相关论文: Nahm's equations and root systems
Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries.…
This paper classifies the splints of the root system of classical Lie superalgebras as a superalgebraic conversion of the splints of classical root systems. It can be used to derive branching rules, which have potential physical application…
As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators…
It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a four-dimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are…
With each semisimple algebra it is possible to connect the system of interacting waves. The number of interacting fields coincides with the number of positive roots of corresponding semisimple algebra. Multisoliton solution of such kind…
We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial…
Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J.\ {\bf 66} (2025), Paper No.\ 62, 31] recently obtained a generalization of Kanade's asymptotic…
This note considers fairly general quasi-homogeneous systems of first-order nonlinear ODEs and homogeneous systems of second-order nonlinear ODEs that contain arbitrary functions of several arguments. It presents several exact solutions to…
Relative algebroids provide a framework that unifies Lie algebroids with partial differential equations. In this set of notes, we explain how relative algebroids arise from geometric problems, and give an introduction to their structural…
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential…
We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian…
The general solution of M\o ller's field equations in case of spherical symmetry is derived. The previously obtained solutions are verified as special cases of the general solution.
We show that solutions of Pentagon equations lead to solutions of the Tetrahedron equation. The result is obtained in the spectral parameter dependent case.
We construct a convenient basis for all real semisimple Lie algebras by means of an adapted Chevalley basis of the complexification. It determines rational and in fact half-integer structure constants which we express only in terms of the…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
Let $ \{d_q, \Lambda^{q} \} $ be de Rham complex on a smooth compact closed manifold $X$ over $ \mathbb{R}^3 $ with Laplacians $\Delta_{q} $. We consider operator equations, associated with the parabolic differential operators $\partial_t +…
All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…
We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three…
Continual Lie algebras are infinite-dimensional generalizations of Lie algebras with discrete root system by considering continual root systems. In this paper we establish the general relation between chain complexes and continual Lie…
We employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the $U_{q}(\hat{gl_n})$ case. With the help of linear intertwining relations involving the…