![]() We propose a Lie-algebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). ![]() Part II considers the additional structures of differential forms and finitely generated quantum Lie algebras - it is devoted to the construction of the Cartan calculus, based on an undeformed Cartan identity. We study this in detail for quasitriangular Hopf algebras, giving the determinant and orthogonality relation for the ``reflection`` matrix. This construction allows the translation of undeformed matrix expressions into their less obvious quantum algebraic counter parts. The pure braid group is introduced as the commutant of U and thereby bicovariant vector fields, casimirs and metrics. Using a generalized semi-direct product construction we combine the dual Hopf algebras A of functions and U of left-invariant vector fields into one fully bicovariant algebra of differential operators. The material is organized in two parts: Part I studies vector fields on quantum groups, emphasizing Hopf algebraic structures, but also introducing a ``quantum geometric`` construction. The construction of a tangent bundle on a more » quantum group manifold and an BRST type approach to quantum group gauge theory are given as further examples of applications. A Cartan calculus that combines functions, forms, Lie derivatives and inner derivations along general vector fields into one big algebra is constructed for quantum groups and then extended to quantum planes. A generalization of unitary time evolution is proposed and studied for a simple 2-level system, leading to non-conservation of microscopic entropy, a phenomenon new to quantum mechanics. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods.įurther information: Winding number Vector field corresponding to (the Hodge dual of) dθ.Ī simple example of a form that is closed but not exact is the 1-form d θ, i.e., that there are no magnetic monopoles.The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related sciences. On a contractible domain, every closed form is exact by the Poincaré lemma. The question of whether every closed form is exact depends on the topology of the domain of interest. Since the exterior derivative of a closed form is zero, β is not unique, but can be modified by the addition of any closed form of degree one less than that of α.īecause d 2 = 0, every exact form is necessarily closed. ![]() The form β is called a "potential form" or "primitive" for α. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.įor an exact form α, α = dβ for some differential form β of degree one less than that of α. ![]() In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero ( dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. ![]()
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