In this paper, two different definitions of the BRST complex are connected. We obtain the BRST co... more In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action of H ~< DiffM on a configuration space M, where H is a compact Lie group representing the gauge symmetry in this model.
In this article, X will denote a C∞ manifold. In a very famous article, Kontsevich ([K]) showed t... more In this article, X will denote a C∞ manifold. In a very famous article, Kontsevich ([K]) showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque ([C1]) extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid over X, he defined the DGLA of E-polydifferential operators, ED∗ poly, and showed that it is formal. Denote by T ∗ poly the DGLA of E-polyvector fields. Considering M , a module over E, we define T ∗ poly(M) the T ∗ poly-module of Epolyvector fields with values in M . Similarly, we define the ED∗ poly-module of E-polydifferential operators with values in M , ED∗ poly(M). We show that there is a quasi-isomorphism of L∞-modules over T ∗ poly from T ∗ poly(M) to ED∗ poly(M). We recover Calaque ’s result as a particular case. Generalizing results of Shoikhet, Dolgushev and Calaque-Dolgushev-Halbout ([S], [D2], [C-D-H]), we prove a similar statement for Hochschild chains with values in M .
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an app... more We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of Van Den Bergh ([VDB]). We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds in [CLYZ].
We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids,... more We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G. Bohm who studied integral theory for Hopf algebroids. We make use of recent results about left Hopf left bialgebroids. We apply our results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.
The notion of Lie algebroids or Lie-Rinehart algebras constantly occurs in my research. It alread... more The notion of Lie algebroids or Lie-Rinehart algebras constantly occurs in my research. It already occured in my Ph. D thesis (written under the direction of M. Duflo [1], [2], [3]) in the particular case of coinduced representations. During my stays at Harvard and at the university of Utrecht, I carried on my research in the direction of my Ph. D thesis ([4]). During these postdoctoral years, discussions with J. Kalkman, Y. Karshon, H. Duistermaat and S. Sternberg allowed me to learn the basis of symplectic geometry . During my stay in Utrecht, I had the idea of looking at induction of Lie algebras as a particular case of direct image. I was convinced that the duality property involving the direct image of D-modules could be extended to the Lie algebroid setting without being able to achieve it. It was only when hired in Paris 6, with the help of P. Schapira, that I overcame the technical difficulties ([5], [7]). Then, benefiting from a visit of M. Kashiwara at Paris 6, I treated t...
In this article, X will denote a C ∞ manifold. In a very famous article, Kontsevich ([K]) showed ... more In this article, X will denote a C ∞ manifold. In a very famous article, Kontsevich ([K]) showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque ([C1]) extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid over X, he defined the DGLA of E-polydifferential operators, E D * poly , and showed that it is formal. Denote by E T * poly the DGLA of E-polyvector fields. Considering M , a module over E, we define E T * poly (M) the E T * poly-module of Epolyvector fields with values in M. Similarly, we define the E D * poly-module of E-polydifferential operators with values in M , E D * poly (M). We show that there is a quasi-isomorphism of L ∞-modules over E T * poly from E T * poly (M) to E D * poly (M). We recover Calaque 's result as a particular case. Generalizing results of Shoikhet, Dolgushev and Calaque-Dolgushev-Halbout ([S], [D2], [C-D-H]), we prove a similar statement for Hochschild chains with values in M. If X is a Poisson manifold, we know from Kontsevitch's work ([K]) that there is a star product on O = Γ(O X). Let M be a Γ(D X)-module, using the star product, we can endow M [[h]] with a O[[h]]⊗ O[[h]] op-module structure. If π ∈ Γ(2 T X) is the bivector defining the Poisson structure on O, hπ defines a Poisson structure on the algebra O[[h]]. As a corollary of our theorem, we get an isomorphism from the Poisson cohomology of the Poisson algebra O[[h]] with coefficients in M [[h]] and the differential Hochschild cohomology of O[[h]] with coefficients in M [[h]]. In [Ts], B.Tsygan conjectured a formality theorem for Hochschild chains. This conjecture was proved by Shoikhet in the case where X = IR n and generalized to any manifold by Dolgushev ([D2]). It was extended to the Lie algebroid setting by Calaque, Dolgushev and Halbout ([C-D-H]). We give a version of the formality theorem for Hochschild chains with coefficients in a D(E)-module. Our proofs are analogous to that of [D1],[C1], [D2], [C-D-H]. We use Kontsevitch's and Shoikhet's formality theorem for IR d f ormal and a Fedosov like globalization techniques.
Some duality properties for induced representations of enveloping algebras involve the character ... more Some duality properties for induced representations of enveloping algebras involve the character T rad g. We extend them to deformation Hopf algebras A h of a noetherian Hopf k-algebra A 0 satistying Ext i A0 (k, A 0) = {0} except for i = d where it is isomorphic to k. These duality properties involve the character of A h defined by right multiplication on the one dimensional free k[[h]]-module Ext d A h (k[[h]], A h). In the case of quantized enveloping algebras, this character lifts the character T rad g. We also prove Poincaré duality for such deformation Hopf algebras in the case where A 0 is of finite homological dimension. We explain the relation of our construction with quantum duality.
Interpreting Lie algebroid theory in terms of D-modules, we define a duality functor for a Lie al... more Interpreting Lie algebroid theory in terms of D-modules, we define a duality functor for a Lie algebroid as well as a direct image functor for a morphism of Lie algebroids. Generalizing the work of Schneiders (see also the work of Schapira-Schneiders) and making assumptions analog to his, we show that the duality functor and the direct image functor commute. As an application, we extend to Lie algebroids some duality properties already known for Lie algebras.
Letk be a field of characteristic 0 and letA be a supercommutative associativek-superalgebra. Let... more Letk be a field of characteristic 0 and letA be a supercommutative associativek-superalgebra. LetL be ak−A-Lie-Rinehart superalgebra. From these data, one can construct a superalgebra of differential operatorsV(A,L) (generalizing the enveloping superalgebra of a Lie superalgebra). We will give a difinition of Lie-Rinehart superalgebra morphisms allowing to generalize the notions of inverse image and direct image. We will prove that
In the first part of this article, we compute the rigid dualizing complex of a quantum enveloping... more In the first part of this article, we compute the rigid dualizing complex of a quantum enveloping algebra. We consider the generic case and the case of a specialization at a non-root of unity. This answers a question of Yekutieli [J. Pure Appl. Algebra 150 (2000) 85]. In [Bull. Soc. Math. France 122 (1994) 371] and [Math. Z. 232 (1999) 367], we generalized D-module theory to Lie algebroids. Using these results, we compute explicitly the rigid dualizing complex of the algebra of differential operators defined by an affine Lie algebroid. This generalizes results of Yekutieli [J. Pure Appl. Algebra 150 (2000) 85].
Extending material from the theory of D-modules to Lie algebroids (see also [S. Chemla, A duality... more Extending material from the theory of D-modules to Lie algebroids (see also [S. Chemla, A duality property for complex Lie algebroids, Math. Z. (1999) 367-388]), we introduce an inverse image functor and show that it preserves coherence under appropriate circumstances. As in the case of D-modules, a suitable non characteristicity notion enables us to give a sufficient condition in order for the duality functor and the inverse image functor to commute. This generalizes a result for D-modules due to Kashiwara, Kawai, and Sato but, even in the case of D-modules, our proof is different from theirs. In particular, we obtain a new duality formula for complexes of modules over an ordinary Lie algebra and, as a special case, we get a new adjunction formula for modules over Lie algebras. Moreover, our result will shed some light on the behaviour of Poisson cohomology (a notion introduced by Lichnerowitz) under a Poisson map (in the analytic case).
Communications in Mathematical Physics, Jun 1, 1994
In this paper, two different definitions of the BRST complex are connected. We obtain the BRST co... more In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action of H x DiffM on a configuration space M, where H is a compact Lie group representing the gauge symmetry in this model.
We explore special features of the pair pU *, U * q formed by the right and left dual over a (lef... more We explore special features of the pair pU *, U * q formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S* from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phùng's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.
In this paper, two different definitions of the BRST complex are connected. We obtain the BRST co... more In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action of H ~< DiffM on a configuration space M, where H is a compact Lie group representing the gauge symmetry in this model.
In this article, X will denote a C∞ manifold. In a very famous article, Kontsevich ([K]) showed t... more In this article, X will denote a C∞ manifold. In a very famous article, Kontsevich ([K]) showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque ([C1]) extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid over X, he defined the DGLA of E-polydifferential operators, ED∗ poly, and showed that it is formal. Denote by T ∗ poly the DGLA of E-polyvector fields. Considering M , a module over E, we define T ∗ poly(M) the T ∗ poly-module of Epolyvector fields with values in M . Similarly, we define the ED∗ poly-module of E-polydifferential operators with values in M , ED∗ poly(M). We show that there is a quasi-isomorphism of L∞-modules over T ∗ poly from T ∗ poly(M) to ED∗ poly(M). We recover Calaque ’s result as a particular case. Generalizing results of Shoikhet, Dolgushev and Calaque-Dolgushev-Halbout ([S], [D2], [C-D-H]), we prove a similar statement for Hochschild chains with values in M .
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an app... more We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of Van Den Bergh ([VDB]). We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds in [CLYZ].
We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids,... more We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G. Bohm who studied integral theory for Hopf algebroids. We make use of recent results about left Hopf left bialgebroids. We apply our results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.
The notion of Lie algebroids or Lie-Rinehart algebras constantly occurs in my research. It alread... more The notion of Lie algebroids or Lie-Rinehart algebras constantly occurs in my research. It already occured in my Ph. D thesis (written under the direction of M. Duflo [1], [2], [3]) in the particular case of coinduced representations. During my stays at Harvard and at the university of Utrecht, I carried on my research in the direction of my Ph. D thesis ([4]). During these postdoctoral years, discussions with J. Kalkman, Y. Karshon, H. Duistermaat and S. Sternberg allowed me to learn the basis of symplectic geometry . During my stay in Utrecht, I had the idea of looking at induction of Lie algebras as a particular case of direct image. I was convinced that the duality property involving the direct image of D-modules could be extended to the Lie algebroid setting without being able to achieve it. It was only when hired in Paris 6, with the help of P. Schapira, that I overcame the technical difficulties ([5], [7]). Then, benefiting from a visit of M. Kashiwara at Paris 6, I treated t...
In this article, X will denote a C ∞ manifold. In a very famous article, Kontsevich ([K]) showed ... more In this article, X will denote a C ∞ manifold. In a very famous article, Kontsevich ([K]) showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque ([C1]) extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid over X, he defined the DGLA of E-polydifferential operators, E D * poly , and showed that it is formal. Denote by E T * poly the DGLA of E-polyvector fields. Considering M , a module over E, we define E T * poly (M) the E T * poly-module of Epolyvector fields with values in M. Similarly, we define the E D * poly-module of E-polydifferential operators with values in M , E D * poly (M). We show that there is a quasi-isomorphism of L ∞-modules over E T * poly from E T * poly (M) to E D * poly (M). We recover Calaque 's result as a particular case. Generalizing results of Shoikhet, Dolgushev and Calaque-Dolgushev-Halbout ([S], [D2], [C-D-H]), we prove a similar statement for Hochschild chains with values in M. If X is a Poisson manifold, we know from Kontsevitch's work ([K]) that there is a star product on O = Γ(O X). Let M be a Γ(D X)-module, using the star product, we can endow M [[h]] with a O[[h]]⊗ O[[h]] op-module structure. If π ∈ Γ(2 T X) is the bivector defining the Poisson structure on O, hπ defines a Poisson structure on the algebra O[[h]]. As a corollary of our theorem, we get an isomorphism from the Poisson cohomology of the Poisson algebra O[[h]] with coefficients in M [[h]] and the differential Hochschild cohomology of O[[h]] with coefficients in M [[h]]. In [Ts], B.Tsygan conjectured a formality theorem for Hochschild chains. This conjecture was proved by Shoikhet in the case where X = IR n and generalized to any manifold by Dolgushev ([D2]). It was extended to the Lie algebroid setting by Calaque, Dolgushev and Halbout ([C-D-H]). We give a version of the formality theorem for Hochschild chains with coefficients in a D(E)-module. Our proofs are analogous to that of [D1],[C1], [D2], [C-D-H]. We use Kontsevitch's and Shoikhet's formality theorem for IR d f ormal and a Fedosov like globalization techniques.
Some duality properties for induced representations of enveloping algebras involve the character ... more Some duality properties for induced representations of enveloping algebras involve the character T rad g. We extend them to deformation Hopf algebras A h of a noetherian Hopf k-algebra A 0 satistying Ext i A0 (k, A 0) = {0} except for i = d where it is isomorphic to k. These duality properties involve the character of A h defined by right multiplication on the one dimensional free k[[h]]-module Ext d A h (k[[h]], A h). In the case of quantized enveloping algebras, this character lifts the character T rad g. We also prove Poincaré duality for such deformation Hopf algebras in the case where A 0 is of finite homological dimension. We explain the relation of our construction with quantum duality.
Interpreting Lie algebroid theory in terms of D-modules, we define a duality functor for a Lie al... more Interpreting Lie algebroid theory in terms of D-modules, we define a duality functor for a Lie algebroid as well as a direct image functor for a morphism of Lie algebroids. Generalizing the work of Schneiders (see also the work of Schapira-Schneiders) and making assumptions analog to his, we show that the duality functor and the direct image functor commute. As an application, we extend to Lie algebroids some duality properties already known for Lie algebras.
Letk be a field of characteristic 0 and letA be a supercommutative associativek-superalgebra. Let... more Letk be a field of characteristic 0 and letA be a supercommutative associativek-superalgebra. LetL be ak−A-Lie-Rinehart superalgebra. From these data, one can construct a superalgebra of differential operatorsV(A,L) (generalizing the enveloping superalgebra of a Lie superalgebra). We will give a difinition of Lie-Rinehart superalgebra morphisms allowing to generalize the notions of inverse image and direct image. We will prove that
In the first part of this article, we compute the rigid dualizing complex of a quantum enveloping... more In the first part of this article, we compute the rigid dualizing complex of a quantum enveloping algebra. We consider the generic case and the case of a specialization at a non-root of unity. This answers a question of Yekutieli [J. Pure Appl. Algebra 150 (2000) 85]. In [Bull. Soc. Math. France 122 (1994) 371] and [Math. Z. 232 (1999) 367], we generalized D-module theory to Lie algebroids. Using these results, we compute explicitly the rigid dualizing complex of the algebra of differential operators defined by an affine Lie algebroid. This generalizes results of Yekutieli [J. Pure Appl. Algebra 150 (2000) 85].
Extending material from the theory of D-modules to Lie algebroids (see also [S. Chemla, A duality... more Extending material from the theory of D-modules to Lie algebroids (see also [S. Chemla, A duality property for complex Lie algebroids, Math. Z. (1999) 367-388]), we introduce an inverse image functor and show that it preserves coherence under appropriate circumstances. As in the case of D-modules, a suitable non characteristicity notion enables us to give a sufficient condition in order for the duality functor and the inverse image functor to commute. This generalizes a result for D-modules due to Kashiwara, Kawai, and Sato but, even in the case of D-modules, our proof is different from theirs. In particular, we obtain a new duality formula for complexes of modules over an ordinary Lie algebra and, as a special case, we get a new adjunction formula for modules over Lie algebras. Moreover, our result will shed some light on the behaviour of Poisson cohomology (a notion introduced by Lichnerowitz) under a Poisson map (in the analytic case).
Communications in Mathematical Physics, Jun 1, 1994
In this paper, two different definitions of the BRST complex are connected. We obtain the BRST co... more In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action of H x DiffM on a configuration space M, where H is a compact Lie group representing the gauge symmetry in this model.
We explore special features of the pair pU *, U * q formed by the right and left dual over a (lef... more We explore special features of the pair pU *, U * q formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S* from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phùng's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.
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Papers by Sophie Chemla