### Article

## Flag varieties, toric varieties, and suspensions: Three instances of infinite transitivity

We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.

Ths is Proceedings of a Conference on Affine Algebraic Geometry that was held at Osaka Umeda Campus of Kwansei Gakuin University during the period 3--6 March, 2011 on the occasion of the seventieth birthday of Professor Masayaoshi Miyanishi.

Let K be a ﬁeld and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient ﬁeld of A then RDerA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerA of rank k over R (i.e. such that dimR RL = k), then the derived length of L is at most k and L is ﬁnite dimensional over its ﬁeld of constants. In case of solvable Lie algebras over a ﬁeld of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector ﬁelds with polynomial, rational, or formal coeﬃcients.

This volume is dedicated to Professor M. Miyanishi on the occasion of his 70th birthday.

The paper continues research into words denoting everyday life objects in the Russian language. This research is conducted for developing a new encyclopedic thesaurus of Russian everyday life terminology. Working on this project brings up linguistic material which leads to discovering new trends and phenomena not covered by the existing dictionaries. We discuss derivation models which gain polularity: clipped forms (komp < komp’juter ‘computer’, nout < noutbuk ‘notebook computer’, vel < velosiped ‘bicycle’, mot<motocikl ‘motorbike’), competing masculine and feminine con- tracted nouns derived from adjectival noun phrases (mobil’nik (m.) / mo- bilka (f.) < mobil’nyj telefon (m.) ‘mobile phone’, zarjadnik (m.) / zarjadka (f.) < zarjadnoe ustrojstvo (n.) ‘AC charger’), hybrid compounds (plat’e- sviter ‘sweater dress’, jubka-brjuki ‘skirt pants’, shapkosharf ‘scarf hat’, vilkolozhka ‘spork, foon’). These words vary in spelling and syntactic behav- iour. We describe a newly formed series of words denoted multifunctional objects: mfushkaZ< MFU < mnogofunkcional’noe ustrojstvo ‘MFD, multi- function device’, mul’titul ‘multitool’, centr ‘unit, set’. Explaining the need to compose frequency lists of word meanings rather than just words, we of- fer a technique for gathering such lists and provide a sample produced from our own data. We also analyze existing dictionaries and perform various experiments to study the changes in word meanings and their comparative importance for speakers. We believe that, apart from the practical usage for our lexicographic project, our results might prove interesting for research in the evolution of the Russian lexical system.

Rapid and automatic processing of grammatical complexity is argued to take place during speech comprehension, engaging a left-lateralized fronto-temporal language network. Here we address how neural activity in these regions is modulated by the grammatical properties of spoken words. We used combined magneto- and electroencephalography to delineate the spatiotemporal patterns of activity that support the recognition of morphologically complex words in English with inflectional (-s) and derivational (-er) affixes (e.g., bakes, baker). The mismatch negativity, an index of linguistic memory traces elicited in a passive listening paradigm, was used to examine the neural dynamics elicited by morphologically complex words. Results revealed an initial peak 130-180 ms after the deviation point with a major source in left superior temporal cortex. The localization of this early activation showed a sensitivity to two grammatical properties of the stimuli: (1) the presence of morphological complexity, with affixed words showing increased left-laterality compared to non-affixed words; and (2) the grammatical category, with affixed verbs showing greater left-lateralization in inferior frontal gyrus compared to affixed nouns (bakes vs. beaks). This automatic brain response was additionally sensitive to semantic coherence (the meaning of the stem vs. the meaning of the whole form) in left middle temporal cortex. These results demonstrate that the spatiotemporal pattern of neural activity in spoken word processing is modulated by the presence of morphological structure, predominantly engaging the left-hemisphere's fronto-temporal language network, and does not require focused attention on the linguistic input.

This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations.

The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.

More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.

A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.

The author shows the productive derivation and new lexical units formation methods in stock and currency market participants slang by the example of the lexico-thematic group "Stock market goods, their lists, properties, exchange documents and rules".

The article deals with endearing names in the German language. The author defines the semantic potency of endearing sememes and substantiates value preferences of society in mutual renaming. The article also covers word-building features of vocatives, and the author analyzes possible combinations of elements in their composition.

We explore algebraic subgroups of the Cremona group Cn over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of Cn that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on An is intimately related to rationality of the invariant fields and, for tori, is equivalent to it. We prove stable linearizability of these actions and show the existence of nonlinearizable actions among them. This is applied to exploring maximal tori in Cn and to proving the existence of nonlinearizable, but stably linearizable elements of infinite order in Cn for n ? 5. Then we consider some subgroups J (x1, . . . ,xn) of Cn that we call the rational de Jonquie`res subgroups. We prove that every affine algebraic subgroup of J (x1, . . . ,xn) is solvable and the group of its connected components is Abelian. We also prove that every reductive algebraic subgroup of J (x1, . . . ,xn) is diagonalizable. Further, we prove that the natural rational action on An of any unipotent algebraic subgroup of J (x1, . . . ,xn) admits a rational cross-section which is an affine subspace of An. We show that in this statement “unipotent” cannot be replaced by “connected solvable”. This is applied to proving a conjecture of A. Joseph on the existence of “rational slices” for the coadjoint representations of finite-dimensional algebraic Lie algebras g under the assumption that the Levi decomposition of g is a direct product. We then consider some overgroup J^ (x1, . . . ,xn) of J (x1, . . . ,xn) and prove that every torus in J^ (x1, . . . ,xn) is linearizable. Finally, we prove the existence of an element g ? C3 of order 2 such that g does not lie in every connected affine algebraic subgroup G of C?; in particular, g is not stably linearizable.

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.