Twisted Cocycle Research Articles

Cocycle twists of algebras

ABSTRACTLet be a connected graded k-algebra over an algebraically closed field k (thus A0 = k). Assume that a finite Abelian group G, of order coprime to the characteristic of k, acts on A by graded automorphisms. In conjunction with a suitable cocycle, this action can be used to twist the multiplication in A. We study this new structure and, in particular, we describe when properties like Artin–Schelter regularity are preserved by such a twist. We then apply these results to examples of Rogalski and Zhang.

Cocycle twists of 4-dimensional Sklyanin algebras

We study cocycle twists of a 4-dimensional Sklyanin algebra A and a factor ring B which is a twisted homogeneous coordinate ring. Twisting such algebras by the Klein four-group G, we show that the twists AG,μ and BG,μ have very different geometric properties to their untwisted counterparts. For example, AG,μ has only 20 point modules and infinitely many fat point modules of multiplicity 2. The ring BG,μ falls under the purview of Artin and Stafford's classification of noncommutative curves, and we describe it using a sheaf of orders over an elliptic curve.

The shadow nature of positive and twisted quandle invariants of knots

Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.

The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podleś 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 Podleś sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.

ON (α, β, γ)-DERIVATIONS OF LIE SUPERALGEBRAS

This paper is primarily concerned with (α, β, γ)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers. Some properties of (α, β, γ)-derivations of the Lie superalgebras are obtained. In particular, two examples for (α, β, γ)-derivations of low-dimensional non-simple Lie superalgebras are presented and the super-spaces of (α, β, γ)-derivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie superalgebras. A special case for the generalization of 1-cocycles with respect to the adjoint representation is exactly (α, β, γ)-derivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.

Quantum affine Schubert cells and FRT-bialgebras: The [formula omitted] case

De Concini, Kac, and Procesi defined a family of subalgebras Uq+[w]⊆Uq(g) associated with elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E6 and E6(1) map isomorphically onto distinguished subalgebras of the Faddeev–Reshetikhin–Takhtajan universal bialgebra associated with the braiding on the quantum half-spin representation of Uq(so10). We identify the quotients as those obtained by factoring out the quantum Schubert cell algebras by ideals generated by certain submodules with respect to the adjoint action of Uq(so10).

Twisted Differential String and Fivebrane Structures

In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the H_3-field, as well as its magnetic dual version for the H_7-field define cocycles in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and Fivebrane(n)-structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U(n) or O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L-infinity-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

A Remark on Twists and the Notion of Torsion-free Discrete Quantum Groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki-Takai duality. The twisted Takesaki-Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld-Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.

Nonassociative Riemannian geometry by twisting

Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the S7 coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.

Twisting the quantum grassmannian

In contrast to the classical and semiclassical settings, the Coxeter element (12... n) which cycles the columns of an m x n matrix does not determine an automorphism of the quantum grassmannian. Here, we show that this cycling can be obtained by means of a cocycle twist. A consequence is that the torus invariant prime ideals of the quantum grassmannian are permuted by the action of the Coxeter element (12 ... n). We view this as a quantum analogue of the recent result of Knutson, Lam and Speyer, where the Lusztig strata of the classical grassmannian are permuted by (12... n).

On cocycle twisting of compact quantum groups

We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C ∗ -algebra of the quantum group changes.

Yetter–Drinfeld modules under cocycle twists

We give an explicit formula for the correspondence between simple Yetter–Drinfeld modules for certain finite-dimensional pointed Hopf algebras H and those for cocycle twists H σ of H. This implies an equivalence between modules for their Drinfeld doubles. To illustrate our results, we consider the restricted two-parameter quantum groups u r , s ( sl n ) under conditions on the parameters guaranteeing that u r , s ( sl n ) is a Drinfeld double of its Borel subalgebra. We determine explicit correspondences between u r , s ( sl n ) -modules for different values of r and s and provide examples where no such correspondence can exist. Our examples were obtained via the computer algebra system Singular::Plural.

Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra On

AbstractThis paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K1-group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.

Matched pairs of generalized Lie algebras and cocycle twists

We introduce the conception of matched pairs of $(H, \beta)$-Lie algebras, and construct an $(H, \beta)$-Lie algebra through them. We prove that the cocycle twist of a~matched pair of $(H, \beta)$-Lie algebras can also be matched.

Generalized Lie Algebras and Cocycle Twists

We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.

Quantisation of Twistor Theory by Cocycle Twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CMh pulls back to the basic instanton on S^4\subset CMh and that this observation quantises to obtain the Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the tautological bundle on our \theta-deformed CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed CP^3 that maps over under the transform to the \theta-deformed instanton.

Representations and Cocycle Twists of Color Lie Algebras

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra \({\text{sl}}^{c}_{2} \).

A Local Index Formula for the Quantum Sphere

For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SU q (2)-equivariant entire cyclic cocycle corresponding to when evaluated on the element The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincides with the volume form and computes the quantum as well as the classical Fredholm indices.

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.

Examples of Twisted Cyclic Cocycles from Covariant Differential Calculi

For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators $$\left[ {\mathcal{F},\rho \left( x \right)} \right]$$ for some self-adjoint operator $$\mathcal{F}$$ and some *-representation ρ of the underlying *-algebra.