186,324 research outputs found
Metric compatibility and Levi-Civita connections on quantum groups
No full textArbitrary connections on a generic Hopf algebra H are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an H-valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on SLq(2)
Quantum Principal Bundles on Projective Bases
Get PDFThe purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study noncommutative principal bundles corresponding to G→ G/ P, where G is a semisimple group and P a parabolic subgroup
Topological t-duality for twisted tori
Get PDFWe apply the C∗-algebraic formalism of topological T-duality due to Mathai and Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative C∗-algebra with an action of Rn. We treat the general class of almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier– Douady classes. We prove that any such solvmanifold has a topological T-dual given by a C∗-algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these C∗-algebras rigorously describe the T-folds from non-geometric string theory.</p
Twisting quantum groups at the roots of unity
Get PDFLet G Be a simply connected compact Lie group and g be its complexified Lie algebra. Building on the work of H. Wenzl, we present a weak tensor structure on the unitary modular categories arising from quantum groups U_q(g) specialised at the root of 1 q, following the work about weak quasi-tensor categories by S. Carpi, S. Ciamprone, C. Pinzari and myself. The theory therein developed allows us to reconstruct these categories as representation categories of discrete unitary coboundary weak Hopf algebras. Furthermore, we consider the twisted categories obtained by modifying the associator using 3-cocycles on the dual of the centre of G and reconstruct them as representation categories of suitable discrete unitary weak Hopf algebras; this is done by adaptation of a result by S. Neshveyev and M. Yamashita in the analogous scenario of the compact quantum group corresponding to U_q(g) for q>1, i.e. for generic values of q
Twisting all the way: from Classical Mechanics to Quantum Fields
No full textWe discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. In particular commutation relations among creation and annihilation operators are deduced
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