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Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Symplectic geometry
 Bulletin of the American Mathematical Society
, 1981
"... This paper is a survey of Poisson geometry, with an emphasis on global questions and the theory of Poisson Lie groups and groupoids. 1 ..."
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This paper is a survey of Poisson geometry, with an emphasis on global questions and the theory of Poisson Lie groups and groupoids. 1
A ringtheorist’s description of Fedosov quantization
"... ABSTRACT We present a formal, algebraic treatment of Fedosov’s argument that the coordinate algebra of a symplectic manifold has a deformation quantization. His remarkable formulas are established in the context of affine symplectic algebras. ..."
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ABSTRACT We present a formal, algebraic treatment of Fedosov’s argument that the coordinate algebra of a symplectic manifold has a deformation quantization. His remarkable formulas are established in the context of affine symplectic algebras.
A DixmierMoeglin equivalence for Poisson algebras and with torus actions
 in Algebra and Its Applications
, 2005
"... Abstract. A Poisson analog of the DixmierMoeglin equivalence is established for any affine Poisson algebra R on which an algebraic torus H acts rationally, by Poisson automorphisms, such that R has only finitely many prime Poisson Hstable ideals. In this setting, an additional characterization of ..."
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Abstract. A Poisson analog of the DixmierMoeglin equivalence is established for any affine Poisson algebra R on which an algebraic torus H acts rationally, by Poisson automorphisms, such that R has only finitely many prime Poisson Hstable ideals. In this setting, an additional characterization of the Poisson primitive ideals of R is obtained – they are precisely the prime Poisson ideals maximal in their Hstrata (where two prime Poisson ideals are in the same Hstratum if the intersections of their Horbits coincide). Further, the Zariski topology on the space of Poisson primitive ideals of R agrees with the quotient topology induced by the natural surjection from the maximal ideal space of R onto the Poisson primitive ideal space. These theorems apply to many Poisson algebras arising from quantum groups. The full structure of a Poisson algebra is not necessary for the results of this paper, which are developed in the setting of a commutative algebra equipped with a set of derivations.
A note on noncommutative Poisson structures
"... Abstract. We introduce a new type of noncommutative Poisson structure on associative algebras. It induces Poisson structures on the moduli spaces classifying semisimple modules. Path algebras of doubled quivers and preprojective algebras have noncommutative Poisson structures given by the necklace L ..."
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Abstract. We introduce a new type of noncommutative Poisson structure on associative algebras. It induces Poisson structures on the moduli spaces classifying semisimple modules. Path algebras of doubled quivers and preprojective algebras have noncommutative Poisson structures given by the necklace Lie algebra. Recall that a Poisson bracket on a commutative ring A is a Lie bracket which satisfies the Leibnitz rule {−, −} : A × A → A {a, bc} = b{a, c} + {a, b}c. The same definition can be used for a Poisson bracket on a noncommutative ring, but it seems that this is too restrictive, as by a theorem of Farkas and Letzter [5] the only Poisson brackets on a genuinely noncommutative prime ring are the commutator bracket [a, b] = ab − ba and multiples of it (in a suitable sense). A notion of a noncommutative Poisson structure has been suggested by Xu [9] and Block and Getzler [1]. It has the property that if A has a noncommutative Poisson structure, then the centre Z(A) has a Poisson bracket, but otherwise the relationship with Poisson brackets is unclear. In this paper we introduce a new type of noncommutative Poisson structure. It is the weakest structure we can find which (when A is a finitely generated Kalgebra and K is an algebraically closed field of characteristic zero) induces Poisson brackets on the coordinate rings of the moduli spaces Mod(A, n) / GLn(K) classifying ndimensional semisimple Amodules. With this notion, path algebras of doubled quivers, preprojective algebras and multiplicative preprojective algebras all have noncommutative Poisson structures. For a much deeper approach, see the work of Van den Bergh [8]. I would like to thank M. Van den Bergh, who raised this problem, and G. Van de Weyer and Pu Zhang for some useful discussions. 1. Definition and Examples Throughout, we work over a commutative base ring K and, where appropriate, maps are assumed to be Klinear. Let A be an associative Kalgebra (with 1). Recall that the zeroth Hochschild homology of A is A/[A, A], where [A, A] is the subset of A spanned by the commutators. We write a for the image of a ∈ A in A/[A, A]. Observe that if d: A → A is a derivation, then since
POISSON STRUCTURES ON MODULI SPACES OF REPRESENTATIONS
"... Abstract. We show that a Poisson structure can be induced on the affine moduli space of (semisimple) representations of an associative algebra from a suitable Lie algebra structure on the zeroth Hochschild homology of the algebra. In particular this applies to necklace Lie algebra for path algebras ..."
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Abstract. We show that a Poisson structure can be induced on the affine moduli space of (semisimple) representations of an associative algebra from a suitable Lie algebra structure on the zeroth Hochschild homology of the algebra. In particular this applies to necklace Lie algebra for path algebras of doubled quivers and preprojective algebras. We call such structures H0Poisson structures, and show that they are wellbehaved for Azumaya algebras and under Morita equivalence. 1.