Results 1  10
of
164
Formal (non)commutative symplectic geometry
 THE GELFAND MATHEMATICAL SEMINARS, 1990–1992”, BIRKHÄUSER
, 1993
"... ..."
CalabiYau algebras
"... Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative ..."
Abstract

Cited by 155 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by ‘matrix integrals ’ over representation varieties. We discuss examples of CY algebras involving quivers, 3dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3manifolds and ChernSimons. Examples related to quantum Del Pezzo surfaces are discussed in [EtGi].
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. ..."
Abstract

Cited by 47 (5 self)
 Add to MetaCart
(Show Context)
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.
Projective module description of the qmonopole
 COMUN.MATH.PHYS
, 1999
"... The Dirac qmonopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The ChernConnes pairing of cyclic cohomology and Ktheory is computed for the winding number −1. The nontriviality of this pairing is used to conclude that the quantum p ..."
Abstract

Cited by 46 (20 self)
 Add to MetaCart
The Dirac qmonopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The ChernConnes pairing of cyclic cohomology and Ktheory is computed for the winding number −1. The nontriviality of this pairing is used to conclude that the quantum principal Hopf fibration is noncleft. Among general results, we provide a leftright symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (HopfGalois extensions) their associated covariant derivatives on projective modules.
Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
Abstract

Cited by 44 (0 self)
 Add to MetaCart
(Show Context)
1.1 A∞algebras as spaces........................ 2
Necklace Lie Algebras and Noncommutative Symplectic Geometry
 MATH. Z
, 2000
"... Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry, [12]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry, [12]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg [ 13]. Using results of W. CrawleyBoevey and M. Holland [ 10], [8] and [9] we give a combinatorial description of all the relevant couples (ff# ) which are coadjoint orbits. We give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness.