103 research outputs found
Tannakian categories in positive characteristic
We determine internal characterisations for when a tensor category is (super) tannakian, for fields of positive characteristic. This generalises the corresponding characterisations in characteristic zero by P. Deligne. We also explore notions of Frobenius twists in tensor categories in positive characteristic
Operator identities in q-deformed Clifford analysis
In this paper, we define a q-deformation of the Dirac operator as a generalization of the one dimensional q-derivative. This is done in the abstract setting of radial algebra. This leads to a q-Dirac operator in Clifford analysis. The q-integration on R(m), for which the q-Dirac operator satisfies Stokes' formula, is defined. The orthogonal q-Clifford-Hermite polynomials for this integration are briefly studied
Integration in superspace using distribution theory
In this paper, a new class of Cauchy integral formulae in superspace is
obtained, using formal expansions of distributions. This allows to solve five
open problems in the study of harmonic and Clifford analysis in superspace
Orthosymplectically invariant functions in superspace
The notion of spherically symmetric superfunctions as functions invariant
under the orthosymplectic group is introduced. This leads to dimensional
reduction theorems for differentiation and integration in superspace. These
spherically symmetric functions can be used to solve orthosymplectically
invariant Schroedinger equations in superspace, such as the (an)harmonic
oscillator or the Kepler problem. Finally the obtained machinery is used to
prove the Funk-Hecke theorem and Bochner's relations in superspace.Comment: J. Math. Phy
Invariant integration on orthosymplectic and unitary supergroups
The orthosymplectic supergroup OSp(m|2n) and unitary supergroup U(p|q) are
studied following a new approach that starts from Harish-Chandra pairs and
links the sheaf-theoretical supermanifold approach of Berezin and others with
the differential geometry approach of Rogers and others. The matrix elements of
the fundamental representation of the Lie supergroup G are expressed in terms
of functions on the product supermanifold G_0 x R^{0|N}, with G_0 the
underlying Lie group and N the odd dimension of G. This product supermanifold
is isomorphic to the supermanifold of G. This leads to a new expression for the
standard generators of the corresponding Lie superalgebra g as invariant
derivations on G. Using these results a new and transparent formula for the
invariant integrals on OSp(m|2n) and U(p|q) is obtained
Hilbert space for quantum mechanics on superspace
In superspace a realization of sl2 is generated by the super Laplace operator
and the generalized norm squared. In this paper, an inner product on superspace
for which this representation is skew-symmetric is considered. This inner
product was already defined for spaces of weighted polynomials (see [K.
Coulembier, H. De Bie and F. Sommen, Orthogonality of Hermite polynomials in
superspace and Mehler type formulae, arXiv:1002.1118]). In this article, it is
proven that this inner product can be extended to the super Schwartz space, but
not to the space of square integrable functions. Subsequently, the correct
Hilbert space corresponding to this inner product is defined and studied. A
complete basis of eigenfunctions for general orthosymplectically invariant
quantum problems is constructed for this Hilbert space. Then the integrability
of the sl2-representation is proven. Finally the Heisenberg uncertainty
principle for the super Fourier transform is constructed
q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials
We define a q-deformation of the Dirac operator, inspired by the one
dimensional q-derivative. This implies a q-deformation of the partial
derivatives. By taking the square of this Dirac operator we find a
q-deformation of the Laplace operator. This allows to construct q-deformed
Schroedinger equations in higher dimensions. The equivalence of these
Schroedinger equations with those defined on q-Euclidean space in quantum
variables is shown. We also define the m-dimensional q-Clifford-Hermite
polynomials and show their connection with the q-Laguerre polynomials. These
polynomials are orthogonal with respect to an m-dimensional q-integration,
which is related to integration on q-Euclidean space. The q-Laguerre
polynomials are the eigenvectors of an su_q(1|1)-representation
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