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Register a new userA proof of Comes-Kujawas conjecture
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Let $kappa$ be a commutative ring containing $2^{-1}$. In this paper, we prove the Comes-Kujawas conjecture on a $kappa$-basis of cyclotomic oriented Brauer-Clifford supercategory. As a by-product, we prove that the cyclotomic walled Brauer-Clifford superalgebra defined by Comes and Kujawa and ours are isomorphic if $kappa$ is an algebraically closed field with characteristic not two.
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We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1}) are given by noncommutative Laurent polynomials.
In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivariant multivector fields extends to an $L_infty$-quasi-isomorphism. We construct this $L_infty$-quasi-isomorphism using the $G$-invariant formality constructed by Dolgushev. This result has immediate consequences in deformation quantization, since it allows to obtain a quantum moment map from a classical momentum map with respect to a $G$-invariant Poisson structure.
Proofs of Tsygans formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygans formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevichs formality quasi-isomorphism for Hochschild cochains of R[[y_1, >..., y_d]] and Shoikhets formality quasi-isomorphism for Hochschild chains of R[[y_1,..., y_d]] I prove Tsygans formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, abla), where M is the manifold and abla is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.
Boris Shoikhet noticed that the proof of lemma 1 in section 2.3 of math.QA/0504420 contains an error. In this note I give a correct proof of this lemma which was suggested to me by Dmitry Tamarkin. The correction does not change the results of math.QA/0504420.
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. In this paper, we prove this conjecture for large $n$.
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