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## Curvature of Poisson pencils in dimension three

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.

We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.

In this article, we give an explicit formula for the universal weight function of the quantum twisted affine algebra Uq(A(2)2 ). The calculations use the technique of projecting products of Drinfeld currents onto the intersection of Borel subalgebras of different types.

We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case the particle flow demonstrates an apparent jamming tendency not typical for the known updates. We solve the master equation for *N* particles on the infinite lattice by the Bethe ansatz. The non-stationary solution for arbitrary initial conditions is obtained in a closed determinant form.

The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is “simpler”. Thus it is natural to ask: can the dynamics of a bihamiltonian system be understood by studying its hamiltonian pair, without studying the associated first integrals?In this paper, the problem of stability of equilibria in bihamiltonian systems is considered and it is shown that the conditions for nonlinear stability can be expressed in algebraic terms of linearization of the underlying Poisson pencil. This is used to study stability of stationary rotations of a free multidimensional rigid body.

A specific model of statistical mechanics on graphs with vertices of valence 6 and 1. It is shown that the model under consideration is invariant with respect to certain Rozman movements if the graph is interpreted as the graph of singular points of the 2-node diagram. The approach uses the technique of cohomology of a tetrahedral complex.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.