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See detailExtensions of superalgebras of Krichever-Novikov type
Kreusch, Marie ULg

in Letters in Mathematical Physics (2013), 103(11),

An explicit construction of central extensions of Lie superalgebras of Krichever-Novikov type is given. In the case of Jordan superalgebras related to the superalgebras of Krichever-Novikov type we ... [more ▼]

An explicit construction of central extensions of Lie superalgebras of Krichever-Novikov type is given. In the case of Jordan superalgebras related to the superalgebras of Krichever-Novikov type we calculate a 1-cocycle with coefficients in the dual space. [less ▲]

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See detailProjectively equivariant quantizations over the superspace R^{p|q}
Mathonet, Pierre ULg; Radoux, Fabian ULg

in Letters in Mathematical Physics (2011), 98(3), 311-331

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the ... [more ▼]

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields. [less ▲]

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See detailAn Explicit Formula for the Natural and Conformally Invariant Quantization
Radoux, Fabian ULg

in Letters in Mathematical Physics (2009), 89

Lecomte (Prog Theor Phys Suppl 144:125–132, 2001) conjectured the existence of a natural and conformally invariant quantization. In Mathonet and Radoux (Existence of natural and conformally invariant ... [more ▼]

Lecomte (Prog Theor Phys Suppl 144:125–132, 2001) conjectured the existence of a natural and conformally invariant quantization. In Mathonet and Radoux (Existence of natural and conformally invariant quantizations of arbitrary symbols, math.DG 0811.3710), we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in Mathonet and Radoux and to tools already used in Radoux [Lett Math Phys 78(2):173–188, 2006] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in Radoux. [less ▲]

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See detailExplicit formula for the natural and projectively equivariant quantization
Radoux, Fabian ULg

in Letters in Mathematical Physics (2006), 78

In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence ... [more ▼]

In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence using the framework of Thomas–Whitehead connections. In [Lett Math Phys 72(3):183–196, 2005], we gave a new proof of the same theorem thanks to the Cartan connections. After these works, there was no explicit formula for the quantization. In this paper, we give this formula using the formula in terms of Cartan connections given in [Lett Math Phys 72(3):183–196, 2005]. This explicit formula constitutes the generalization to any order of the formulae at second and third orders soon published by Bouarroudj in [Lett Math Phys 51(4):265–274, 2000] and [C R Acad Sci Paris Sér I Math 333(4):343–346, 2001]. [less ▲]

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See detailNatural and projectively equivariant quantizations by means of Cartan connections
Mathonet, Pierre ULg; Radoux, Fabian ULg

in Letters in Mathematical Physics (2005), 72

The existence of a natural and projectively equivariant quantization in the sense of Lecomte was proved recently by M. Bordemann, using the framework of Thomas-Whitehead connections. We give a new proof ... [more ▼]

The existence of a natural and projectively equivariant quantization in the sense of Lecomte was proved recently by M. Bordemann, using the framework of Thomas-Whitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an sl(m+1,R)-equivariant quantization exists in the flat situation, thus solving one of the problems left open by M. Bordemann. [less ▲]

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See detailEquivariant symbol calculus for differential operators acting on forms
Boniver, Fabien; Hansoul, Sarah ULg; Mathonet, Pierre ULg et al

in Letters in Mathematical Physics (2002), 62(3), 219-232

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces D_p of differential operators transforming p-forms into functions, over ... [more ▼]

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces D_p of differential operators transforming p-forms into functions, over R^n. As an application, we classify the Vect(M)-equivariant maps from D_p to D_q over a smooth manifold M, recovering and improving earlier results of N. Poncin. This provides the complete answer to a question raised by P. Lecomte about the extension of a certain intrinsic homotopy operator. [less ▲]

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See detailProjectively Equivariant Symbol Calculus for Bidifferential Operators
Boniver, Fabien ULg

in Letters in Mathematical Physics (2000), 54(2), 83-100

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See detailInvariant bidifferential operators on tensor densities over a contact manifold
Mathonet, Pierre ULg

in Letters in Mathematical Physics (1999), 48(3), 251-261

The spaces of tensor densities over a manifold M are modules over the Lie algebra Vect (M) of vector fields over the manifold. When M is a contact manifold, one can consider the algebra C(M) of vector ... [more ▼]

The spaces of tensor densities over a manifold M are modules over the Lie algebra Vect (M) of vector fields over the manifold. When M is a contact manifold, one can consider the algebra C(M) of vector fields which preserves the contact structure. If the manifold is endowed with a contact projective structure, there is an embedding of the linear symplectic algebra sp (2n+2,R) in C(M). In this Letter, we determine the C(M)- and the sp(2n+2,R)-invariant bidifferential operators on tensor densities. [less ▲]

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