Quantum Groups and Twisted Spectral Triples

On twisted reality conditions

We study the twisted reality condition of Math. Phys. Anal. Geom. 19 (2016),no. 3, Art. 16, for spectral triples, in particular with respect to the product and the commutant. Motivated by this we present the procedure, which allows one to "untwist" the twisted spectral triples studied in Lett. Math. Phys. 106 (2016), 1499-1530. We also relate this construction to conformally rescaled real twisted spectral triples, and discuss the untwisting of the `minimal twist' procedure of an even spectral triple.

2018-04-19math-phmath.MPmath.QA

Gauge transformations for twisted spectral triples

We extend to twisted spectral triples the fluctuations of the metric, as well as their gauge transformations. The former are bounded perturbations of the Dirac operator that arise when a spectral triple is exported between Morita equivalent algebras; the later are obtained by the action of the unitary endomorphisms of the module implementing the Morita equivalence. It is shown that the twisted gauged Dirac operators, previously introduced to generate an extra scalar field in the spectral description of the standard model of elementary particles, in fact follow from Morita equivalence between t…

2018-05-22hep-thmath-phmath.MP

Coideal algebras from twisted Manin triples

We propose a new approach to study coideal algebras. It is well-known that Manin triples (or equivalently Lie bi-algebra structures) are the requirement to deform Lie algebras and to obtain quantum groups. In this paper, introducing some particular automorphisms of Manin triples, we define new structures that we call Lie bi-ideal structures. A link with coisotropic subalgebras is explained. We show that their deformation provide coideal algebras. As examples, we recover from our general construction the twisted Yangians, the q-Onsager algebra and the augmented q-Onsager algebra. As an importan…

2012-06-26math-phmath.MPmath.QA

Twisted entire cyclic cohomology, J-L-O cocycles and equivariant spectral triples

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [KMT]. With very similar definitions and techniques as those used in [jlo], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator-theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry ([Con]) and an additional positive operator having specified properties. Furthermore, it is shown that given a spectral triple (in the …

2002-04-05math-phmath.KTmath.MP

Gauge transformations of spectral triples with twisted real structures

Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative 1-forms. We study the coupling of spectral triples with twisted real structures to gauge fields, adopting Morita equivalence via modules and bimodules as a guiding principle and paying special attention to modifications to the inner fluctuations of the Dirac operator. In particular, we analyse the twisted first-order condition as a possible alternative to abandoning the first-order condition in order to go beyon…

2021-07-29hep-thmath-phmath.MP

Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples

In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present paper generalized this result to the case of dynamical r-matrices, and showed that they correspond to generalized Belavin-Drinfeld triples. The dynamical r-matrix corresponding a triple is given by a certain explicit formula, which works for any symmetrizable Kac-Moody algebra. In special cases, this formula gives the Felder and the Belavin r-matrix. In 1994…

1999-08-20math.QA

Twisted reality condition for Dirac operators

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones we propose a new twisted reality condition for the Dirac operator.

2016-06-22hep-thmath-phmath.MP

Twisted Dirac Operators over Quantum Spheres

We construct new families of spectral triples over quantum spheres, with a particular attention focused on the standard Podles quantum sphere and twisted Dirac operators.

2007-04-20math-phmath.MPmath.QA

Twisted reality and the second-order condition

An interesting feature of the finite-dimensional real spectral triple (A,H,D,J) of the Standard Model is that it satisfies a ``second-order'' condition: conjugation by J maps the Clifford algebra Cl_D(A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence Cl_D(A)-bimodule). This resembles a property of the canonical spectral triple of a closed oriented Riemannian manifold: there is a dense subspace of H which is a self-Morita equivalence Cl_D(A)-bimodule. In this paper we argue that on manifolds, in order for the self-Morita equivalenc…

2020-11-23math-phmath.MPmath.QA

Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere

A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.

2007-07-23math.OAmath.QA

On inductive limit spectral triples

Given an inductive system of spectral triples $\{(A_j,\H_j,D_j)\}_j$, we find conditions under which the triple $(\limind A_j,\limind H_j,\limind D_j)$ is a spectral triple. We also analyze and describe some classical examples of spectral triples in terms of these conditions.

2017-12-27math.OAmath.QA

The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere

We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class w…

2011-11-24math.KTmath.OAmath.QA

Spectral triples on the Jiang-Su algebra

We construct spectral triples on a class of particular inductive limits of matrix-valued function algebras. In the special case of the Jiang-Su algebra we employ a particular $AF$-embedding.

2018-02-01math.OAmath.QA

Discrete spectral triples and their symmetries

We classify 0-dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure. We investigate also whether such spectral triples admit a symmetry arising from the Hopf algebra structure of the finite algebra. We discuss examples of commutative algebras and groups algebras.

1997-02-20math.QAq-alg

Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)

In [CPR2], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2), and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [CPR2], the computatio…

2008-02-04math.OAmath.QA

The Parthasarathy formula and a spectral triple for the quantum Lagrangian Grassmannian of rank two

We show that the Dolbeault--Dirac operator on the quantum Lagrangian Grassmannian of rank two, an example of a quantum irreducible flag manifold, satisfies an appropriate version of the Parthasarathy formula. We use this result to complete the proof that the candidate spectral triple for this space, as defined by Krähmer, is a spectral triple.

2018-12-23math.KTmath.QA

On twisting real spectral triples by algebra automorphisms

We systematically investigate ways to twist a real spectral triple via an algebra automorphism and in particular, we naturally define a twisted partner for any real graded spectral triple. Among other things we investigate consequences of the twisting on the fluctuations of the metric and possible applications to the spectral approach to the standard model of particle physics.

2016-09-19hep-thmath-phmath.MP

A modular spectral triple for $κ$-Minkowski space

We present a spectral triple for $κ$-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the $κ$-Poincaré algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavor: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, the second is a weight replacing the usual operator trace, used to measure the growth o…

2013-06-13hep-thmath-phmath.MP

Morita "equivalences" of equivariant torus spectral triples

In general, Morita equivalence of spectral triples need not be a symmetric relation. In this paper, we show that Morita equivalence of spectral triples is an equivalence relation for equivariant torus spectral triples.

2012-02-02math-phmath.MPmath.OA

Real structures on almost-commutative spectral triples

We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying, in the process, both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almost-commutative spectral triple algebraically encodes the commutative *-algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KO-dimension. Moreover, we define a notion of smooth family of real…

2013-02-10hep-thmath-phmath.MP

Twists and spectral triples for isospectral deformations

We construct explicitly the symmetries of the isospectral deformations as twists of Lie algebras and demonstrate that they are isometries of the deformed spectral triples.

2001-03-02math-phmath.MPmath.QA

Real Spectral Triples over Noncommutative Bieberbach Manifolds

We classify and construct all real spectral triples over noncommutative Bieberbach manifolds, which are restrictions of irreducible real equivariant spectral triple over the noncommutative three-torus. We show that in the classical case the constructed geometries correspond exactly to spin structures over Bieberbach manifolds and the Dirac operators constructed for a flat metric.

2013-01-10math-phmath.MPmath.QA

Equivariant spectral triples on the quantum SU(2) group

We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L_2-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SU_q(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p<4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dim…

2002-10-03math.KTmath.OAmath.QA

The geometry of quantum lens spaces: real spectral triples and bundle structure

We study almost real spectral triples on quantum lens spaces, as orbit spaces of free actions of cyclic groups on the spectral geometry on the quantum group $SU_q(2)$. These spectral triples are given by weakening some of the conditions of a real spectral triple. We classify the irreducible almost real spectral triples on quantum lens spaces and we study unitary equivalences of such quantum lens spaces. Applying a useful characterization of principal $U(1)$-fibrations in noncommutative geometry, we show that all such quantum lens spaces are principal $U(1)$-fibrations over quantum teardrops.

2015-02-27math-phmath.MPmath.OA

A reconstruction theorem for almost-commutative spectral triples

We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of Connes's reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in the reconstruction theorem for commutative spectral triples, and following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectra…

2012-06-28hep-thmath-phmath.MP

Spectral triples with multitwisted real structure

We generalize the notion of spectral triple with reality structure to spectral triples with multitwisted real structure, the class of which is closed under the tensor product composition. In particular, we introduce a multitwisted order one condition (characterizing the Dirac operators as an analogue of first-order differential operator). This provides a unified description of the known examples, which include rescaled triples with the conformal factor from the commutant of the algebra and (on the algebraic level) triples on quantum disc and on quantum cone, that satisfy twisted first order co…

2020-11-12hep-thmath-phmath.MP

Extensions and degenerations of spectral triples

For a unital C*-algebra A, which is equipped with a spectral triple and an extension T of A by the compacts, we construct a family of spectral triples associated to T and depending on the two positive parameters (s,t). Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact quantum metric spaces it is possible to define a metric on this family of spectral triples, and we show that the distance between a pair of spectral triples varies continuously with respect to the parameters. It turns out that a spectral triple associated to the unitarization of the algebra of compact …

2008-07-19math-phmath.MPmath.OA

Finite Pseudo-Riemannian Spectral Triples and The Standard Model

Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard Model spectral triple in this formalism and interpret the physical symmetry preserving the lepton number as a shadow of a finite pseudo-Riemannian structure.

2018-06-19hep-thmath-phmath.MP