Alberto De Sole
This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory of generalized Drinfeld-Sokolov hierarchies.
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We show that one can construct a classical affine W-algebra via a classical BRST complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible $λ$-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we d…
Vertex representations for twisted affine Lie algebra of type A_{2l}^{(2)}
本文构造了A_{2l} ^{（2）}型扭仿射李代数的一个不可约顶点模.
Covering AlgebrasI: Extended Affine Lie Algebras
研究了扩张仿射李代数相对于有穷级自同构的覆盖代数。条件时，所得到的代数再次是一个EALA。本文研究了EALA代数关于有穷级自同构的仿射化，这些代数是仿射Kac-Moody李代数的推广.
Smooth modules for affine Kac-Moody algebras have a prime importance for the quantum field theory as they correspond to the representations of the universal affine vertex algebras. But, very little is known about such modules beyond the category of positive energy representations. We construct a new class of smooth modules over affine Kac-Moody algebras. In a particular case, these modules are isomorphic to those induced from generalized Whittaker modules for Takiff Lie algebras. We establish the irreducibility criterion for constructed modules in the case of the affine sl(2) Lie algebra.
In this paper we have computed all the affine Kac-Moody symmetric spaces which are tame Frechet manifolds starting from the Vogan diagrams related to the affine untwisted Kac-Moody algebras. The detail computation of affine Kac-Moody symmetric spaces associated with A_1^(1) and A_2^(1) are shown algebraically to corroborate our method.
We introduce an explicit representation of the double affine Hecke algebra (of type $A_1$) at $q=1$ that gives rise to a periodic counterpart of a well-known Fourier transform associated with the affine Hecke algebra.
In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2,R) + Ω_R/dR, where R=C[t,t^{-1},u|u^2=t^3 - 2b t^2 + t]. The first realization gives an analogue of Wakimoto's construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new type of representations analogous to Imaginary Verma modules in the Affine setting.
Quantized Algebras of Functions on Affine Hecke Algebras
定义了A型仿射Hecke代数上的量子化函数代数和相应的q-Schur代数，并对它们的不可约酉化表示进行了分类.
We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W-algebras.
We apply the theory of $ϕ$-coordinated modules, developed by H.-S. Li, to the Etingof--Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. We prove, for a certain associate $ϕ$ of the one-dimensional additive formal group, that any $ϕ$-coordinated module for the level $c\in\mathbb{C}$ quantum affine vertex algebra is naturally equipped with a structure of restricted level $c$ module for the quantum affine algebra in type $A$ and vice versa. Moreover, we show that any $ϕ$-coordinated module is irreducible with respect to the action of the quantum affi…
Quantum difference equations and Maulik-Okounkov quantum affine algebras of affine type $A$
本文通过 Dubrovin 联络的单值表示，证明了量子环面代数与仿射型 $A$ 的 Maulik-Okounkov 量子仿射代数正半部分同构，其核心在于证明代数与几何量子差分方程具有相同的退化极限。
The Drinfled realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig's twisted braid group action.
We introduce the two-parameter quantum affine algebra $U_{r,s}(\widehat{gl}_n)$ via the RTT realization. The Drinfeld realization is given and the type A quantum affine algebra is proved to be a special subalgebra of our extended algebra.
We construct all fundamental modules for the two parameter quantum affine algebra of type $A$ using a combinatorial model of Young diagrams. In particular we also give a fermionic realization of the two-parameter quantum affine algebra.
We consider a quantum affine algebra realized in two-dimensional non-linear sigma models with target space three-dimensional squashed sphere. Its affine generators are explicitly constructed and the Poisson brackets are computed. The defining relations of quantum affine algebra in the sense of the Drinfeld first realization are satisfied at classical level. The relation to the Drinfeld second realization is also discussed including higher conserved charges. Finally we comment on a semiclassical limit of quantum affine algebra at quantum level.
Principal Realization for the extended affine Lie algebra of type $sl_2$ with coordinates in a simple quantum torus with two generators
本文构造了$sl_2$型广义仿射代数在量子环面上的不可约表示。我们明确地给出使用顶点算子的公式类似于那些发现在理论的无限秩仿射代数$A_{\infty}$。
Meromorphic braided category arising in quantum affine algebras
构造了一类椭圆曲线上的平坦向量丛。它起源于量子仿射代数的表示论，并且在曲线的平方的对角线上具有奇异性的亚纯辫状结构。
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