On continuum theories of suspensions of deformable particles

Zusammenfassung Das vonAllen, DeSilva undKline [1] vorgeschlagene Kontinuumsmodell richtungsorientierter Flüssigkeiten wird auf die TheorieEricksens [2, 3] undHands [4] zurückgeführt. Es wird die Suspension fester Drehkörper in Flüssigkeiten und solche sphärischer Partikel unter elastischer Verformung behandelt.     

Quantum integrable systems of particles as gauge theories

We study quantum integrable systems of interacting particles from the point of view proposed by A. Gorsky and N. Nekrasov. We obtain the Sutherland system by a Hamiltonian reduction of an integrable system on the cotangent bundles to an affine algebra and show that it coincides with the Yang-Mills theory on a cylinder.We point out that there exists a tower of 2d quantum field theories. The top of this tower is the gauged G/G WZW model on a cylinder with an inserted Wilson line in an appropriate representation, which in our approach corresponds to Ruijsenaars' relativistic Calogero model. Its degeneration yields the 2d Yang-Mills theory, whose small radius limit is the Calogero model itself. We make some comments about the spectra and eigenstates of the models, which one can get from their equivalence with the field theories. Also we point out some possibilities of elliptic deformations of these constructions.Institute of Theoretical and Experimental Physics, 117259, Bol. Cheremushkinskaya 25, Moscow, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 97–103, July, 1994.     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

Indecomposable representations of generalized weyl algebras

For a class of generalized Weyl algebras which includes the Weyl algebras A_n a criterion is given as to when the category of indecomposable weight and generalized weight modules with supports from a fixed orbit is tame. In tame cases indecomposable modules are described.     

Left almost split morphisms and generalized weyl algebras

It is proved that the category of modules of finite length over a broad class of generalized Weyl algebras contains no left almost split morphism starting from a simple module. It is shown that a similar assertion holds for the algebraUsl₂(k) over an algebraically closed field k of characteristic 0. As a by-product, a new series of simple modules for such algebras is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 5 pp. 734–740, November, 1999.     

Simple lie color algebras of weyl type

A class of graded simple associative algebras are constructed, and from them, simple Lie color algebras are obtained. The structure of these simple Lie color algebras is explicitly described. More precisely, for an (ε, Γ)-color-commutative associative algebraA with an identity element over a fieldF of characteristic not 2, and for a color-commutative subalgebraD of color-derivations ofA, denote byA[D] the associative subalgebra of End (A) generated byA (regarded as operators onA via left multiplication) andD. It is easily proved that, as an associative algebra,A[D] is Γ-graded simple if and only ifA is Γ-gradedD-simple. SupposeA is Γ-gradedD-simple. Then, (a)A[D] is a free leftA-module; (b) as a Lie color algebra, the subquotient [A[D],A[D]]/Z(A[D])∩[A[D],A[D]] is simple (except one minor case), whereZ(A[D]) is the color center ofA[D]. This work was supported by NSF of China, National Educational Department of China, Jiangsu Educational Committee, and Hundred Talents Program of Chinese Academy of Sciences. These authors were partially supported by Academy of Mathematics and System Sciences during their visit to this academy.     

Affine weyl groups and conjugacy classes in Weyl groups

We define a map from an affine Weyl group to the set of conjugacy classes of an ordinary Weyl group. Supported in part by the National Science Foundation.