Effacement des dérivations et spectres premiers des algèbres quantiques

Quantum Groups and Integrable Models

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

Fourier restriction estimates on quantum Euclidean spaces

Unitary representations of kinematical symmetry groups of quantum systems are\nfundamental in quantum theory. We propose in this paper its generalization to\nquantum kinematical groups. Using the method, proposed by us in a recent paper\n(olmo01), to induce representations of quantum bicrossproduct algebras we\nconstruct the representations of the family of standard quantum inhomogeneous\nalgebras $U_\\lambda(iso_{\\omega}(2))$. This family contains the quantum\nEuclidean, Galilei and Poincar\\'e algebras, all of them in (1+1) dimensions. As\nbyproducts we obtain the actions of these quantum algebras on regular co-spaces\nthat are an algebraic generalization of the homogeneous spaces and $q$--Casimir\nequations which play the role of $q$--Schr\\"odinger equations.\n

We define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements of a particular Schubert cell, indexed by an element of the Weyl group. They are generally not fully automorphic. We will develop some results and methods for ${\rm GL}_3$ that may be suggestive about the general case. The six Schubert Eisenstein series are shown to have meromorphic continuation and some functional equations. The Schubert Eisenstein series $E_{s_1s_2}$ and $E_{s_2s_1}$ corresponding to the Weyl group elements of order three are particularly interesting: at the point where the full Eisenstein series is maximally polar, they unexpectedly become (with minor correction terms added) fully automorphic and related to each other.

This paper extends an algorithm and canonical embedding in [6] to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon’s original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space A ′ from the original quantum algebra A via a series of change of variables within the division ring of fractions Frac(A). The canonical embedding takes a completely prime ideal P⊲A to a completely prime ideal Q⊲A ′ such that when A is a PI algebra, PI-deg(A/P)=PI-deg(A ′ /Q). When the quantum parameter is a root of unity, combining our construction with results from [2] allows us to state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of A/P given a suitable irreducible representation of A ′ /Q when A is PI.

Cauchon diagrams for quantized enveloping algebras

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper computes such Xi and Ti in the case of S3 where S3 is the symmetric group Sn, n=3.

Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac–Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type Ã2 on a Euclidean space V≅ℝ2 to study the Bruhat order on W. We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of Boe and Graham (which is a conjectural simplification of the Carrell–Peterson criterion for rational smoothness) to type Ã2. Computational evidence suggests that the only Schubert varieties in type Ã2 where the "nontrivial" case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety X(w) of type Ã2. The proof uses descriptions we obtain of the elements x≤w and of the rationally smooth locus of X(w) in terms of the W-action on V. As a consequence we describe the maximal nonrationally smooth points of X(w). The results of this paper are used in a sequel to describe the smooth locus of X(w), which is different from the rationally smooth locus.

We define a class of quantum linear Galois algebras which include the universal enveloping algebra the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand–Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum Weyl algebra for symmetric groups are, in fact, Galois orders over an adequate commutative subalgebras and free as right (left) modules over these subalgebras. In the rank 1 cases the results hold for an arbitrary finite group of automorphisms when the field is

Field of Fractions of Certain Quantum Algebras

In this paper, the paradox on the elastic wedge is restudied under the Hamiltonian system. This research shows that a special paradox in Euclidean space under the Lagrange system is just the Jordan form solutions in symplectic space under the Hamiltonian system, so they can be solved directly and rationally by normal mathematical physics methods.

On the symplectic superposition method for new analytic free vibration solutions of side-cracked rectangular thin plates

The Saint-Venant problem and principle in elasticity

We study solvability of convolution equations for functions with discrete support in , a special case being functions with support in the integer points. The more general case is of interest for several grids in Euclidean space, like the body-centred and face-centred tessellations of 3-space, as well as for the non-periodic grids that appear in the study of quasicrystals. The theorem of existence of fundamental solutions by de Boor et al is generalized to general discrete supports, using only elementary methods. We also study the asymptotic growth of sequences and arrays using the Fenchel transformation.