Related papers: Solvable lattice models labelled by Dynkin diagram…
Reported are two applications of the functional relations ($T$-system) among a commuting family of row-to-row transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra $X_r$, we determine the correlation…
We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the…
It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the…
We consider the $A$ series and exceptional $E_6$ Restricted Solid-On-Solid lattice models as prototypical examples of the critical Yang-Baxter integrable two-dimensional $A$-$D$-$E$ lattice models. We focus on type I theories which are…
A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…
We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…
We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic…
The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the…
For a simple algebraic group $G$ over an algebraically closed field, we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of $G$. We use two types of labels, a binary marking and a labeling with non-negative…
Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the…
We present the solution of a linear Restricted Solid--on--Solid (RSOS) model confined to a slit. We include a field-like energy, which equivalently weights the area under the interface, and also include independent interaction terms with…
A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation…
A new set of exact scattering matrices in 1+1 dimensions is proposed by solving the bootstrap equations. Extending earlier constructions of colour valued scattering matrices this new set has its colour structure associated to non…
We investigate $N=2$ supersymmetric sigma model orbifolds of the sphere in the large radius limit. These correspond to $N=2$ superconformal field theories. Using the equations of topological-anti-topological fusion for the topological…
A linkage diagram is obtained from the Carter diagram $\Gamma$ by adding an extra root $\gamma$, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector…
The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have…
Dotted graphs are certain finite graphs with vertices of degree 2 called dots in the $xy$-plane $\mathbb{R}^2$, and a dotted graph is said to be admissible if it is associated with a lattice polytope in $\mathbb{R}^2$ each of whose edge is…
We represent Feigin's construction [11] of lattice W algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For simplest case $g=sl(2)$ we introduce whole $U_q(sl(2))$ quantum group on this lattice. We find simplest…
Over the past few years it has been discovered that an "observable" can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which…
The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been…