We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070.
Abstract Let R be an integral domain, let ħ ∈ R\{0} be such that 𝕂 := R/ħR is a field, and let ℋ𝒜 be the category of torsionless (or flat) Hopf algebras over R. We call H ∈ ℋ𝒜 a ‘quantized function algebra’ (= QFA), resp. ‘quantized restricted universal enveloping algebra’ (= QrUEA), at ħ if—roughly speaking—H/ħH is the function algebra of a connected Poisson group, resp. the (restricted, if R/ħR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an ‘inner’ Galois' correspondence on ℋ𝒜, via two endofunctors, ( )∨ and ( )′, of ℋ𝒜 such that H ∨ is a QrUEA and H′ is a QFA (for all H ∈ ℋ𝒜). In addition: (a) the image of ( )∨, resp. of ( )′, is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(𝕂) = 0, the restrictions ( )∨|QFAs and ( )′|QrUEAs yield equivalences inverse to each other; (c) if p = 0, starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra 𝔤, the functor ( )∨, resp. ( )′, gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [F. Gavarini, The global quantum duality principle: theory, examples, and applications, preprint 2003, http://arxiv.org/abs/math.QA/0303019] and [F. Gavarini, The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries, J. Algebra 285 (2005), 399–437] and [F. Gavarini, Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms, Commun. Math. Phys. 253 (2005), 121–155].
Abstract Multipotential stem cells can be detected in bone marrow cells preparations by injecting these cells into syngeneic irradiated hosts. Colonies (CFU-s) are visible macroscopically on the spleen after 8 days. Pretreatment of A-Tlab (Qa-1.2) bone marrow with A-anti-A-Tlab serum (anti-Qa-1.2), in the presence of complement, reduced CFU-s by 80% whereas A-Tlab anti-A serum (anti-Qa-1.1) treatment had no effect. Pretreatment of A(Qa-1.1) bone marrow with A-Tlab anti-A serum reduced macroscopic splenic colonies by 78%. Similar results were obtained by using C57BL/6 (Qa-1.2) and B6-Tlaa (Qa-1.1) bone marrow. Thus, both Qa-1.1 and Qa-1.2 are expressed on CFU-s. Qa-2 expression on CFU-2 was examined in the same manner. B6.KI anti-B6 (anti-Qa-2,3) serum reduced CFU-s numbers by 87% on B6 (Qa-2,3+) while having little or no effect on B6.KI (Qa-2,3-). Treatment with D3.262, a monoclonal anti-Qa-2 antibody, reduced B6 CFU-s by 80%, and had no effect on B6.KI CFU-s. Detection of Qa-1 and Qa-2 on CFU-s underscores the wide distribution of the Qa antigens in the hematopoietic system.
We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions on periodic diagrams, and in particular, compute some generating functions concerning plane partitions, where Kostka polynomials and their level restricted generalization appear. On representation side, we study representations of Cherednik algebras which admit weight decomposition with respect to a certain commutative subalgebra. All the irreducible representations of this class are constructed combinatorially through standard tableaux on periodic diagrams, and this realization as "tableaux representations" provides a new combinatorial approach to the investigation of these representations. As consequences, we describe the decomposition of a tableaux representation as a representation of the degenerate affine Hecke algebra, which is a subalgebra of the Cherednik algebra, and also describe the spectral decomposition of the spherical subspace (the invariant subspace under the action of the Weyl group) of a tableaux representation with respect to the center of the degenerate affine Hecke algebra, In particular, the computation of the character of the spherical subspace is reduced to the computation of the generating function for the set of column strict plane partitions, and we obtain an expression of the characters in terms of Kostka polynomials as announced in math.QA/0508274.
In this work, we begin to uncover the architecture of the general family of zeta functions and multiple zeta values as they appear in the theory of integrable systems and conformal field theory. One of the key steps in this process is to recognize the roles that zeta functions play in various arenas using transform methods. Other logical connections are provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of conformal field theory and braid groups. These recurring themes are subtly linked in a vast scheme of a logically woven tapestry. An immediate application of this framework is to provide an answer to a question of Kontsevich regarding the appearance of Drinfeld type integrals and in particular, multiple zeta values in: a) Drinfeld's work on the KZ equation and the associator; b) Etingof-Kazhdan's quantization of Poisson-Lie algebras; c) Tamarkin's proof of formality theorems; d) Kontsevich's quantization of Poisson manifolds. Combinatorial arguments relating Feynman diagrams to Selberg integrals, multiple zeta values, and finally Poisson manifolds provide an additional step in this framework. Along the way, we provide additional insight into the various papers and theorems mentioned above. This paper represents an overall introduction to work currently in progress. More details to follow. See our paper Math.QA/[ ] for a proof of the Connes Kreimer Conjecture.