Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

In this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$ g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone $$C_x$$ C x . Assuming that $${\mathfrak g}$$ g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying $$[h,x]=x$$ [ h , x ] = x for which $$C_x$$ C x pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$ ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.

Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation

Using the theory of cycle sets and braces, non-degenerate indecomposable involutive set-theoretic solutions to the Yang–Baxter equation are classified in terms of their universal coverings and their fundamental group. The universal coverings are characterized as braces with an adjoint orbit generating the additive group. Using this description, all coverings of non-degenerate indecomposable cycle sets are classified. The method is illustrated by examples.

Geometric aspects of self-adjoint Sturm–Liouville problems

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

ON THE EXACTNESS OF KOSTANT–KIRILLOV FORM AND THE SECOND COHOMOLOGY OF NILPOTENT ORBITS

We give a criterion for the Kostant–Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebra.

ON THE STABILITY PROBLEM FOR THE $\mathfrak{so}(5)$ FREE RIGID BODY

In the general case of the [Formula: see text] free rigid body, we will give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of [Formula: see text], we prove that there are 15 coordinate-type Cartan subalgebras which on a regular adjoint orbit give 15 Weyl group orbits of equilibria. These coordinate-type Cartan subalgebras are the analogs of the three axes of equilibria for the classical rigid body on [Formula: see text]. The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are 10 other continuous families of equilibria.

INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH TWO ISOTROPY SUMMANDS

Let M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices

Let n = 2m be even and denote by Spn(F) the symplectic group of rank m over an infinite field F of characteristic different from 2. We show that any n × n symmetric matrix A is equivalent under symplectic congruence transformations to the direct sum of m × m matrices B and C, with B diagonal andC tridiagonal. Since the Spn(F)-module of symmetric n × n matrices over F is isomorphic to the adjoint module spn(F), we infer that any adjoint orbit of Spn(F) in spn(F) has a representative in the sum of 3m − 1 root spaces, which we explicitly determine.

Quantization of the 4-Dimensional Nilpotent Orbit of Sl(3, ℝ)

We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit Oℝof SL(3,ℝ). The space of quantization is the space of holomorphic functions on 𝕔2- {0}) which are square integrable with respect to a signed measure defined by a Meijer G-function. We construct the quantization out a non-flat Kaehler structure on 𝕔2- {0}) (the universal cover of Oℝ) with Kaehler potential ρ |z|4.

ANISOTROPIC AFFINE ALGEBRA AND ANISOTROPIC WZW MODEL

Invariant action for a new kind of (anisotropic) WZW model has been constructed using the co-adjoint orbit approach and the affine anistropic algebra suggested by Sidorenko. The Alekseev-Shatashvili type of equation is used to construct the corresponding symplectic form. The present model is similar in many aspects to the WZW model on a Riemann surface.