From Koszul duality to Poincaré duality

We discuss the notion of Poincaré duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincaré duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.     

Quantum groups on fibre bundles

It is shown that the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. Within the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the endq-deformed instanton models are introduced for every integral index.     

Differential calculus and gauge transformations on a deformed space

We consider a formalism by which gauge theories can be constructed on noncommutative space time structures. The coordinates are supposed to form an algebra, restricted by certain requirements that allow us to realise the algebra in terms of star products. In this formulation it is useful to define derivatives and to extend the algebra of coordinates by these derivatives. The elements of this extended algebra are deformed differential operators. We then show that there is a morphism between these deformed differential operators and the usual higher order differential operators acting on functions of commuting coordinates. In this way we obtain deformed gauge transformations and a deformed version of the algebra of diffeomorphisms. The deformation of these algebras can be clearly seen in the category of Hopf algebras. The comultiplication will be twisted. These twisted algebras can be realised on noncommutative spaces and allow the construction of deformed gauge theories and deformed gravity theory. Dedicated to the 60th birthday of Prof. Obregon.     

Spectral flow invariants and twisted cyclic theory for the Haar state on

In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], 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 [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index.     

Formality and Deformations of Universal Enveloping Algebras

We describe enveloping algebras of finite-dimensional Lie algebras which are formal in the sense that their Hochschild complex as a differential graded Lie algebra is quasi-isomorphic to its Hochschild cohomology. For Abelian Lie algebras this is true thanks to the Kontsevich formality theorem. We are using his formality map twisted by the group-like element generated by the linear Poisson structure to simplify the problem, and then study examples. For instance, the universal enveloping algebras of the Lie algebras are formal. We also recover our rigidity results for enveloping algebras from this new angle and present some explicit deformations of linear Poisson structure in low dimensions.     

Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone. Received: 28 October 1998 / Accepted: 7 March 1999     

Second Quantized Frobenius Algebras

We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n^th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n^th tensor powers and the natural permutation action. We also show that naturally graded symmetric group twisted Frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super–grading. Furthermore we discuss several notions of discrete torsion and show that indeed a non–trivial discrete torsion leads to a non–trivial super structure on the second quantization.This work was partially supported by NSF grant #0070681.     

Graded quasi-Lie algebras

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

Construction of Perfect Crystals Conjecturally Corresponding to Kirillov-Reshetikhin Modules over Twisted Quantum Affine Algebras

Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Twisted WZW branes from twisted REA's

Quantum geometry of twisted Wess–Zumino–Witten branes is formulated in the framework of twisted reflection equation algebras. It is demonstrated how the representation theory of these algebras leads to the correct classification of branes. A semiclassical formula for quantised brane positions is derived and shown to be consistent with earlier string-theoretic analyses.     

On the variational noncommutative poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.     

An operator formulation of orbifold conformal field theory

In many applications of conformal field theory one encounters twisted conformal fields, i.e. fields which have branch cut singularities on the relevant Riemann surfaces. We present a geometrical framework describing twisted conformal fields on Riemann surfaces of arbitrary genus which is alternative to the standard method of coverings. We further illustrate the theory of twisted Grassmannians and its relation with the representation theory of the twisted oscillator algebras. As an application of the above, we expound an operator formalism for orbifold strings.     

Twisted group algebras II

Wendel showed that norm non-increasing isomorphisms between the group algebras of locally compact groups could be expressed in terms of group characters and topological isomorphisms. His results are extended to twisted group algebras. In particular, by applying a generalisation ofWendel's main result to twisted group algebras over the same group, it is shown that the number of such algebras is equal to the number of orbits in a 2-cohomology group overG under the action of the automorphism group ofG. An application to the twisted group algebra defined byWeyl's form of the canonical commutation relations is considered.     

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

Reflection equation- and FRT-type algebras

We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan algebras associated with a quasitriangular Hopf algebraH. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted tensor square ofH. We present some applications to the equivariant quantization. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Twisted Logarithmic Modules of Vertex Algebras

Motivated by logarithmic conformal field theory and Gromov–Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.     

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.     

Orbifold compactification of heterotic string theories

The internal degrees of freedom of twisted heterotic strings are discussed using the theory of Kac-Moody algebras.     

Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.