Related papers: On sl3 Knizhnik-Zamolodchikov equations and W3 nul…
Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of…
In the present paper, using a modification of the method of vector fields $Z_i$ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-St\"ackel variable separation for three-dimensional extension of the…
In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra.…
The $SL(3)$ Kuznetsov formula exists in several versions, and has been employed with some success to study automorphic forms on $SL(3)$. In each version, the weight functions on the geometric side are given by multiple integrals with…
It is shown, at the level of the classical action, that the Wess-Zumino-Witten-Novikov model is equivalent to a combined BF theory and a Chern-Simons action in the presence of a unique boundary term. This connection relies on the techniques…
We present a new conjecture relating the minimal polynomial solution of the level-one $U_q(\frak{sl}(2))$ quantum Knizhnik-Zamolodchikov equation for generic values of $q$ in the link pattern basis and some $q$-enumeration of Totally…
A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3) = so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of…
Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the…
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the…
We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. This class of…
In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts \cite[Conjecture 0.3]{H} stating…
We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field k with three generators x,y,z given by three relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0, where p,q,r are parameters from the…
Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite…
We show how the method of separation of variables can be used to construct integrable models corresponding to curves describing vacuum structure of four-dimensional ${\cal N} = 2$ SUSY Yang-Mills theories. We use this technique to construct…
We study MNLS related to the D.III-type symmetric spaces. Applying to them Mikhailov reduction groups of the type $\mathbb{Z}_r\times \mathbb{Z}_2$ we derive new types of 2-component NLS equations. These are {\bf not} counterexamples to the…
The quasiclassical asymptotics of the Knizhnik-Zamolodchikov system is studied. Solutions to this system in this limit are related naturally to Bethe vectors in the Gaudin model of spin chains.
We discuss the hypergeometric solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation at level zero and show that they give all solutions of the qKZ equation. We completely describe linear relations between the hypergeometric…
We show that the $SU(N)$, level-1 Wess-Zumino-Witten conformal field theory provides a natural realization of the Yangian $Y(sl_N)$ for $N\geq 3$. We also construct a hamiltonian $H_2$ which commutes with the Yangian generators and study…
In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with…
Quantum mechanical systems whose symmetry is given by $\mathbb{Z}_2^3$-graded version of superconformal algebra are introduced. This is done by finding a realization of a $\mathbb{Z}_2^3$-graded Lie superalgebra in terms of a standard Lie…