A Recent Look at the Quantum Berezinian in the Yangian Y(gl_{m|n})

Twisted Yangians and symmetric pairs

We describe recent work on the twisted Yangians Y(g,h) which arise as boundary remnants of Yangians Y(g) in 1+1D integrable field theories, bringing out the special role played by the requirement that (g,h) be a symmetric pair.

关于超 Yangian 奇反射与 Bethe ansatz 的一个注记

A note on odd reflections of super Yangian and Bethe ansatz

本文从新视角重新推导超 Yangian 的奇反射转换规则，建立其与 XXX 型 Bethe ansatz 费米子再生程序的联系，并给出 q-特征标变化算法及斜表示的显式计算。

The PBW theorem for the affine Yangians

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.

A Note on Quiver Quantum Toroidal Algebra

Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian $\mathfrak{gl}_{1}$. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations. In this paper, we propose a $q$-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consist…

Quiver Yangians and Crystal Melting: A Concise Summary

The goal of this short article is to summarize some of the recent developments in the quiver Yangians and crystal meltings. This article is based on a lecture delivered by the author at International Congress on Mathematical Physics (ICMP), Geneva, 2021.

Yangian变形的S-交换量子顶点代数与Bethe子代数

Yangian deformations of $\mathcal{S}$-commutative quantum vertex algebras and Bethe subalgebras

本文构造了基于标准化Yang R矩阵的新类量子顶点代数，探讨其表示论及交换性与Bethe子代数的关系。

Quiver Yangian and Supersymmetric Quantum Mechanics

The statistical model of crystal melting represents BPS configurations of D-branes on a toric Calabi-Yau three-fold. Recently it has been noticed that an infinite-dimensional algebra, the quiver Yangian, acts consistently on the crystal-melting configurations. We physically derive the algebra and its action on the BPS states, starting with the effective supersymmetric quiver quantum mechanics on the D-brane worldvolume. This leads to remarkable combinatorial identities involving equivariant integrations on the moduli space of the quantum mechanics, which can be checked by numerical computation…

A Quantum Affine Algebra for the Deformed Hubbard Chain

The integrable structure of the one-dimensional Hubbard model is based on Shastry's R-matrix and the Yangian of a centrally extended sl(2|2) superalgebra. Alcaraz and Bariev have shown that the model admits an integrable deformation whose R-matrix has recently been found. This R-matrix is of trigonometric type and here we derive its underlying exceptional quantum affine algebra. We also show how the algebra reduces to the above mentioned Yangian and to the conventional quantum affine sl(2|2) algebra in two special limits.

经典型李代数中的移位扭 Yangian 与 Slodowy 切片

Shifted twisted Yangians and Slodowy slices in classical Lie algebras

本文引入 AI 型移位扭 Yangian，证明其半经典极限与 gl_n 半经典移位 Yangian 的 Dirac 约化一致，并由此得到 B、C、D 型非矩形幂零元 Slodowy 切片的截断同构、抛物型表示及偶幂零元情形的泊松表示。

A Lax type operator for quantum finite W-algebras

For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g,f). We show that for the classical linear Lie algebras gl_N, sl_N, so_N and sp_N, the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.

T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently the T-systems were generalized to quantum affinizations of a wide class of quantum Kac-Moody algebras by Hernandez. In this note we introduce the corresponding Y-systems and establish a relation between T and Y-systems. We also introduce the T and Y-systems associated with a class of cluster algebras, which include the former T and Y-systems of simply laced type as special cases.

T-systems, Y-systems, and cluster algebras: Tamely laced case

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently they were generalized to quantum affinizations of quantum Kac-Moody algebras associated with a wide class of generalized Cartan matrices which we say tamely laced. Furthermore, in the simply laced case, and also in the nonsimply laced case of finite type, they were identified with relations arising from cluster algebras. In this note we generalize such an identification to any tamely laced Cartan matrices, especially to the nonsimply laced ones of nonfinite t…

类型A的大代数与矩阵空间坐标环

Big Algebra in Type $A$ for the Coordinate Ring of the Matrix Space

本文研究了Hausel引入的大代数，通过微分算子得到生成元公式，并将其与Yangian的Bethe子代数关联，应用于对称幂的标准表示的大代数。