### Article

## Infinite transitivity, finite generation, and Demazure roots

An affine algebraic variety *X* of dimension ≥2 is called *flexible* if the subgroup SAut(X)⊂Aut(X) generated by the one-parameter unipotent subgroups acts *m*-transitively on reg(X) for any m≥1. In the previous paper we proved that any nondegenerate toric affine variety *X* is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property, provided the toric variety *X* is smooth in codimension two. For X=A^n with n≥2, three such subgroups suffice.

In my article, I address the factors which favor using a verb as labile (both transitive and intransitive, with no formal change required).

Haspelmath (1993) proposes that the key feature which conditions a way of marking (in)transitivity of verbs in the transitive / intransitive verb pair is the spontaneity parameter.

However, the statistical analysis of Haspelmath’s data shows that for labile / ambitransitive verb the main parameter is the lexical semantic class of the verb, not the degree of spontaneity. This lets us discover a more general principle: phenomena which are not purely grammatical, but rather lexico-grammatical (as lability) depend on lexical features, not on generalized grammatical or semantic parameters.

The paper looks into the contemporary state of the problem of decision-making and preference of some alternatives over others, discussing intransitivity of relations of superiority: one object is superior to another in a certain aspect, while the second is superior to the third and the third is superior to the first (A>B, B>C, C>A). The authors analyze two broad groups of theories and empirical studies reflecting opposite views on the nature of the relations and rationality of intransitivity of relations of superiority. The authors argue that understanding of intransitivity of superiority relations is no less important line of cognitive development than understanding of transitivity; they should be studied as complementary subjects. Thus it is necessary to study individual differences in cognitive sets with regard to transitivity/intransitivity of superiority, as well as individual characteristics of solving problems of that kind.

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.