Wakimoto modules, opers and the center at the critical level

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In this paper, we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc, giving another proof of the theorem of B. Feigin and the author.     

Coding the principal character formula for affine Kac-Moody lie algebras

In this paper, an algorithm for computing the principal character for affine Lie algebras is discussed and presented. The principal characters discovered using this program are given and/or proven. Results include level 2 and 3 character formulas in and the sole existence of the Rogers-Ramanujan products in , , , , , , .     

The Nilpotency Properties of the Leibniz Algebra Mn(C)D

We study the nilpotency properties of the Leibniz algebras constructed by means of D-mappings on the algebra of complex square matrices M _n (C). In particular, we obtain a criterion for nilpotency of these algebras in terms of the properties of a D-mapping. We prove also that the Leibniz algebras under consideration cannot be simple.     

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦₂ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦₂ algebra would be another 𝒦₂ algebra. We show that this is not necessarily the case by constructing a monomial 𝒦₂ algebra for which the corresponding Yoneda algebra is not 𝒦₂.     

Analytic approach to the (an)harmonic crystal chains with self-consistent stochastic reservoirs

We consider the harmonic and anharmonic chains of oscillators with self-consistent stochastic reservoirs and derive an integral representation (à la Feynman-Kac) for the correlations, in particular, for the heat flow. For the harmonic chain, we give a new proof that its thermal conductivity is finite in the steady state. Based on this integral representation for the correlations and a perturbative analysis, the approach is quite general and can be extended to more intricate systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 138–146, July, 2008.     

Large‐time behavior of solutions to a 1D liquid crystal system

In this paper, we prove the large‐time behavior, as time tends to infinity, of solutions in $H^i\times H_0^i\times H^{i+1}(i=1,2)$ and $H^4\times H_0^4\times H^4$ for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier‐Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.     

On the Affine Schur Algebra of Type A II

By studying certain centralizer subalgebras of the affine Schur algebra we show that is Noetherian and we determine its center. Assuming n ≥ r, we show that is Morita equivalent to , and the Schur functor is an equivalence under certain conditions. The author acknowledges support by National Natural Science Foundation of China No.10131010.     

Complex brackets and balanced complex 1st-order difference polynomials in 4-dimensional Minkowski space

This paper investigates complex brackets and balanced complex 1st-order di?erence (BCD) polynomials. Then we propose an algorithm of O(n log n) complexity to check the equality of brackets. It substitutes exponential algorithms before. Also, BCD polynomials have some usages in geometric calculation.     

n+1维n-Lei代数的导子代数

本文给出了n 1维n-Lie代数的导子代数的结构，以及导子的具体表示形式．     

HBa(D)空间的内插定理

研究了一类由H^p空间构成的Ba空间的内插性质,给出了H^Ba(D)空间的三个内插定理。     

Constructions of Metric $(n+1)$-Lie Algebras

Metric n-Lie algebras have wide applications in mathematics and mathematical physics. In this paper, the authors introduce two methods to construct metric (n+1)-Lie algebras from metric n-Lie algebras for n≥2. For a given m-dimensional metric n-Lie algebra(g, [, ···, ], B_g), via one and two dimensional extensions ￡=g+IFc and g0= g+IFx~(-1)+IFx~0 of the vector space g and a certain linear function f on g, we construct(m+1)-and (m+2)-dimensional (n+1)-Lie algebras(￡, [, ···, ]cf) and(g0, [, ···, ]1), respectively.Furthermore, if the center Z(g) is non-isotropic, then we obtain metric(n + 1)-Lie algebras(L, [, ···, ]cf, B) and(g0, [, ···, ]1, B) which satisfy B|g×g = Bg. Following this approach the extensions of all(n + 2)-dimensional metric n-Lie algebras are discussed.     

Principal subspaces for the quantum affine vertex algebra in type A1(1)

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof–Kazhdan quantum affine vertex algebra of integer level $k?1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers–Ramanujan-type identities.     

Monomial Base for Little q-Schur Algebra uk(2, r) at Even Roots of Unity

Let uk(2, r) be a little q-Schur algebra over k, where k is a field containing an l-th primitive root ε of 1 with l ≥ 4 even, the author constructs a certain monomial base for little q-Schur algebra uk(2, r).     

Deformations of the algebra of abstract vector variables

In this paper we present two types of deformation of the algebra of abstract vector variables, which may be useful for quantum geometry.