41 research outputs found
Lectures on Duflo isomorphisms in Lie algebra and complex geometry
International audienceDuflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds.All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details.The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory
Universal KZB equations I: the elliptic case
We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB)
connection in genus 1. This is a flat connection over a principal bundle on the
moduli space of elliptic curves with marked points. It restricts to a flat
connection on configuration spaces of points on elliptic curves, which can be
used for proving the formality of the pure braid groups on genus 1 surfaces. We
study the monodromy of this connection and show that it gives rise to a
relation between the KZ associator and a generating series for iterated
integrals of Eisenstein forms. We show that the universal KZB connection
realizes as the usual KZB connection for simple Lie algebras, and that in the
sl_n case this realization factors through the Cherednik algebras. This leads
us to define a functor from the category of equivariant D-modules on sl_n to
that of modules over the Cherednik algebra, and to compute the character of
irreducible equivariant D-modules over sl_n which are supported on the
nilpotent cone.Comment: Correction of reference of Thm. 9.12 stating an equivalence of
categories between modules over the rational Cherednik algebra and its
spherical subalgebr
On the universal ellipsitomic KZB connection
We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of -structured elliptic curves with marked points, where , and are two integers. It restricts to a flat connection on -twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical -matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras
Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants
In this paper we prove a conjecture of B. Shoikhet. This conjecture states that the tangent isomorphism on homology, between the Poisson homology associated to a Poisson structure on Rd and the Hochschild homology of its quantized star-product algebra, is an isomorphism of modules over the (isomorphic) respective cohomology algebras. As a consequence, we obtain a version of the Duflo isomorphism on coinvariants
Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
In this paper we prove, with details and in full generality, that the
isomorphism induced on tangent homology by the Shoikhet-Tsygan formality
-quasi-isomorphism for Hochschild chains is compatible with
cap-products. This is a homological analog of the compatibility with
cup-products of the isomorphism induced on tangent cohomology by Kontsevich
formality -quasi-isomorphism for Hochschild cochains.
As in the cohomological situation our proof relies on a homotopy argument
involving a variant of {\bf Kontsevich eye}. In particular we clarify the
r\^ole played by the {\bf I-cube} introduced in \cite{CR1}.
Since we treat here the case of a most possibly general Maurer-Cartan
element, not forced to be a bidifferential operator, then we take this
opportunity to recall the natural algebraic structures on the pair of
Hochschild cochain and chain complexes of an -algebra. In particular
we prove that they naturally inherit the structure of an -algebra
with an -(bi)module.Comment: The first and second Section on -algebras and modules have
been completely re-written, with new results; partial revision of Section 3;
the proofs in Section 4 and 5 have been re-formulated in a more general
context; we added Section 8 on globalisatio
From Atiyah Classes to Homotopy Leibniz Algebras
A celebrated theorem of Kapranov states that the Atiyah class of the tangent
bundle of a complex manifold makes into a Lie algebra object in
, the bounded below derived category of coherent sheaves on .
Furthermore Kapranov proved that, for a K\"ahler manifold , the Dolbeault
resolution of is an
algebra. In this paper, we prove that Kapranov's theorem holds in much wider
generality for vector bundles over Lie pairs. Given a Lie pair , i.e. a
Lie algebroid together with a Lie subalgebroid , we define the Atiyah
class of an -module (relative to ) as the obstruction to
the existence of an -compatible -connection on . We prove that the
Atiyah classes and respectively make and
into a Lie algebra and a Lie algebra module in the bounded below
derived category , where is the abelian
category of left -modules and is the universal
enveloping algebra of . Moreover, we produce a homotopy Leibniz algebra and
a homotopy Leibniz module stemming from the Atiyah classes of and ,
and inducing the aforesaid Lie structures in .Comment: 36 page
Formality theorems for Hochschild chains in the Lie algebroid setting
In this paper we prove Lie algebroid versions of Tsygan's formality
conjecture for Hochschild chains both in the smooth and holomorphic settings.
In the holomorphic setting our result implies a version of Tsygan's formality
conjecture for Hochschild chains of the structure sheaf of any complex manifold
and in the smooth setting this result allows us to describe quantum traces for
an arbitrary Poisson Lie algebroid. The proofs are based on the use of
Kontsevich's quasi-isomorphism for Hochschild cochains of R[[y_1,...,y_d]],
Shoikhet's quasi-isomorphism for Hochschild chains of R[[y_1,...,y_d]], and
Fedosov's resolutions of the natural analogues of Hochschild (co)chain
complexes associated with a Lie algebroid.Comment: 40 pages, no figure
M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
We show that the zeroth cohomology of M. Kontsevich's graph complex is
isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is
explicitly described. This result has applications to deformation quantization
and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber
operad. They are parameterized by grt_1, up to one class (or two, depending on
the definitions). More generally, the homotopy derivations of the (non-unital)
E_n operads may be expressed through the cohomology of a suitable graph
complex. Our methods also give a second proof of a result of H. Furusho,
stating that the pentagon equation for grt_1-elements implies the hexagon
equation
Derived coisotropic structures I: affine case
We define and study coisotropic structures on morphisms of commutative dg
algebras in the context of shifted Poisson geometry, i.e. -algebras.
Roughly speaking, a coisotropic morphism is given by a -algebra acting
on a -algebra. One of our main results is an identification of the space
of such coisotropic structures with the space of Maurer--Cartan elements in a
certain dg Lie algebra of relative polyvector fields. To achieve this goal, we
construct a cofibrant replacement of the operad controlling coisotropic
morphisms by analogy with the Swiss-cheese operad which can be of independent
interest. Finally, we show that morphisms of shifted Poisson algebras are
identified with coisotropic structures on their graph.Comment: 49 pages. v2: many proofs rewritten and the paper is split into two
part
Comments about quantum symmetries of SU(3) graphs
For the SU(3) system of graphs generalizing the ADE Dynkin digrams in the
classification of modular invariant partition functions in CFT, we present a
general collection of algebraic objects and relations that describe fusion
properties and quantum symmetries associated with the corresponding Ocneanu
quantum groupo\"{i}ds. We also summarize the properties of the individual
members of this system.Comment: 36 page