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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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TL;DR: In this paper, it was shown that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to non-commutative geometries.
Abstract: Our geometric concepts evolved first through the discovery of Non-Euclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. The basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation θ → 1/θ for the noncommutative torus T θ 2 which are unseen in perturbative expansions in θ such as star or Moyal products. We discuss the foundational problem of “what is a manifold in NCG” and explain the fundamental role of Poincare duality in K-homology which is the basic reason for the spectral point of view. This leads us, when specializing to 4-geometries to a universal algebra called the “Instanton algebra”. We describe our joint work with G. Landi which gives noncommutative spheres S θ 4 from representations of the Instanton algebra. We show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to noncommutative geometries. We give a survey of several recent developments. First our joint work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant (rather than the usual covariant one) geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Second, our joint work with D. Kreimer on renormalization and the Riemann-Hilbert problem. Finally we describe the spectral realization of zeros of zeta and L-functions from the noncommutative space of Adele classes on a global field and its relation with the Arthur-Selberg trace formula in the Langlands program. We end with a tantalizing connection between the renormalization group and the missing Galois theory at Archimedean places.
142 citations
TL;DR: In this article, the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group associated with a Weyl group element w has the structure of a quantum cluster algebra.
Abstract: We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac–Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory \({\fancyscript{C}_{w}}\) of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities that can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.
142 citations
TL;DR: In this article, the coherent states for a quantum particle on a circle are introduced and the Bargmann representation within the actual treatment provides the representation of the algebra, where U is unitary, which is a direct consequence of the Heisenberg algebra.
Abstract: The coherent states for a quantum particle on a circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra , where U is unitary, which is a direct consequence of the Heisenberg algebra , but it is more adequate for the study of circular motion.
140 citations
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TL;DR: This paper reports on the work on the “fuzzification” of the four-dimensional CP 2 and its QFT’s, which is not spin, but spinc, and has many unique features.
139 citations
TL;DR: In this article, it was shown that there is no 4D bicovariant differential calculus which are Lorentz covariant, however, there exists a five-dimensional differential calculus, which satisfies both requirements.
Abstract: Following the construction of the $\kappa$-Minkowski space from the bicrossproduct structure of the $\kappa$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $\kappa$-Minkowski space and and we briefly discuss the main properties of its differential calculi.
139 citations