Global Pseudo-differential Operators on the Quantum Group SUq(2)SU_q(2)SUq(2) (original) (raw)
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International Mathematics Research Notices, 2012
Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere S 3 and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hörmander's classes of operators Ψ m (G), Ψ m (S 3 ) and Ψ m (SU ). Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as L 2 -boundedness and Sobolev L 2 -boundedness of operators in this global quantization are established on general compact Lie groups.
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2012
For an arbitrary pseudo-differential operator A : S(R n) −→ S ′ (R n) with Weyl symbol a ∈ S ′ (R 2n), we consider the pseudo-differential operators A : S(R n+k) −→ S ′ (R n+k) associated with the Weyl symbols a = (a ⊗ 1 2k) • s, where 1 2k (x) = 1 for all x ∈ R 2k and s is a linear symplectomorphism of R 2(n+k). We call the operators A symplectic dimensional extensions of A. In this paper we study the relation between A and A in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of A in terms of those of A. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes HG m1,m0 ρ of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in HG m1,m0 ρ but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.
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The universal enveloping algebra U (G) of a Lie algebra G acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl 2 .
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Communications in Mathematical Physics, 1989
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A symplectic extension map and a new Shubin class of pseudo-differential operators
Journal of Functional Analysis, 2014
For an arbitrary pseudo-differential operator A : S(R n ) −→ S ′ (R n ) with Weyl symbol a ∈ S ′ (R 2n ), we consider the pseudo-differential operators A : S(R n+k ) −→ S ′ (R n+k ) associated with the Weyl symbols a = (a ⊗ 1 2k ) • s, where 1 2k (x) = 1 for all x ∈ R 2k and s is a linear symplectomorphism of R 2(n+k) . We call the operators A symplectic dimensional extensions of A. In this paper we study the relation between A and A in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of A in terms of those of A. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes HG m1,m0 ρ of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in HG m1,m0 ρ but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.
Banach Journal of Mathematical Analysis, 2011
The relevance of modulation spaces for deformation quantization, Landau-Weyl quantization and noncommutative quantum mechanics became clear in recent work. We continue this line of research and demonstrate that Q s (R 2d ) is a good class of symbols for Landau-Weyl quantization and propose that the modulation spaces M p vs (R 2d ) are natural generalized Shubin classes for the Weyl calculus. This is motivated by the fact that the Shubin class Q s (R 2d ) is the modulation space M 2 vs (R 2d ). The main result gives estimates of the singular values of pseudodifferential operators with symbols in M p vs (R 2d ) for the standard Weyl calculus and for the Landau-Weyl calculus.
Spectral Invariance for Certain Algebras of Pseudodifferential Operators
Journal of the Institute of Mathematics of Jussieu, 2005
We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.
Gradient-type differential operators and unitary highest weight representations of SU(p, q)
Journal of Functional Analysis, 1988
Let CP+* be equipped with a hermitian form of signature (p, 4) and let SU(p, q) denote the subgroup of the corresponding invariance group U(p, q) consisting of matrices with determinant 1. To certain highest weights 1, we associate a first-order group invariant linear differential operator gA whose kernel contains a unitary highest weight representation with highest weight 1. The Fock model realization of unitary highest weight representations of U(p, q) is the fundamental tool used to implement this construction. The operator & is shown to be equivalent to an operator a, which acts on Hol(G/K, H"), the space of holomorphic vector valued functions defined on G/K. We identify a set A, of highest weights such that Ker($) is a proper subspace of Hol(G/E(, H") and show that those 1 in A, correspond to points occurring at the far right of the discrete set in the classification scheme of Enright, Howe, and Wallach. First-order differential equations arising from this proper containment are explicitly derived from the operator a,. We illustrate the fundamental nature of these first-order equations by deriving from them a system which completely determines the irreducible spaces for ladder representations of su(p, 4).
ON THE HERMITICITY OF q-DIFFERENTIAL OPERATORS AND FORMS ON THE QUANTUM EUCLIDEAN SPACES
Reviews in Mathematical Physics, 2006
We show that the complicated ⋆-structure characterizing for positive q the U q so(N )-covariant differential calculus on the non-commutative manifold R N q boils down to similarity transformations involving the ribbon element of a central extension of U q so(N ) and its formal square rootṽ. Subspaces of the spaces of functions and of p-forms on R N q are made into Hilbert spaces by introducing non-conventional "weights" in the integrals defining the corresponding scalar products, namely suitable positive-definite qpseudodifferential operatorsṽ ′±1 realizing the action ofṽ ±1 ; this serves to make the partial q-derivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the 'radial coordinate' r. Unless we choose a constant m, then the square-integrables functions/forms mustfulfill an additional condition, namely their analytic continuations to thecomplex r plane can have poles only on the sites of some special lattice.Among the functions naturally selected by this condition there are q-special functions with 'quantized' free parameters.
Communications in Mathematical Physics, 2009
The spectral action on the equivariant real spectral triple over A SU q (2) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S 3 .