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[more ▼]It is shown that the normalized trace of Fedosov star product for quantum moment map depends only on the path component in the cohomology class of the symplectic form and the cohomology class of the closed formal 2-form required to define Fedosov connections (Theorem 1.3). As an application we obtain a family of obstructions to the existence of closed Fedosov star products naturally attached to symplectic manifolds (Theorem 1.5) and Kähler manifolds (Theorem 1.6). These obstructions are integral invariants depending only on the path component of the cohomology class of the symplectic form. Restricted to compact Kähler manifolds we re-discover an obstruction found earlier in La Fuente-Gravy (2019). [less ▲]Detailed reference viewed: 48 (1 UL) Pre-Courant algebroidsBruce, Andrew ; Grabowski, Januszin Journal of Geometry and Physics (2019), 142Pre-Courant algebroids are ‘Courant algebroids’ without the Jacobi identity for the Courant–Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some ... [more ▼]Pre-Courant algebroids are ‘Courant algebroids’ without the Jacobi identity for the Courant–Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof. In particular, we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. We give the definition of (sub-)Dirac structures and study the naïve quasi-cochain complex within the setting of supergeometry. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids and our approach drastically simplifies working with them. [less ▲]Detailed reference viewed: 51 (1 UL) Atiyah classes and dg-Lie algebroids for matched pairsBatakidis, Panagiotis; Voglaire, Yannick in Journal of Geometry and Physics (2017)For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\ZZ$-graded manifold $\M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \M$ and the projection $p:\M\to L[1 ... [more ▼]For every Lie pair$(L,A)$of algebroids we construct a dg-manifold structure on the$\ZZ$-graded manifold$\M=L[1]\oplus L/A$such that the inclusion$\iota: A[1] \to \M$and the projection$p:\M\to L[1]$are morphisms of dg-manifolds. The vertical tangent bundle$T^p\M$then inherits a structure of dg-Lie algebroid over$\M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion$\iota\$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras. [less ▲]Detailed reference viewed: 186 (20 UL) Splitting theorem for Z_2^n-supermanifoldsCovolo, Tiffany; Grabowski, Janusz; Poncin, Norbert in Journal of Geometry and Physics (2016), 110Detailed reference viewed: 153 (24 UL) A bicategory of reduced orbifolds from the point of view of differential geometryTommasini, Matteo in Journal of Geometry and Physics (2016), 108Detailed reference viewed: 94 (1 UL) Commutative n-ary superalgebras with an invariant skew-symmetric formVishnyakova, Elizaveta in Journal of Geometry and Physics (2015), 98We study nn-ary commutative superalgebras and L∞L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie ... [more ▼]We study nn-ary commutative superalgebras and L∞L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their nn-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative mm-dimensional (m−3)(m−3)-ary algebras with an invariant form, and a classification of real simple mm-dimensional Lie (m−3)(m−3)-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for L∞L∞-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric nn-ary algebras. [less ▲]Detailed reference viewed: 76 (4 UL) From hypercomplex to holomorphic symplectic structuresHong, Wei in Journal of Geometry and Physics (2015), 96Detailed reference viewed: 163 (12 UL) SUSY-structures, representations, and the Peter-Weyl theorem for S^1|1Kwok, Stephen ; Carmeli, C.; Fioresi, R.in Journal of Geometry and Physics (2015)Detailed reference viewed: 71 (2 UL) Graded geometry in gauge theories and beyondSalnikov, Vladimir in Journal of Geometry and Physics (2015)We study some graded geometric constructions appearing naturally in the context of gauge theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to ... [more ▼]We study some graded geometric constructions appearing naturally in the context of gauge theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q -manifolds introducing thus the concept of equivariant Q -cohomology. Using this concept we describe a procedure for analysis of gauge symmetries of given functionals as well as for constructing functionals (sigma models) invariant under an action of some gauge group. As the main example of application of these constructions we consider the twisted Poisson sigma model. We obtain it by a gauging-type procedure of the action of an essentially infinite dimensional group and describe its symmetries in terms of classical differential geometry. We comment on other possible applications of the described concept including the analysis of supersymmetric gauge theories and higher structures. [less ▲]Detailed reference viewed: 106 (11 UL) On Mpc-structures and symplectic Dirac operatorsCahen, Michel; Gutt, Simone; La Fuente-Gravy, Laurent et alin Journal of Geometry and Physics (2014), 86We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite ... [more ▼]We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute these kernels explicitly for complex projective spaces and show that the remaining Dirac–Dolbeault operator has infinite dimensional kernels on these finite rank sub-bundles. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabiliser of a Lagrangian subspace) in the Mpc group and classify G-invariant Mpc-structures on symplectic manifolds with a G-action. We prove a variant of Parthasarathy’s formula for the commutator of two symplectic Dirac-type operators on general symmetric symplectic spaces. [less ▲]Detailed reference viewed: 43 (0 UL) On the category of Lie n-algebroidsBonavolontà, Giuseppe ; Poncin, Norbert in Journal of Geometry and Physics (2013), 73Detailed reference viewed: 262 (33 UL) Higher trace and Berezinian of matrices over a Clifford algebraCovolo, Tiffany ; Ovsienko, Valentin; Poncin, Norbert in Journal of Geometry and Physics (2012), 62(11), 22942319We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n-graded commutative associative algebra A. The applications include a new approach to the classical ... [more ▼]We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n-graded commutative associative algebra A. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z_2)^n-graded matrices of degree 0 is polynomial in its entries. In the case of the algebra A = H of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (Z_2)^n-graded version of Liouville’s formula. [less ▲]Detailed reference viewed: 246 (55 UL) Superizations of Cahen-Wallach symmetric spaces and spin representations of the Heisenberg algebraSanti, Andrea in Journal of Geometry and Physics (2010), 60(2), 295--325Detailed reference viewed: 171 (2 UL) Equivariant quantization of orbifoldsPoncin, Norbert ; Radoux, Fabian; Wolak, Robertin Journal of Geometry and Physics (2010), 60(9), 1103--1111Detailed reference viewed: 163 (3 UL) A first approximation for quantization of singular spacesPoncin, Norbert ; Radoux, Fabian; Wolak, Robertin Journal of Geometry and Physics (2009), 59(4), 503--518Detailed reference viewed: 182 (4 UL) Coherent state embeddings, polar divisors and Cauchy formulasBerceanu, Stefan; Schlichenmaier, Martin in Journal of Geometry and Physics (2000), 34Detailed reference viewed: 89 (1 UL) 1