the appearanceof such non-linear terms. This depends on whether an invariant, symmetrictensor of the algebra can vanish or not. This proposal is studied in detail forthe fundamental representation of $\mathfrak{u}(2)$. We find a knottedstructure for the attractor, a bimodal distribution for the largest Lyapunovexponent and that the dynamics takes place within the Cartan subalgebra, thatdoes not contain only the identity matrix, thereby can describe the quantumfluctuations.

Knotted Strange Attractors and Matrix Lorenz Systems