Cladistics of Double Yangians and Elliptic Algebras

Deformed Double Yangian Structures

Scaling limits when q tends to 1 of the elliptic vertex algebras A_qp(sl(N)) are defined for any N, extending the previously known case of N=2. They realise deformed, centrally extended double Yangian structures DY_r(sl(N)). As in the quantum affine algebras U_q(sl(N)), and quantum elliptic affine algebras A_qp(sl(N)), these algebras contain subalgebras at critical values of the central charge c=-N-Mr (M integer, 2r=ln p/ln q), which become Abelian when c=-N or 2r=Nh for h integer. Poisson structures and quantum exchange relations are derived for their abstract generators.

Quantum elliptic algebras and double Yangians

Quantum universal enveloping algebras, quantum elliptic algebras and double (deformed) Yangians provide fundamental algebraic structures relevant for many integrable systems. They are described in the FRT formalism by R-matrices which are solutions of elliptic, trigonometric or rational type of the Yang--Baxter equation with spectral parameter or its generalization known as the Gervais--Neveu--Felder equation. While quantum groups and double Yangians appear as quasi-triangular Hopf algebras, this is no longer the case for elliptic algebras and the various deformations of Yangian type algebras.…

将h-Yangian和量子椭圆代数在类型A中的模块与h-adic量子顶点代数关联

Associating modules for the $h$-Yangian and quantum elliptic algebra in type $A$ with $h$-adic quantum vertex algebras

本文研究了将h-Yangian和量子椭圆代数在类型A中的模块与h-adic量子顶点代数关联，并在临界水平构造了新的中心元素。

Yangian Algebras and Classical Riemann Problems

We investigate different Hopf algebras associated to Yang's solution of quantum Yang-Baxter equation. It is shown that for the precise definition of the algebra one needs the commutation relations for the deformed algebra of formal currents and the specialization of the Riemann problem for the currents. Two different Riemann problems are considered. They lead to the central extended Yangian double associated with ${sl}_2$ and to the degeneration of scaling limit of elliptic affine algebra. Unless the defining relations for the generating functions of the both algebras coincide their properties…

Toroidal and Elliptic Quiver BPS Algebras and Beyond

The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $\mathcal{N}=2$ supersymmetr…

箭图Yangian代数表示的算法

Algorithms for representations of quiver Yangian algebras

本文详细回顾了构造quiver Yangians晶体表示的算法，指出其仅需线性代数即可实现并易于机器化，可推广至环面和椭圆代数，并通过$\mathsf{Y}(\mathfrak{sl}_2)$等简单案例进行了说明。