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.
We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.
Since the ($β$-deformed) Hurwitz Kontsevich model corresponds to the special case of affine Yangian of ${\mathfrak{gl}}(1)$. In this paper, we construct two general cases of the $β$-deformed Hurwitz Kontsevich model. We find that the $W$-operators of these two models can be represented by the generators $e_k,\ f_k,ψ_k$ of the affine Yangian of ${\mathfrak{gl}}(1)$, and the eigenstates (the symmetric functions $Y_λ$ and 3-Jack polynomials) can be obtained from the 3D Young diagram representation of affine Yangian of ${\mathfrak{gl}}(1)$. Then we can see that the $W$-operators and eigenstates ar…
定义了椭圆顶点代数Aqp（sl（N））在q趋于1时的标度极限，推广了已知的N=2的情形.它们实现了变形的、中心扩展的双Yangian结构DY_r（sl（N））。如量子仿射代数U_q（sl（N））和量子椭圆仿射代数A_qp（sl（N）），这些代数在中心荷c=-N-Mr（M整数，2 r = lnp/lnq）的临界值处含有子代数，当c=-N或2 r =Nh（h整数）时，这些子代数成为阿贝尔代数.泊松结构和量子交换关系的抽象生成元导出。
The deformed $\mathcal W$ algebras of type $\textsf{A}$ have a uniform description in terms of the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal E$. We introduce a comodule algebra $\mathcal K$ over $\mathcal E$ which gives a uniform construction of basic deformed $\mathcal W$ currents and screening operators in types $\textsf{B},\textsf{C},\textsf{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $\mathcal K$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine…
The algebraic engineering technique is applied to a class of 3D $\mathcal{N}=2$ gauge theories on the omega-deformed background $\mathbb{R}_ε^2\times S^1$. The vortex partition function and the fundamental qq-character are obtained from a network of intertwiners between representations of the shifted (or asymptotic) quantum affine $\mathfrak{sl}(2)$ algebra. This network involves two types of representations, the prefundamental representation of Hernandez-Jimbo, and a new vertex representation acting on a bosonic Fock space. The brane system associated to this network is identified: D3 branes …
Integrable boundaries in AdS/CFT: revisiting the Z=0 giant graviton and D7-brane
在AdS/CFT对应中，我们考虑了Z=0巨引力子和Z=0 D 7-膜的世界片边界散射和相应的边界代数.我们考虑两种方法的边界散射，通常由（广义）扭曲的杨氏和q-变形模型的这些边界所管辖的量子仿射余理想子代数。我们表明，q-变形的方法导致边界代数是一个更紧凑的形式比相应的扭曲的杨格，从而有利于使用明确的计算。我们得到的q-变形的反射矩阵的两个边界，在q->1的限制专门使用扭曲的杨格。
We study the $q$-analogue of the Haldane-Shastry model, a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of Uglov, via 'freezing' from the affine Hecke algebra. We obtain explicit expressions for the spin-analogue of Macdonald operators. Through freezing these yield the higher Hamiltonians of the spin chain, including a Hamiltonian of the opposite chirality. The sum of the two chiral Hamiltonians has a real spectrum also for $q$ …
Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
本文用两类三角超对称 R 矩阵构造对合差分算子，证明分次 R 矩阵满足结合 Yang-Baxter 方程，并借助 Polychronakos 冻结技巧得到 q 形变、可含各向异性的新型长程自旋链。
