Volume 45, Issue 10, October 2004
 SPECIAL ISSUE: QUANTUM GROUPS AND NONCOMMUTATIVE GEOMETRY
 I


Quantum groups and deformation quantization: Explicit approaches and implicit aspects
View Description Hide DescriptionDeformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

Mirror symmetry and noncommutative geometry of categories
View Description Hide DescriptionHomological mirror symmetry aims to explain the phenomenon of mirror symmetry in the language of categories and their deformation theory. In these notes I discuss various aspects of this approach from the point of view of noncommutative algebraic geometry in the tensor category of graded vector spaces.
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 II


Integrable renormalization I: The ladder case
View Description Hide DescriptionIn recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e., for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical matrix, coming from a Rota–Baxter structure underlying the target space of the regularized Hopf algebra characters. Working in the rooted tree Hopf algebra, the simple case of the Hopf subalgebra of ladder trees is treated in detail. The extension to the general case, i.e., the full Hopf algebra of rooted trees or Feynman graphs, is briefly outlined.

On the Fock space for nonrelativistic anyon fields and braided tensor products
View Description Hide DescriptionWe realize the physical anyon Hilbert spaces, introduced previously via unitary representations of the group of diffeomorphisms of the plane, as fold braidedsymmetric tensor products of the 1particle Hilbert space. This perspective provides a convenient Fock space construction for nonrelativistic anyonquantum fields along the more usual lines of boson and fermion fields, but in a braided category, and clarifies how discrete (lattice) anyon fields relate to anyon fields in the continuum. We also see how essential physical information is encoded. In particular, we show how the algebraic structure of the anyonic Fock space leads to a natural anyonic exclusion principle related to intermediate occupation number statistics, and obtain the partition function for an idealized gas of fixed anyonic vortices.

conformal invariant equations and plane wave solutions
View Description Hide DescriptionWe give new solutions of the quantum conformal deformations of the full Maxwell equations in terms of deformations of the plane wave. We study the compatibility of these solutions with the conservation of the current. We also start the study of quantum linear conformal (Weyl) gravity by writing the corresponding deformed equations.

Degenerate principal series of quantum Harish–Chandra modules
View Description Hide DescriptionIn this paper we study a quantum analog of a degenerate principal series of modules related to the Shilov boundary of the quantum matrix unit ball. We give necessary and sufficient conditions for the modules to be simple and unitarizable and investigate their equivalence. These results are analogs of known classical results on reducibility and unitarizability of modules obtained by Johnson, Sahi, Zhang, Howe, and Tan.

A combinatorial approach to the settheoretic solutions of the Yang–Baxter equation
View Description Hide DescriptionA bijective map : , where is a finite set, is called a settheoretic solution of the Yang–Baxter equation (YBE) if the braid relation holds in . A nondegenerate involutive solution satisfying , for all , is called squarefree solution. There exist close relations between the squarefree settheoretic solutions of YBE, the semigroups of Itype, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of squarefree solutions and the associated Yang–Baxter algebraic structures—the semigroup , the group and the algebra over a field , generated by and with quadratic defining relations naturally arising and uniquely determined by . We study the properties of the associated Yang–Baxter structures, and prove a conjecture of the present author that the three notions: a squarefree solution of (settheoretic) YBE, a semigroup of I type, and a semigroup of skewpolynomialtype, are equivalent. This implies that the Yang–Baxter algebra is a Poincaré–Birkhoff–Witttype algebra, with respect to some appropriate ordering of . We conjecture that every squarefree solution of YBE is retractable, in the sense of Etingof–Schedler–Solovyev.
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 III


Recoupling Lie algebra and universal algebra
View Description Hide DescriptionWe formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an algebra defined in this paper. algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping algebra of recoupling Lie algebras and prove a generalized Poincaré–Birkhoff–Witt theorem. As an example we consider the algebras over an arbitrary recoupling of graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure.

Division algebras: Family replication
View Description Hide DescriptionThe known link of the division algebras to 10dimensional spacetime and one leptoquark family is extended to encompass three leptoquark families.

Braided cyclic cocycles and nonassociative geometry
View Description Hide DescriptionWe use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeldtype cochain twist. These are the socalled quasialgebras and include the octonions as braidedcommutative but nonassociative coordinate rings, as well as quasialgebra versions of the standard deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalized automorphism , and obtain the required generalization of cyclic cohomology. We show that this braided cyclic cohomology is invariant under a cochain twist. We also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra. The paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry, as well as the algebraic noncommutative torus as an associative example.

QuasiHopf algebras and representations of octonions and other quasialgebras
View Description Hide DescriptionModules over a quasialgebra (here, by quasialgebra we mean a left module algebra, where is a quasiHopf algebra), as defined by Albuquerque and Majid, coincide with modules over a certain associative algebra, a quasiHopf smash product. As a consequence of this, we get that the category of modules over the octonions is isomorphic to the category of modules over the algebra of real matrices. We provide a new approach to the endomorphism quasialgebra associated to a left module, which in the finite dimensional case yields the same results as the one of Albuquerque and Majid. We discuss possible definitions as endomorphism quasialgebras for Heisenberg doubles of a finite dimensional quasiHopf algebra.

Frobenius monads and pseudomonoids
View Description Hide DescriptionSix equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What it means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to “strongly separable” Frobenius algebras and “weak monoidal Morita equivalence.” Wreath products of Frobenius algebras are discussed.

Higher gauge theory—differential versus integral formulation
View Description Hide DescriptionThe term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1 and 2forms. So far, there have been two approaches to this subject. The differential picture uses nonAbelian 1 and 2forms in order to generalize the connection 1form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of nonAbelian groups and generalizes the formulation of gauge theory in terms of parallel transports. We recall how to circumvent the classic nogo theorems in order to define nonAbelian surface ordered products in the integral picture. We then derive the differential picture from the integral formulation under the assumption that the curve and surface labels depend smoothly on the position of the curves and surfaces. We show that some aspects of the nogo theorems are still present in the differential (but not in the integral) picture. This implies a substantial structural difference between nonperturbative and perturbative approaches to higher gauge theory. We finally demonstrate that higher gauge theory provides a geometrical explanation for the extended topological symmetry of theory in both pictures.
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 IV


Homological mirror symmetry, deformation quantization and noncommutative geometry
View Description Hide DescriptionWe discuss the possible relationship of homological mirror symmetry with deformation quantization. We speculate that after certain nonlinear “twist” the Fukaya category becomes equivalent to the category of holonomic modules over a quantized algebra of functions.

Deformation quantization in singular spaces
View Description Hide DescriptionWe present a method of quantizing analytic spaces immersed in an arbitrary smooth ambient manifold. Remarkably our approach can be applied to singular spaces. We begin by quantizing the cotangent bundle of the manifold. Using a supermanifold framework we modify the Fedosov construction in a way such that the ⋆product of the functions lifted from the base manifold turns out to be the usual commutative product of smooth functions on . This condition allows us to lift the ideals associated to the analytic spaces on the base manifold to form left (or right) ideals on in a way independent of the choice of generators and leading to a finite set of PDEs defining the functions in the quantum algebra associated with . Some examples are included.

Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel
View Description Hide DescriptionThe equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a wellknown result. A remarkable generalization of this equivalence, given by Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. In this paper the existence of suitably equivariant transition functions is proved for such groupoids, generalizing consequently the classification of principal bundles by means of their transition functions, to extensions of principal bundles by an equivariant form of Čech cohomology.
