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## Lie algebras of vertical derivations on semiaffine varieties with torus actions

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper is a classification of vertical additive group actions on X under the assumption that X is proper over an affine variety. Then we establish a criterion as to when the infinitesimal generators of a finite collection of additive group actions on X generate a finite-dimensional Lie algebra inside the Lie algebra of derivations of X.

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety $\Hor$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. As an application we recover the description of the image of the moment map of $T^*X$ obtained by Knop. In our proofs we use only geometric methods which do not involve differential operators.

The author draws a linguistic view of the intimate communication of lovers, determines the specificity of this type of interpersonal communication and considers some features of the linguistic code of lovers: the group of pet names, the formula of love declarations. In the analysis of endearment sememe the author determines their semantic potency, proves value preferences of society in mutual renaming, highlights derivational features of vocatives. Syntactical features of speech lovers are conditioned by the aim of mutual emotional impact, they are syntactic repetitions, which increase the emotional density of the utterance.

Lie theory, inaugurated through the fundamental work of Sophus Lie during the late nineteenth century, has proved central in many areas of mathematics and theoretical physics. Sophus Lie’s formulation was originally in the language of analysis and geometry; however, by now, a vast algebraic counterpart of the theory has been developed. As in algebraic geometry, the deepest and most far-reaching results in Lie theory nearly always come about when geometric and algebraic techniques are combined. A core part of Lie theory is the structure and representation theory of complex semisimple Lie algebras and Lie groups, which is an exemplary harmonious field in modern mathematics. It has deep ties to physics, and many areas of mathematics, such as combinatorics, category theory, and others. This field has inspired many generalizations, among them the representation theories of affine Lie algebras, vertex operator algebras, locally finite Lie algebras, Lie superalgebras, etc. This volume originates from a pair of sister conferences titled “Algebraic Modes of Representations” held in Israel in July 2017. The first conference took place at the Weizmann Institute of Science, Rehovot, July 16–18, and the second conference took place at the University of Haifa, July 19–23. Both conferences were dedicated to the 75th birthday of Anthony Joseph, who has been one of the leading figures in Lie Theory from the 1970s until today. The conferences were supported by the United States–Israel Binational Science Foundation and the Chorafas Institute for Scientific Exchange (Weizmann part) and by the Israel Science Foundation (Haifa part). Joseph has had a fundamental influence on both classical representation theory and quantized representation theory. A detailed description of his work in both areas has been given in the articles by W. McGovern and D. Farkash–G. Letzter in the volume “Studies in Lie theory,” Progress in Mathematics, vol. 243, Birkhauser. Concerning Joseph’s contribution to classical representation theory, it is impossible not to mention his classification of primitive ideals of the universal enveloping algebra of sl(n). The essential new ingredient here is the introduction of a partition of the Weyl group into left cells, corresponding to the Robinson map from the symmetric group to the standard Young tableaux. Joseph further extended this result to other simple Lie algebras using similar techniques, and this has since then become a powerful tool in Lie theory. As for quantized representation theory, Joseph’s monograph “Quantum Groups and Their Primitive Ideals,” Ergebnisse der Mathematik und Ihrer Grenzgebiete, 3rd series, vol. 29, has had a fundamental influence over the field since its appearance in 1995. The present volume contains 14 original papers covering a broad spectrum of current aspects of Lie theory. The areas discussed include primitive ideals, invariant theory, geometry of Lie group actions, crystals, quantum affine algebras, Yangians, categorification, and vertex algebras. The authors of this volume are happy to dedicate their works to Anthony Joseph.

Fascinating and surprising developments are taking place in the classification of algebraic varieties. Work of Hacon and McKernan and many others is causing a wave of breakthroughs in the Minimal Model Program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the field. Inspired by this exciting progress, the editors organized a meeting at Schiermonnikoog and invited leading experts to write papers about the recent developments. The result is the present volume, a lively testimony of the sudden advances that originate from these new ideas. This volume will be of interest to a wide range of pure mathematicians, but will appeal especially to algebraic and analytic geometers.

Macedonian possesses a rich system of affixes, some of them are considered to be completely synonymous. This is the case for diminutive suffixes serving to build diminutives from feminine nouns: there are four suffixes, three of them are of protoslavonic origin (namely, suffixes -ka, -ca, -ica) and one - -ichka - is considered to be of a more recent descent. As those suffixes have virtually the same range of meanings, a question arises as to their rivalry. Our research revealed that not all of the suffixes in question have the potential to combine with all the stem types. Their combinability is restrained by their morphonological properties and these constraints form a system of suffixal distribution. Having studied a corpus of over 500 diminutives, we came to the conclusion that suffix -ka primarily serves as a substitute suffix in the cases where other suffixes cannot combine, suffix -ca can solely combine with a single stem type (stems of feminine nouns ending with a consonant, former i-stems), and suffixes -ichka and -ica, possessing similar properties, are engaged in a rivalry which apparently is settled in favour of the former.

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.

Preface

The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.

Recently, there have been remarkable developments in algebraic geometry and related fields, especially, in the area of (birational) automorphism groups and algebraic group actions.

The purpose of this workshop was to provide the experts and young researchers with the opportunities to interact in the fields of affine and complete algebraic geometry, group actions and commutative algebra related to the topics listed below as well as to publicize the new results. We are confident that these purposes were achieved by the endeavors of the participants.

The main topics of the workshop were the following:

Algebraic varieties containing An-cylinders; Algebraic varieties with fibrations; Algebraic group actions and orbit stratifications on algebraic varieties; Automorphism groups and birational automorphism groups of algebraic varieties.There were 19 talks on the above and related topics by experts from the viewpoints of (affine) algebraic geometry, transformation groups, and commutative algebra. Inspired by the talks, there were active discussions and communication among participants during and after the workshop.

The present volume is the proceedings of the workshop and contains 15 articles on the workshop topics. We hope that this volume will contribute to the progress in the theories of algebraic varieties and their automorphism groups.

The workshop was financially supported by the RIMS and Grant- in-Aid for Scientific Research (B) 24340006, JSPS. We wish to thank all those who supported us in organizing the workshop and preparing this volume.

June, 2016

Kayo Masuda, Takashi Kishimoto, Hideo Kojima, Masayoshi Miyanishi, Mikhail Zaidenberg

All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere.

Exploring Bass’ Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass’ Triangulability Problem in the affirmative. To this end we prove a theorem on invariant subfields of 1-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.