Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from B. Feigin et al. (1994) [17] to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P¹ with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.     

The quantum Gaudin system

In this paper, we explicitly construct the quantum $\mathfrak{g}\mathfrak{l}_n $ Gaudin model for general n and for a general number N of particles. To this end, we construct a commutative family in $U(\mathfrak{g}\mathfrak{l}_n )^{ \otimes N} $ . When passing to the classical limit (which is the projection onto the associated graded algebra), our family gives the entire family of classical Gaudin Hamiltonians. The construction is based on the special limit of the Bethe subalgebra in the Yangian $Y(\mathfrak{g}\mathfrak{l}_n )$ .     

Primitive wonderful varieties

Mathematische Zeitschrift - We complete the classification of wonderful varieties initiated by D. Luna. We review the results that reduce the problem to the family of primitive varieties, and...     

Gaudin magnet and off-mass-shell Bethe wave functions

The general Bethe equation (or off-shell Bethe ansatz equation) is proved for the Gaudin magnet for a broad class of simple complex Lie algebras. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 13–28, October, 1997.     

The Elliptic Gaudin System with Spin

We study the algebraic–geometric structure of the elliptic Gaudin two-puncture model previously obtained. We identify this system with the system of pole dynamics of finite-gap solutions of the matrix Davey–Stewartson equation. We also obtain the action–angle variables and construct explicit solutions of this system in terms of theta functions. We discuss the geometry of degenerations of this system.     

Automorphisms of wonderful varieties

Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group Aut⁰(X) and we calculate Luna’s invariants of X under its action.     

Independence of subalgebras and partitions

In the paper conditions are considered under which a given system of subalgebras becomes independent after passing from the basic probability measure to an absolutely continuous one.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 30–38, 1979.     

The argument shift method and the Gaudin model

We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U ( ) of a semisimple Lie algebra . This family is parameterized by finite sequences μ, z ₁, ..., z _n , where μ ∈ ^* and z _i ∈ ℂ. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n = 1, the corresponding commutative subalgebras in the Poisson algebra S( ) were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional -modules and the Gaudin model. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 40, No. 3, pp. 30–43, 2006 Original Russian Text Copyright ? by L. G. Rybnikov     

Maximal subalgebras of the general linear Lie algebra containing Cartan subalgebras

Let gln(R) be the general linear Lie algebra of all n × n matrices over a unital commutative ring R with 2 invertible, dn(R) be the Cartan subalgebra of gln(R) of all diagonal matrices. The maximal subalgebras of gln(R) that contain dn(R) are classified completely.     

Elementary operators and invariant subalgebras

We provide an example of an elementary operator which leaves invariant a nest algebra but which cannot be written as a finite sum of multiplications each of which leaves the nest algebra invariant. We also prove that the given operator lies in the completely bounded norm closure of the linear span of the multiplications which leave the nest algebra invariant.

     

Inverse closed ultradifferential subalgebras

In previous work we have shown that classical approximation theory provides methods for the systematic construction of inverse-closed smooth subalgebras. Now we extend this work to treat inverse-closed subalgebras of ultradifferentiable elements. In particular, Carleman classes and Dales–Davie algebras are treated. As an application the result of Demko, Smith and Moss, and Jaffard on the inverse of a matrix with exponential decay is obtained within the framework of a general theory of smoothness.     

Relative subalgebras of MV-algebras

Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\). These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra.     

Completely reducible Lie subalgebras

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H ⊂ G. In this paper, we give a notion of G-complete reducibility—G-cr for short—for Lie subalgebras of Lie(G), and we show that if the closed subgroup H ⊂ G is G-cr, then Lie(H) is G-cr as well.