Opérateurs différentiels bi-invariants sur un groupe de Lie

Duflo, Michel

doi:10.24033/asens.1327

Cited by 124 publications

(138 citation statements)

References 6 publications

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“…Note that the combination of the Weyl map with the obtained gauge operator is exactly the Duflo quantization map for polynomial functions on the dual of su(2) [25,26], extended to a larger class of functions in [22] by means of Fourier expansion…”

Section: Jhep08(2015)024mentioning

confidence: 99%

“…The Duflo isomorphism [25,26] establishes that the algebra of invariant polynomials on the dual of finite dimensional Lie algebras is isomorphic to the center of the corresponding universal enveloping algebra. It is obtained on composing the Poincaré-Birkhoff-Witt isomorphism (which is only an isomorphism at the level of vector spaces) with an automorphism of the space of polynomials.…”

Section: Jhep08(2015)024mentioning

confidence: 99%

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Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Kupriyanov

Vitale

2015

J. High Energ. Phys.

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Abstract:We consider linear star products on R d of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, Tr (f g) = Tr (f · g). We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibniz rule holds true up to a total derivative. As a particular example we study the space R 3 θ with su(2) type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibniz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents.

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Section: Jhep08(2015)024mentioning

confidence: 99%

Section: Jhep08(2015)024mentioning

confidence: 99%

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Kupriyanov

Vitale

2015

J. High Energ. Phys.

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“…On the other hand, it was shown by Duflo that the PBW map does have this property if it is pre-composed with a certain infinite order differential operator known as the "Duflo factor". [13].…”

Section: In Which the Vertical Maps Are Injective Algebra Homomorphismentioning

confidence: 99%

Lie theory and the Chern–Weil homomorphism

Alekseev

Meinrenken

2005

Annales Scientifiques de l’École Normale Supérieure

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Abstract. Let P → B be a principal G-bundle. For any connection θ on P , the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra W g = Sg * ⊗ ∧g * into the algebra of differential forms A = Ω(P ). Invariant polynomials (Sg * )inv ⊂ W g map to cocycles, and the induced map in cohomology (Sg * )inv → H(A basic ) is independent of the choice of θ. The algebra Ω(P ) is an example of a commutative g-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map W g → A for possibly noncommutative g-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism "up to g-homotopy". Hence, the induced map (Sg * )inv → H basic (A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.

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“…The Duflo map [39] is a generalization of the universal quantization map proposed by Harish-Chandra for semi-simple Lie algebras. The latter provides a prescription to quantize polynomials of commuting variables (the classical triad fields e) which after quantization acquire Lie algebra commutation relations (the flux operatorsÊ).…”

Section: The Duflo Mapmentioning

confidence: 99%

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Noui

Pérez

Pranzetti

2011

J. High Energ. Phys.

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In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant Λ > 0 in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection A and the canonically conjugate triad field e. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A ± = A± √ Λe. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A λ = A + λe (for λ ∈ R) on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity (with q = exp(i λ/2)). The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.

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Opérateurs différentiels bi-invariants sur un groupe de Lie

Cited by 124 publications

References 6 publications

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Lie theory and the Chern–Weil homomorphism

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

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