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On universal enveloping algebras in a topological setting

Daniel Beltiţă, Mihai Nicolae (2015)

Studia Mathematica

We study some embeddings of suitably topologized spaces of vector-valued smooth functions on topological groups, where smoothness is defined via differentiability along continuous one-parameter subgroups. As an application, we investigate the canonical correspondences between the universal enveloping algebra, the invariant local operators, and the convolution algebra of distributions supported at the unit element of any finite-dimensional Lie group, when one passes from finite-dimensional Lie groups...

On Witten multiple zeta-functions associated with semisimple Lie algebras I

Kohji Matsumoto, Hirofumi Tsumura (2006)

Annales de l’institut Fourier

We define Witten multiple zeta-functions associated with semisimple Lie algebras 𝔰𝔩 ( n ) , ( n = 2 , 3 , ... ) of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case 𝔰𝔩 ( 4 ) , we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove...

One-dimensional infinitesimal-birational duality through differential operators

Tomasz Maszczyk (2006)

Fundamenta Mathematicae

The structure of filtered algebras of Grothendieck's differential operators on a smooth fat point in a curve and graded Poisson algebras of their principal symbols is explicitly determined. A related infinitesimal-birational duality realized by a Springer type resolution of singularities and the Fourier transformation is presented. This algebro-geometrical duality is quantized in appropriate sense and its quantum origin is explained.

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra 𝒜 = 𝕂 [ 𝔸 n ] is said to be of Kostant type, if its centre Z ( 𝒜 ) is freely generated by homogeneous polynomials F 1 , ... , F r such that they give Kostant’s regularity criterion on 𝔸 n ( d x F i are linear independent if and only if the Poisson tensor has the maximal rank at x ). If the initial Poisson algebra is of Kostant type and F i satisfy a certain degree-equality, then the contraction is also of Kostant...

One-parameter subgroups and the B-C-H formula

Wojciech Wojtyński (1994)

Studia Mathematica

An algebraic scheme for Lie theory of topological groups with "large" families of one-parameter subgroups is proposed. Such groups are quotients of "𝔼ℝ-groups", i.e. topological groups equipped additionally with the continuous exterior binary operation of multiplication by real numbers, and generated by special ("exponential") elements. It is proved that under natural conditions on the topology of an 𝔼ℝ-group its group multiplication is described by the B-C-H formula in terms of the associated...

Orbit functions.

Klimyk, Anatoliy, Patera, Jiri (2006)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Orbit measures, random matrix theory and interlaced determinantal processes

Manon Defosseux (2010)

Annales de l'I.H.P. Probabilités et statistiques

A connection between representation of compact groups and some invariant ensembles of hermitian matrices is described. We focus on two types of invariant ensembles which extend the gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal....

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