The aim of this project is to investigate reductive groups over arbitrary fields and asscociated linear and homological structures. Related linear structures are for example Lie algebras, Azumaya algebras, Jordan algebras, associative algebras with involution, quadratic and hermitean forms, related homological structures are those (co)homological functors and tools which are used or can be used to study those objects. Among them are the Galois or etale cohomology ring (e.g of a field or of a scheme), the cohomology functors describing Lie algebras, the Witt ring describing quadratic forms, the functors of algebraic K-theory and so on. More specialized objects in this context are e.g. the Brauer group of a field or of a variety, which is a particular instance of the functors being subsummized in the Galois cohomology ring of that object.
3.1.3 Background & Justification for Undertaking the Project
Linear algebraic groups have been investigated for a very long time. In particular semisimple Lie groups or Lie algebras over the complex field have been studied and classified by Killing as early as 1888, by Cartan, Weyl, v.d. Waerden, Witt, and in particular, Chevalley, who discovered their integral structure and showed that these groups exist as schemes over arbitrary fields and can be classified by Killings scheme over arbitrary algebraically closed fields as well.
This discovery of Chevalley (in the 60ths of this century) initiated many branches of research: K-theory originates from this, the theory of buildings (by Tits and others) was developed, Kac-Moody Lie algebras and Kac-Moody groups were detected, representation theory of linear groups was studied and developed.
All these theories, however, are based on the facts that the reductive groups involved contain unipotent elements, i.e. are isotropic.
One can say that in the theory of semisimple groups, the first half of our century was devoted to the investigation of these groups and structures over the complex numbers, while in the second half one studied those groups over arbitrary fields, but just in those cases, in which unipotent elements in the groups occured already over the field of definition. For example, the subgroup structure of an arbitrary semisimple group over some field is quite satisfactorily described by the general Bruhat decomposition, but only up to its so called "anisotropic kernel".
The most extreme and also the most general case of a semisimple group is the "anisotropic" case, when the group does not contain unipotent elements over its field of definition, like the special orthogonal group of an anisotropic quadratic form.
Very little is known in general about these groups. E.g., the Tits building of such a group is trivial, as well as its Bruhat decomposition. In a survey article on linear algebraic groups, T. A. Springer writes in 1984:
"The most difficult part of a classification of reductive k-groups is the classification of semi-simple anisotropic k-groups ... A complete classification of all anisotropic k-groups seems out of reach." (Perspectives in Math., Anniv. of Oberwolfach, 1984, p. 477)
The situation still is similar. However, parallel to the development of the theory of reductive groups, other branches of mathematics have been developed, and partially formed their own "research industries", which can certainly be helpful in this context.
Linear algebraic groups occur as automorphism groups of certain linear structures as mentioned above, and these structures have been traditionally studied by (co-) homological tools of many kinds.
It ist the goal of this project to bring together experts of these areas in order to gain understanding of the interaction between the groups and the related structures in order to finally get some better understanding of "anisotropic situations" of various kinds.
Of course there have been classical approaches to these questions over special fields. Anisotropic groups over the complex field are just the compact Lie groups, which are well understood. Over local and global fields, those groups have been studied, and so called "local-global" or "Hasse"-principles - mostly based on class field theory - are the method of choice here to get a deeper understanding.
For slightly more complicated fields (function fields over curves or surfaces, say) the situation is not understood. The subprojects initiated by the Minsk and Gent team of this proposal undertake to study situations like this. Galois cohomological methods for arbitrary fields will be developed and applied by the projects of Bielefeld and St.-Petersburg, as well as by joint projects of Besancon, Bielefeld and Minsk.
Quadratic forms as a key ingredient for studying orthogonal groups are, over arbitrary fields, investigated by scientist from St. Petersburg/Bielefeld/Besancon,
Hochschild homology will be studied by St. Petersburg, Tbilisi and Bielefeld, and Lie algebras respective right symmetric algebras together with their cohomology are studied by the Almaty group.
It is expected that the research in this project will bring fruitful results for all fields and disciplines involved, like linear algebraic groups, Azumaya algebras, quadratic forms, Brauer groups, Lie algebras and associated (co-)homology theories.
In this sense, the project is of inner mathematical interdiciplinary type.
Some of the scientists involved here have already fruitfully cooperated in the Intas project
This project ends on December 31, 1999.
3.1.4 Scientific Description
3.1.4.1 Research Programmme
The project is broken into tasks to be performed by single teams or by a subgroups of them. Typically, mathematical results will be written down in manuscripts and submitted to refereed international journals for publication. Since publication may take some time, the results will be made available as preprints via some web preprint server, for example in the server on
Significant milestones will particular be reported on the occasion of the annual workshops, which will take place at the end of the first, second and third year of the project in one of the INTAS partner universities.
3.1.4.2 Deliverables, Exploitation & Dissemination of Results
The details of the deliverables can be obtained from the following task by task description:
P1 Bielefeld:-----------------------------------------------------------
Properties of algebraic groups and related structures over arbitrary fields. Generic splitting properties for algebraic groups and quadratic forms have recently been investigated by memeber sof this group (see given Scientific References). On the basis of these achievements the following problems will be investigated.
Tasks: Research in the generic theory of quadratic and hermitean form and cohomological descriptions of certain anisotropic groups.
T1.1 Research on quadratic forms and excellent groups (mon 1-12, jointly with St. Petersburg, Kersten, Rehmann, Panin),
T1.2 Cohomological properties of orthogonal groups (mon 12-24, Rehmann) The output will be a Galois cohomological description of certain anisotropic orthogonal grooups whic is uniform for all fields.
T1.3 Generic theory of quadratic and hermitean forms and cohomological properties of their unitary groups (mon 25-36, Kersten, Rehmann) The output should be an attempt to understand cohomological properties of unitary groups.
T1.4 Cooperation with Team 6 (Tbilisi) on Hochschild homology (Waldhausen)
T1.5 Workshop for the whole project in Bielefeld (one week in 22th month) This workshop will represent the scientific output of the second year of the project.
T1.6 Description of normal structure of simply connected algebraic groups defined over number fields (V. Chernousov, mon 1-36). The main purpose is to prove Platonov--Margulis' Conjecture for trialitarian groups and exceptional groups of type $E_6$.
P2 Besancon: -----------------------------------------------------------
Galois cohomological investigations of algebraic groups, hermitean and quadratic forms have been successfully studied in the past by this team (cf. Scientific References). Therefore this team is enabled to successfully untertake the following:
Tasks: Research in the splitting properties of quadratic forms over arbitrary fields and cohomolgical investigations.
T2.1 Splitting behavior of quadratic forms, embedding of quadratic forms in Pfister forms (Izhboldin, Hoffmann) (mon 1-36)
T2.2 Cohomological properties of trialitarian groups (E. Bayer and V. Chernousov, mon 1-36). The main purpose is to prove Serre's Conjecture II and the Hasse principle.
T2.3 Workshop for the whole project in Besancon (one week in 12th month) This workshop will represent the scientific output of the first year of the project.
P3 Gent: -----------------------------------------------------------------
Division algebras and quadratic forms over function fields of curves over local fields In recent years there has been some progress in the study of division algebra's, respectively quadratic forms, over function fields of curves C over local fields. V.I. Yanchevskii (Team 5) started a large project with the aim to obtain an explicit description of the elements in the Brauergroups Br(C) and Br(K). We refer to work of him, Rehmann and Margolin. The research group in Gent and Yanchevskii studied in the past years the kernel of the natural map Br(k) to Br(k(C) for certain radical coverings C of the projective line. This research is not finished and the continuation forms part of the program proposed.
Tasks:
T3.1: General hyperelliptic curves: Two classes, one to be studied in Gent, the other in Minsk (Mon 1-18)
T3.2: Prove: Index of the curve = the cardinality of Br(k(C)/k) (Mon 1-18).
T3.3: Structure of elements in Br(K(C)) (Mon 19-27).
T3.4: Investigate u-invariant of function fields over local fields including function fields over power series fields over real closed fields. (Mon 19-27). See also task T7.3.
T3.5: Workshop at Gent, Report, preparing papers on T3.3, T3.4 This workshop will represent the scientific output of the third year of the project.
P4 Almaty: -----------------------------------------------------------------
Research Project on Cohomologies of Right-Symmetric Algebras and Applications: The Tasks are directed and supervised by A. S. Dzhumadil'daev.
Tasks: T4.1: (S. Abdukasimova): To develop an extension theory of Novikov and Leibniz algebras and to classify simple Leibniz algebras connected with Zassenhaus (Witt) algebras Subtasks: T4.1.1 (Month 1-12): Constructing of cocycles; T4.1.2 (Month 13-24.) Proving that obtained list of cocycles is complete. T4.1.3 (Month 25-36). Classification of simple Leibniz algebra of rank 1. Results will be published according to the progress. At the end there should be a complete description of all simple Leibniz algebras with a Lie factor isomorphic to Zassenhaus (Witt) algebras.
T4.2: Polynomial identities for Witt algebras (I.Irgalieva and E.Bekmuchambetov):
T4.2.1 (I.Irgalieva):T4.2.1.1 (Month 1-12): Find prolongations for two-parametrical local deformations of Witt algebra of rank 1. T4.2.1.2 (Month 13-24.): Prove that three-parametrical deformations of right-symmetric Witt algebra $W_1$ have obstructions T4.2.1.3 (Month 25-36): Find isomorphism classes for right-symmetric deformations of W_1(m). Expected results: Description of Witt algebras as right-symmetric algebras in terms of local deformations. Publication at the end of the 3. year.
T4.2.2 (E.Bekmuchambetov): T4.2.2.1(Month 1-24): Calculations for polynomial identity using exterior forms and Grassman algebra calculus. T4.2.2.2(Month 24-36). Writing programs on Mathematica or Maple for generalised commutators on the space of linear differential operators. Results should appear in a master thesis in the third year and will be included in the paper dedicated to polynomial identities of differential operators. Here computer equipment is necessary.
T4.3: Irreducible representations, cohomologies and spectra of Casimir operators T4.3.(S.Ibraev and A. Kungojin): T4.3.1 (Month 1-12): Construct Casimir elements and eigenvalue functions. T4.3.2 (Month 13-24): Describe all weights with trivial Casimir spectra. T4.3.3 (Month 25-36): Construct irreducible modules with weights with trivial Casimir spectra and calculate cohomology groups. Results should be a description of the spectra of Casimir operators as well as some facts about the cohomology of classical Lie algebras. They will be published according to the research progress.
P5 Minsk: ------------------------------------------------------------------------- Tasks:
T5.A: Investigation of the structure of division algebras defined over function fields of p-adic curves (especially unramified division algebras and algebras with involutions) in connection with obtaining further applications in linear algebraic groups.
T5.A1: It is supposed that the a structure of the important classes of the above division algebras will be described.
T5.A2: The above description will be used for description of the multiplicative structure of of such division algebras.
T5.A3: Results from T5.A1 and T5.A2 will be applied to description of groups of R-equivalences, excellence property , etc. Time schedule: T5.A1: 1-12month, publications T5.A2: 13-24month, publicatios T5.A3. 25-36month, publications Visits to Bielefeld., cooperation with Bielefeld team.
Partial cases of well known conjectures of Lang and Pfister state that for a field F of transcendence degree 2 over real closed field, the u-invariant of F is 6. Unfortunately, both conjectures still aren't proved despite of numerous attempts to do this. We suppose to attack similar problems which seem closely related to the conjectures above (for example we look at the case of a function field F of curves defined over the formal power with real coefficients). The above problems are closely related to some problems in the theory of Brauer groups which we also suppose to solve. As a possible applications one can keep in mind for example the following: If the field F is of virtual cohomological dimension 2, then the group of R-equivalences of SL_n(A) is trivial.
T5.B: Computation of u-invariants of some special fields. Obtaining applications for linear algebraic groups. (See Team 3)
T5.B1. Investigation of the problem of computation of the u-invariant for some special fields (for example for function fields of curves defined over the formal power series with coefficients in real closed field) in connection with related problems in the theory of Brauer groups.
T5.B2. Obtaining applications of the above results to linear algebraic groups. Time schedule: T5.B1. 1-24month, publications T5.B2. 25-36month, publications Visits to Bielefeld and Gent, cooperation with Bielefeld and Gent teams.
T5.C: Investigation of the structure of unramified division algebras over function fields of some arithmetical surfaces with further applications to linear algebraic groups.
In many problems of arithmetic geometry one is lead to the investigation of the Brauer group of some smooth projective algebraic variety. The important role in such problems very often play the knowledge of unramified division algebras representing its elements. The general description of such algebras is very difficult for varieties of arbitrary dimensions. For curves over arithmetic fields we have numerous results giving in some cases more or less satisfactory descriptions. The next step is the case of surfaces defined over arithmetic fields. We investigate the case of some special surfaces (e.g. elliptic surfaces over local fields) and hope to find some applications to linear algebraic groups.
T5.C1. Obtain a description of the structure of the Brauer group of some interesting classes of arithmetical surfaces.
T5.C2. To find a complete list of representing unramified division algebras (especially for 2-torsion).
T5.C3. Obtain applications of the above results to linear algebraic groups (similar to T5.A3). Time schedule: T5.C1. 6-12month, publications T5.C2. 13-30month, publications T5.C3. 31-36month, publications.
T5.D. Investigation of indicies of hyperelliptic curves defined over local fields
The index of a curve C defined over a local field k coincides with cardinality of the kernel of the natural map from the Brauer group of k to the Brauer group of the function field k(C) of a curve C (Roquette,Lichtenbaum). Its computation is one of the important problems in arithmetic geometry (see e.g.B.Poonen, M.Stoll, The Cassels-Tate pairing in polirized abelian varieties, Isaak Newton Institute for Mathematical sciences, preprint NI98005-AMG).Taken in its generality this problem is very dificult an strongly depends on the curve under consideration. Some results on this problem were obtained recently by Colliot-Thelene and van Geel-Yanchevskii. We assume to investigate the above problem for hyperelliptic curves (and more generally for cyclic coverings of projective line). As an application we expect to obtain some examples of interesting division algebras which are useful for the solution of some problems in the theory of linear algebraic groups.
T5.D1. Computation of indicies of hyperelliptic curves defined over local fields with defining affine equations y^2=f(x) for irreducible f(x).
T5.D2. Generalization of the above results to case of arbitrary f(x).
T5.D3. Obtaining similar results to cyclic coverings of the projective line. Time schedule: T5.D1. 1-12month, publications T5.D2. 13-24month, publications T5.D3. 25-36month, publications. Visits to Gent, cooperation with Gent team.
P6 Tbilisi -----------------------------------------------------------------------
Tasks: Dealing with the investigation of topological Hochschild cohomology. Some background: Coefficients of MacLane cohomology of integers can be defined to be functors from finitely generated free abelian groups to abelian groups. Any commutative group scheme G gives rise to such a functor F(G). We are going to calculate MacLane cohomology of Z with coefficients in F(G) when G is the group scheme corresponding to Witt vectors over a perfect field. We are going to describe polynomial functors in terms of pre-Mackey functors. This will give an application in characteristic free representation theory. The MacLane obstruction in the third MacLane cohomology of a ring with coefficients in a bimodule will be studied with the aim of giving it an interpretation in terms of categories with ring structures up to isomorphism. It is expected that there is an interpretation of third cohomology groups of general algebraic theories, and discrepancy between the third Shukla and MacLane cohomology measured by the MacLane obstruction can be given explanation in terms of this interpretation. Each convex lattice polytope P determines a graded k-algebra whose generators are the lattice points of P, all of degree 1, and the relations are binomials in these generators reflecting affine dependencies between the lattice points. Assigning to a k-vector space its symmetric algebra, i.~e. the polytopal algebra of a unit simplex, one obtains a full embedding of the category of finite-dimensional vector spaces into the category of polytopal algebras and graded algebra homomorphisms. The goal is to develop a new kind of K-theory, with the category of polytopal k-algebras playing the role of the category of finite-dimensional k-vector spaces. The investigation of A-infinity algebras, and in particular of the multiplicative structure of the Hochschild complex of a homotopy commutative DGA. In particular, construction of an explicit A-infinity algebra structure on the Hochschild complex of a homotopy Gerstenhaber algebra.
T6.1: Calculation of the MacLane cohomology of integers with coefficients in functors determined by Witt vectors.
T6.1a (Subtask): Investigation of behavior of the MacLane cohomology with respect to functors obtained from short exact sequences of abelian group schemes.
T6.2: Interpretation of the obstruction in the third MacLane cohomology in terms of multiplicative Picard categories.
T6.2a (Subtask): Definition and properties of singular extensions of categories with ring structure up to isomorphism.
T6.3: Construction and investigation of a new kind of K-theory of a ring based on the category of polytopal algebras and graded algebra homomorphisms.
T6.4: Investigation of the multiplicative structure of the Hochschild complex of a homotopy commutative differential graded algebra.
T6.4a: (Subtask): Construction of an explicit A-infinity algebra structure on the Hochschild complex of a homotopy Gerstenhaber algebra
Results of the subtasks will be published as preprints by the end of the first year.
P7 St. Petersburg ----------------------------------------------------------------------
T7.1 Stable K-theory and topological Hochschild homology (Suslin, Skorichenko). A well-known theorem due to Dundas and Mc Carty shows that topological Hochschild homology coincides with stable K-theory in case of additive functors. A purely algebraic proof of this result was given recently by E.Friedlander and A.Suslin. A.Suslin generalized this result to the case of arbitrary finite functor, assuming that the base field is finite. We expect to prove that topological Hochschild homology coincides with stable K-theory for arbitrary finite functors over arbitrary base rings.
T7.1.1 (Mon 1-24) Research
T7.1.2 (Mon 25-36) Further study, writing reports and papers
T7.2 Motivic homology and K-theory of low dimensional fields (Pushin, Suslin). We expect to give a clean and direct computation of motivic cohomology and K-theory for fields of cohomological dimension one ( and hopefully of cohomological dimension two)
T7.2.1 (Mon 1-18) fields of cohomological dimension 1
T7.2.2 (Mon 19-36) fields of cohomological dimension 2
T7.3 The Kaplansky problem on u-invariant (Izhboldin) The famous Kaplansky conjecture (1953) stated that U-invariant of a field is either a power of 2 or infinite. In 1992 Merkurev disproved this conjecture constructing fields with U-invariant equals to a given even number. After that it is still mysterious are there a field with odd U-invariant or not. We are going to construct a field with U-invariant 9
T7.3.1 (Mon 1-12) Fields with odd u-invariant
T7.3.2 (Mon 13-36) Splitting behavior of quadratic forms, various visits of Bielefeld
T7.4 The Gersten conjecture in K-theory (Panin) The Gersten conjecture in K-theory was solved by Quillen in 1973 in geometric case. Since that time different versions of Gersten conjecture for other cohomology theories were proved by Bloch-Ogus, Gabber and others. But there was no progress in the original Gersten conjecture. We are going to solve the original Gersten conjecture in equi-characteristic case. T7.4.1 (Mon 1-24) Research T7.4.2 (Mon 25-36) Generalizations, writing reports and papers
T7.5 A Conjecture of Grothendieck (Rationally trivial principal homogeneous spaces over a reductive group are locally trivial in smooth case). Study various versions of purity problems for wide class of groups (Mon 1-36). Reports and preparing papers (Panin, Zainoulline, Ojanguren).
T7.6 Suslin's rigidity theorem (Yagunov, Panin). Suslin's famous rigidity theorem states in particular that an inclusion of k in K of algebraically closed field induces an isomorphism of K-groups with finite coefficients. We expect to extend this result to arbitrary generalized oriented cohomology theory in the sense of Morel and Voevodsky Study of general properties of oriented cohomology theories (Months 1-12). Reports and preparing papers.
T7.7 K-theory and the Weil transfer functor (Jouhovitsky). We plan to construct a Weil transfer functor for the motivic category of Merkurev - Panin and use this construction to compute K-theory of the Weil transfers of various classical varieties
T7.7.1 Research (Mon 1-18)
T7.7.2 Constructing of Weil transfer functor for the several types of motivic categories (Months 19-36).
T7.8 Multicommutators and multiproducts of conjugacy classes of simple algebraic groups (Gordeev, Rehmann). The well-known Ore's commutators problem was recently solved by Ellers and Gordeev for split algebraic groups. Here we suppose to investigate the problem of representation of sequences of elements of a simple algebraic group as commutators with a given element. We also suppose to investigate related problems concerning products of conjugacy classes, generation of free subgroups and identities in a simple (nonsplit) algebraic group.
T7.8.1 Nonsplit but isotropic groups, reporting (Mon 1-18)
T7.8.2 Anisotropic groups (Mon 19-36)
3.1.5 Description of the Consortium
3.1.5.1 Research Teams
P1 Bielefeld:
Name of Institute: Fakultaet fuer Mathematik, Universitaet Bielefeld
Names of Scientists: U. Rehmann, Prof., (Co-ordinator). H. Abels, Prof., V. Chernousov (PhD, from Minsk, has position at Bielefeld till End of 2000) I. Kersten, Prof., Senior Research Scientist. O. Izhboldin (PhD, from St. Petersburg, has position at Bielefeld till End of 2000). F. Waldhausen, Prof.
This team has expertise in algebraic groups, quadratic forms over arbitrary fields, generic splitting of algebraic structures, transformation groups as well as in topology and algebraic K-theory, Hochschild homology.
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P2 Besancon:
Name of Institute: Laboratoire de Mathematiques Universite de Franche-Comte, Besancon
Names of Scientists: E. Bayer-Fluckiger, Prof., Senior Research Scientist (Team Leader) G. Berhuy, A. Cortella, L. Fainsilber, D.W. Hoffmann, M. Monsurro, A. Queguine, Several PhD students also participate here. Total Number of team members: 14
This team has expertise in classical groups, quadratic and hermitean forms, in particular their generic theory, algebras with involution, Hasse principles for classical groups and is suited to collaborate with Bielefeld and St. Petersburg
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P3 Gent:
Name of Institute: Department Pure Mathematics and Computer Algebra University of Gent
Names of Scientists: J. Van Geel, Prof., Head of the research group algebra (Team Leader) Francis Gardeyn (aspirant-FWO, univeristy of Gent), Karim Zahidi (assistent, university of Gent)
This team is very experienced in quadratic forms, Azumaya algebras, Brauer groups, division algebras and is particularly suited to collaborate with Minsk as well as with other teams of the project.
Requested funding for each of the three INTAS teams:
For the INTAS teams, there is a uniform request on resources:
For the one week workshops on each institute it is desirable to invite 3-4 scientists who are not member of the project. For this purpose ca. 3000 Euro is necessary. Moreover, for travel of the scientists in INTAS teams, each year 1000 Euro is necessary, hence a total of 6000 Euro for travel and subsistence is requested for each of the INTAS teams. Also, an overhead of 2000 ECU for each INTAS team is requested as well.
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P4 Almaty:
Name of Institute: Institute of Mathematics Academy of Sciences of Kazakhstan
Names of Scientists: (5 Scientists with Age < 35) 1)A.S. Dzhumadil'daev, Prof., Head of algebra laboratory (Team Leader). 2)S. Abdukasimova, aspirant, Age 30. 3) S. Ibraev, aspirant, Age 34. 4) I. Irgalieva, aspirant, Age 28. 5) A. Kungojin, student (probably will be soon aspirant) Age 21. 6) E. Bekmuchamedov, student, Age 16.
This team has particular experience in Lie algebras and their cohomological properties. Techniques used here will be instructive for other teams working with cohomological methodes. The planned investigations on commutators, in particular the computatons with computers may be very useful for cooperation with the joint study of commutator investigations by team 1 and 7 (Task T7.8).
Requested Funding:
Travel: 3 weeks each year = 3 x 2020 Euro = 6060 Euro
Labor: Prof.Dr.of Science = 36 x 150 Euro = 5400 Euro
Aspirant (4 persons, 4 months/year each ) 12 x 120 Euro = 5760 Euro
Technical staff (1 person, Computer prog.) 36 x 80 Euro = 2880 Euro
Equipment: Computer Pentium III (used for Task T4.2.2) = 1500 Euro
Overhead: = 2400 Euro
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P5 Minsk:
Name of Institute: Institute of Mathematics National Academy of Sciences of Belarus Algebra department
Names of Scientists: (2 Scientists of Age < 35) V.I.Yanchevskii, Prof., Head of Algebra department (Team Leader) V. Chernousov (PhD, has position at Bielefeld till end of 2000) S.V.Tikhonov (Age=26) (Junior research) D.F.Bazylev (Age=24) (Aspirant)
This team is experienced in Brauer groups, Azumaya algebras over classical (local/global fields and function fields over varieties). Useful cooperation is expected with team 1, 3, 7.
Requested Funding:
Travel: 2 x 4 weeks/year travel (e.g. to Bielefeld and Gent): 10800 Euro
Labor: 1 Senior Scientist 36 x 150 Euro = 5400 Euro
1 Scientist 36 x 90 Euro = 3240 Euro
1 Scientist 36 x 60 Euro = 2160 Euro
Overhead: = 2400 Euro
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P6 Tbilisi:
Name of Institute: Department of Algebra Razmadze Mathematical Institute Georgian Academy of Sciences
Names of Scientists: (None of Age < 35) T. Pirashvili, Prof., Senior Research Scientist (Team Leader) J. Gubeladze (D. Sc.) M. Jibladze T. Kadeishvili (D. Sc.) S. Saneblidze (D. Sc.)
This team is particularly experienced in toplogical Hochschild homology. Cooperation with team 1 (Waldhausen) will be very useful. Their methods, in particular their knowledge in homological methodes, are of methodological interest for most other teams as well.
Requested Funding:
Travel: 3 x 1 week/year (including visit of the Congress of the
European Math Society to represent the project) 6048 Euro
Labor: 3 Senior Scientists: 3 x 18 x 280 Euro = 9720 Euro
2 Scientists: 2 x 18 x 162 Euro = 5832 Euro
Overhead: = 2400 Euro
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P7 St.Petersburg:
Name of Institute: V.A.Steklov Institute of Mathematics at St.Petersburg, RAS Department of Algebra
Names of Scientists: (5 of Age <35) I. Panin, (Team leader). A. Suslin, Prof. (has also position at Northwestern, USA). O. Pushin, Age 25, PhD student of A.Suslin. A. Skorichenko, Age 25, PhD student of A.Suslin. S.Yagunov Age 30, K.Zainoulline, Age 22, PhD student of I.Panin. O. Izhboldin, Bielefeld and St. Petersburg State University. V. Joukhovitsky, Age 25, PhD student of A.Suslin, St. Petersburg State University, N. Gordeev, St.Petersburg State Pedagogical University
Remark: O. Pushin, A. Skorichenko, V. Joukhovitsky have temporary positions at Northwestern (USA) for part of the proposed running time of this project.
This team is very experienced in algebraic K-theory, algebraic geometry, K-theory as well as with the generic theory of algebraic structures. Cooperation and exchange with them will be of great benefit for all other teams.
Requested Funding:
Travel: 5 x 1 month visit of one of the INTAS labs: = 10620 Euro
Labor: 1 Senior Scientist: 36 x 80 Euro = 2880 Euro
1 Scientist: 36 x 70 Euro = 2520 Euro
1 Scientist: 24 x 70 Euro = 1680 Euro
1 Scientist: 30 x 70 Euro = 2100 Euro
1 Scientist: 36 x 50 Euro = 1800 Euro
Overhead: = 2400 Euro
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3.1.5.2 Scientific References
P1 Bielefeld:
1. Chernousov, V. Merkurjev, A., R-equivalence and special unitary groups. J. Algebra 209 (1998), no. 1, 175--198.
2. Chernousov, V., An alternative proof of Scheiderer's theorem on the Hasse principle for principal homogeneous spaces. Doc. Math. 3 (1998), 135--148
3. Chernousov, Vladimir I.; Platonov, V.P., The rationality problem for semisimple group varieties. J. Reine Angew. Math. 504 (1998), 1--28.
4. Ryzhkov, A. A.; Chernousov, V. I., On the classification of maximal arithmetic subgroups of simply connected groups. (Russian) Mat. Sb. 188 (1997), no. 9, 127--156; translation in Sb. Math. 188 (1997), no. 9, 1385--1413
5. Benyash-Krivets, V. V.; Chernousov, V. I., Varieties of representations of fundamental groups of compact nonoriented surfaces. (Russian) Mat. Sb. 188 (1997), no. 7, 47--92; translation in Sb. Math. 188 (1997), no. 7, 997--1039
6. G. Margolin, U. Rehmann and V. Yanchevskii, Quaternion Generation of the 2-Torsion Part of the Brauer Group of a Local Quintic, In: H. Bass, A.O.Kuku, and C. Pedrini. (Ed). Algebraic K-theory and its Applications. Proc. of the Workshop and Symposium, (1997), ICTP, Trieste, Italy. World Scientific, Singapore, New Jersey, London, Hongkong (1999). 503--535.
7. J. Hurrelbrink and U. Rehmann, Splitting patterns and linear combinations of two Pfister forms, J. Reine Angew. Math. 495, (1998), 163--174.
8. I. Kersten and U. Rehmann, Excellence properties of algebraic groups. I. J. Algebra 200 (1998), no.~1, 334--346.
9. J. Hurrelbrink and U. Rehmann, Splitting patterns of quadratic forms. Math. Nachr. 176 (1995), 111--127.
10. Kersten, I., Rehmann, U., Generic splitting of reductive groups. Tôhoku Math. J. (2) 46 (1994), no. 1, 35--70.
P2 Besancon:
1. E. Bayer-Fluckiger, L. Fainsilber, Non-unimodular hermitian forms, Inventiones Mathematicae, (1996), 233-240.
2. E. Bayer-Fluckiger, R. Parimala, Galois cohomology of linear algebraic groups over fields of cohomological dimension < 2, Inventiones Mathematicae, 122 (1995), 195-229.
3. E. Bayer-Fluckiger, R. Parimala, Classical groups and the Hasse principle, Ann. of Math., 147 (1998), 651-693.
4. E. Bayer-Fluckiger, M. Monsurro, Doubling trace forms, St.-Petersburg Math. J., accepted.
5. G. Berhuy, Characterization of hermitian trace forms, Journal of Algebra, 210 (1998), 690-696.
6. A. Cortella, The Hasse principle for similarities of bilinear forms, St.-Petersburg Math. Math. J., 9 (1998), 743-762.
7. D.W. Hoffmann, On the dimensions of anisotropic quadratic forms in I^4, Inventiones Mathematicae, 131 (1998), 185-198.
8. D.W. Hoffmann, On Elman and Lam's filtration of the u-invariant, J. reine angew. Math.}, 495 (1998), 175-186.
9. D.W. Hoffmann, J.-P. Tignol, On 14-dimensional quadratic forms in I^3, 8-dimensional forms in I^2, and the common value property, Doc. Math. J. DMV, 3 (1998), 189-214.
10. D.W. Hoffmann, Pythagoras numbers of fields, J. Amer. Math. Soc., accepted.
11. N. Karpenko, A. Queguiner, A criterion of decomposability for degree 4 algebras with unitary involution, Journal of Pure and Applied Algebra, accepted.
P3 Gent:
1. Lewis D.W. - Van Geel, J., Quadratic forms isotropic over the function field of a conic Indag. Mathem. 6 (3), 325-339, (1994)
2. Hoffmann D.W., Lewis D.W., Van Geel, J., Minimal forms for function fields of conics. Proceedings AMS Summer Institute, Quadratic forms and division algebras: Connections with algebraic K-theory and algebraic geometry. Proceedings of Symposia in Pure Mathemaics, Vol. 58.2, 227-237, (1995).
3. Hoffmann D.W., Van Geel, J., Minimal forms with respect to function fields of conics, Manuscripta Math. 86, 23-48 (1995).
4. Van Geel, J., Yanchevskii V.I., Indices of hyperelliptic curves over p-adic fields, manuscripta math. 96, (1998), 317-333.
5. Hoffmann D., Van Geel, J., Zeroes and norm groups of quadratic forms over function fields in one variable over a local non-dyadic field, J. Ramanujan Math. Soc. 13 nr. 2, (1998), 85-110.
6. Van Geel, J., Yanchevskii V.I., Indices of double coverings of genus 1 over p-adic fields, Annales de la Faculte des Sciences de Toulouse. Vol. VIII, nr. 1, (1999), 155-172.
P4 Almaty
1. A.S. Dzhumadil'daev, On the cohomology of modular Lie algebras, Math.USSR Sb., 47(1984), No.1, p.127-143.
2. A.S. Dzhumadil'daev, Generalised Casimir elements, Math.USSR Izv., 27(1986), p.391-400.
3. A.S. Dzhumadil'daev, Cohomology and nonsplit extensions of modular Lie algebras, Contemp.Math. 131(1992), part 2, p.31-43.
4. A.S. Dzhumadil'daev, Quasi-Lie bialgebra structures of sl_2, Witt and Virasoro algebras, Proc. Israel Math. Conf., 7(1993), p.13-24.
5. A.S. Dzhumadil'daev, Central extensions of infinite-dimensional Lie algebras, Funct.Anal.Appl., 26(1992), No.4, p.21-29, engl.transl 246-253.
6. A.S. Dzhumadil'daev, Virasoro Type Lie algebras and deformations, Zeitschrift fuer Physik, Ser.C, 72(1996), 509-517.
7. A.S. Dzhumadil'daev, Symmetric (co)homologies of Lie algebras, Comptes Rendus Acad.Sci. Paris, Ser. Mathem., 324(1997), p.497-502.
8. A.S. Dzhumadil'daev, Cohomologies and deformations of right-symmetric algebras, J.Math.Sciences, 93(1999), No. 6, 1836-1876.
9. A.S. Dzhumadil'daev, Minimal polynomials for right-symmetric algebras, J.Algebra, to appear 1999.
10.A.S. Dzhumadil'daev, A.I. Kostrikin, Modular Lie algebras: new trends, Proc. Kurosh Conference, may, 1998, Walter de Greuter, 1999.
P5 Minsk
1. G.Margolin, U.Rehmann, V.Yanchevskii. Generation of 2-torsion part of Brauer group of local quintic by quaternion algebras, the totally splitting case. Proc. of Workshop and Symposium "Algebraic K-theory and its applications", (ICTP,Trieste,Italy,1997)503-535.
2. V.I.Guletskii, V.I.Yanchevskii. Ramification and reciprocity laws in the Brauer groups of fields of functions on curves of genus one over number fields. St-Petersburg Math. J. 8, N5 (1997)863-877.
3. J. van Geel, V.I.Yanchevskii. Indices of hyperelliptic curves over p-adic fields. Manuscripta Math.96 (1998)317-333.
4. J. van Geel, V.I.Yanchevskii. Indices of double coverings of genus 1 over p-adic fields. Ann. de la Fac. de Sciences de Toulouse Math. 8, N1(1999)155-172.
5. V.I.Guletskii, V.I.Yanchevskii. Reciprocity laws for simple algebras over function fields of number curves. Comm. Algebra (to appear).
6. V.I.Yanchevskii. P-primary torsion of Brauer groups of local elliptic curves with bad reduction and unramified division algebras overtheir function fields. Journees Arithmetiques-99, Abstracts of talks ,Vatican, July 12-16,(1999)63.
7. S.V.Tikhonov, V.I.Yanchevskii. 3-primary torsion of Brauer groups of local elliptic curves with additive reduction. Bull.Nat.Acad.Sci.Belarus, 42, N1(1998)50-549 (in Russian).
8. S.V.Tikhonov. Brauer groups of elliptic curves defined over the field of complex power series. Bull.Nat.Acad.Sci.Belarus,43,N3(1999)17-21 (in Russian).
9. S.V.Tikhonov. Brauer groups of ellipticcurves with bad reduction defined over the field of complex power series. Bull.Nat.Acad.Sci.Belarus 43, N4(1999)17-21(in Russian).
10. V.I.Yanchevskii. Index reduction formulae for local hyperelliptic curves with bad reduction. Bull.Nat.Acad.Sci.Belarus 41, N1 (1997)16-21(in Russian). Work in Preparation: 11. D.F.Bazylev, V.I.Yanchevskii. A Generalization of the Nagell-Lutz Theorem for Gauss Points of Finite Order for Integer Elliptic Curves. (To appear)
P6 Tbilisi
1. J. Gubeladze (with W. Bruns and Ngo Viet Trung), Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123-160 (1997).
2. J. Gubeladze, $K$-theory of affine toric varieties. Homology Homotopy Appl. 1, 135-146 (1999)
3. M. Jibladze, T. Pirashvili, Cohomology of algebraic theories. J. Algebra 137 (1991), no. 2, 253-296.
4. T. Kadeishvili, S. Saneblidze, Iterating the bar construction, Georgian Math. J., 5(1998), 441-452.
5. T. Kadeishvili, S. Saneblidze, Permutohedral complex model for the double loop space, Proc. Of the International Meeting, ISPM-98, Mathematical Methods in Modern Theoretical Physics, School and Workshop, Tbilisi, Georgia, September 5-18, 1998.
6. T. Pirashvili, Simplicial degrees of functors. Math. Proc. Cambridge Philos. Soc. 126 (1999) 45-62.
7. T. Pirashvili, On the topological Hochschild homology of $Z/p^kZ$. Comm. Algebra 23 (1995), no. 4, 1545-1549.
8. T. Pirashvili (with V. Franjou), On the Mac Lane cohomology for the ring of integers. Topology 37 (1998), no. 1, 109-114.
9. T. Pirashvili (with Waldhausen, F.) Mac Lane homology and topological Hochschild homology. J. Pure Appl. Algebra 82 (1992), no. 1, 81-98.
10. Saneblidze, S., On the homotopy classification of sections in the free loop fibration, J. of Pure and Appl. Algebra, 91 (1994), 317-327.
P7 St.Petersburg
1. I.A.Panin, A.A.Suslin, On a Grothendieck Conjecture for Azumaya algebras, St. Petersbg. Math. J. 9, No.4, 851-858 (1998)
2. M. Ojanguren, I.A.Panin, The Purity Theorem for the Witt group, Ann. Sci. Ecole Norm. Sup. (4) 32 (1999), no. 1, 71--86.
3. V. Franjou, E. M. Friedlander, A. Skorichenko and A. Suslin General Linear and Functor Cohomology over Finite Fields, to appear.
4. O. Izhboldin, Motivic equivalence of quadratic forms. Doc.Math.J.DMV 3 (1998) 341-351.
5. O. Izhboldin, N. Karpenko, Isotropy of Six-dimensional forms over Function Fields of Quadrics,Jour. of Algebra, 209, 65--93 (1998).
6. S.Yagunov, Homology of Bi-Grassmannian Complexes, K-theory 12, No.3: 277-292, 1997
7. V.Joukhovitsky, K-theory of the Weil restriction functor, University Franche-Comte Besancon, Preprint n 99/01.
8. N.L.Gordeev, Freedom in Conjugacy Classes of Simple Algebraic Groups and Identities with Constants, St. Petersbg. Math. J. 9, No.4, (1998), 709-723.
9. K.Zainoulline, On a Grothendieck conjecture about principal homogeneous spaces for some classical algebraic groups, St.Petersbg. Math. J., to appear
10. E. Ellers; N. Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3657--3671.
-----------------------------------------------
3.1.6 Management
The day-to-day management of the project will be by using email. There will be three project workshops at the end of the first, second and third year, which will take place at the western teams Besancon, Bielefeld and Gent, in order to exchange the results of the project and to give the occasion for personal scientific exchange. Many exchanges of scientists will be arranged according to the needs of the various research projects, for that purpose, also financial means of sources other than Intas will be used at the western partners.
The following diagram explains the prject, including task attribution. schedules and relationsships among the partners and projects:
3.1.6.1 Planning & Task Allocation
Tasks |Participants|Mon 1-6|Mon 7-12|Mon 13-18|Mon 19-24|Mon 25-30|Mon 31-36|
T1.1 | P1,P7 |<-----T1.1----->| | | | |
T1.2 | P1,P7 | | |<------T1.2------->| | |
T1.3 | P1 | | | | |<------T1.3------->|
T1.4 | |<-----------------------T1.4--------------------------->|
T1.5 | P1-7 | | | | Workshop | |
T1.6 | |<-----------------------T1.6--------------------------->|
| | | | | | | |
T2.1 | P2,P1,P7 |<-----------------------T2.1--------------------------->|
T2.2 | P2,P1 |<-----------------------T2.1--------------------------->|
T2.3 | | | Workshop | | | |
| | | | | | | |
| | | | | | | |
T3.1 | P3,P5 |<---------T3.1----------->| | | |
T3.2 | P3,P5 |<---------T3.2----------->| | | |
T3.3 | P3,P5 | | | |<------------T3.2----------->|
T3.4 | P3,P5 | | | |<------------T3.2----------->|
T3.5 | P1-P7 | | | | | Workshop |
| | | | | | | |
T4.1 | P4 |<----T4.1.1---->|<-----T4.1.2------>|<------T4.1.3----->|
T4.2.1| P4 |<---T4.2.1.1--->|<----T4.2.1.2----->|<-----T4.2.1.3---->|
T4.2.2| P4 |<---T4.2.2.1--->|<----T4.2.2.2----->|<-----T4.2.2.3---->|
T4.3 | P4 |<----T4.3.1---->|<-----T4.3.2------>|<------T4.3.3----->|
| | | | | | | |
T5.A | P5 |<----T5.A1----->|<------T5.A2------>|<-------T5.A3----->|
T5.A | P5,P1 | Several 1-onth visits at Bielefeld required |
T5.B | P5 |<-----------------T5.B1------------>|<-------T5.B2----->|
T5.B | P5,P1,P3 | Several 1-onth visits at Bielefeld and Gent required |
T5.C | P5 | | T5.C1 |<-----------T5.C2----------->| T5.C3 |
T5.D | P5 |<----T5.D1----->|<------T5.D2------>|<------T5.D3------>|
T5.D | P5,P3 | Several- month visits at Gent required |
| | | | | | | |
T6.1 | P6,P1 |<----T6.1a----->|<-----------------T6.1---------------->|
T6.2 | P6,P1 |<----T6.2a----->|<-----------------T6.2---------------->|
T6.3 | P6,P1 |<------------------------T6.3-------------------------->|
T6.4 | P6,P1 |<----T6.4a----->|<-----------------T6.4---------------->|
| P6 | Presentation at ECM Barcelona July 10-14, 2000 |
| | | | | | | |
T7.1 | P7 |<------------T7.1.1---------------->|<-----T7.1.2------>|
T7.2 | P7 |<--------7.2.1----------->|<----------T7.2.2----------->|
T7.3 | P7 |<----T7.3.1---->|<-----------T7.3.2-------------------->|
T7.4 | P7 |<------------T7.4.1------>|<----------T7.4.2----------->|
T7.5 | P7 |<-------------------T7.5------------------------------->|
T7.6 | P7 |<----T7.6------>| | | | |
T7.7 | P7 |<--------T7.7.1---------->|<-----------T7.7.2---------->|
T7.8 | P7,P1 |<--------T7.8.1---------->|<-----------T7.8.2---------->|
T7.9 | P7,P5 |<------------------------T7.9-------------------------->|
| | | | | | | |
3.1.6.2 Cost Table
MAIN COST TABLE
INTAS MEMBER TEAMS:
TEAM NAME|STATUS| COST CATEGORIES | TOTAL |
| |Labour|Over-|Travel+ |Consum-|Equip-|Other| Total |
| |Costs |heads|subsist.| ables | ment |Costs| Euro |
| | | | | | | | |
P1 Bielefeld| CO | | 2000| 6000 | | | | 8000 |
P2 Besancon | CR | | 2000| 6000 | | | | 8000 |
P3 Gent | CR | | 2000| 6000 | | | | 8000 |
| | | | | | | | |
Subtotal |(Euro)| | 6000| 18000 | | | | 24000 |
NIS TEAMS:
TEAM NAME|STATUS| COST CATEGORIES | TOTAL|
| |Labour|Over-|Travel+ |Consum-|Equip-|Other| Total |
| |Costs |heads|subsist.| ables | ment |Costs| Euro |
| | | | | | | | |
P4 Almaty | CR | 14040| 2180| 6280 | | 1500 | | 24000|
P5 Minsk | CR | 10800| 2180| 11020 | | | | 24000|
P6 Tbilisi | CR | 15552| 2180| 6268 | | | | 24000|
P7 St.Peters| CR | 10980| 2180| 10840 | | | | 24000|
burg | | | | | | | | |
| | | | | | | | |
Subtotal |(Euro)| 51372| 8720| 34408 | | 1500 | | 96000|
| | | | | | | | |
| | | | | | | | |
Total |(Euro)| 51372|14720| 52408 | | 1500 | | 120000|
P5 Minsk:
NIS Labor Cost Summary Table
Team Name |Number of |Cost/month|Number of|Total Cost|
|indiv.grants| (Euro) | Months | (Euro) |
| | | each | |
| | | | |
P4 Almaty | 1 | 150 | 36 | 5400 |
P4 Almaty | 4 | 120 | 12 | 5760 |
P4 Almaty | 1 | 80 | 36 | 2880 |
P5 Minsk | 1 | 150 | 36 | 5400 |
P5 Minsk | 1 | 90 | 36 | 3240 |
P5 Minsk | 1 | 60 | 36 | 2160 |
P6 Tbilisi | 3 | 180 | 18 | 9720 |
P6 Tbilisi | 2 | 162 | 18 | 5832 |
P7 St.Petersburg | 1 | 80 | 36 | 2880 |
P7 St.Petersburg | 1 | 70 | 36 | 2520 |
P7 St.Petersburg | 1 | 70 | 24 | 1680 |
P7 St.Petersburg | 1 | 70 | 30 | 2100 |
P7 St.Petersburg | 1 | 50 | 36 | 1800 |
| | | | |
Totals | sum | | | 51372 |
3.1.7 SUMMARY
Linear algebraic groups occur as automorphism groups of certain linear structures like Azumaya algebras, quadratic and hermitean forms, Algebras with involutions, Jordan and Lie algebras and similar.
Therefore the study of algebraic groups gives insight into these linear structures and vice versa. Algebraic groups are very well understood up to a classification and an understanding of the so called anisotropic groups, which occur as automorphism groups of anisotropic linear structures. While groups which are not anisotropic have been studied in extenso during the last century, anisotropic groups and structures are in general not yet understood.
Classical tools to get information about such linear structures or about linear groups are very often of (co)homological nature.
It is the goal of this project to bring together experts of various fields of algebra (like algebraic groups, quadratic and hermitean forms, Brauer groups, Lie algebras and homology) in order to combine their joint expertise to develop methods which allow to classify these structures as well as of their automorphism groups and to get to a better understanding of them.
It is intended to do research in the generic theory of quadratic an hermitean forms, of their associated automorphism groups, in particular, for certain classes of such groups it is intended to give a description in terms of Galois cohomology (i.e. in the sense of a complete system of invariants). Moreover, the splitting behaviour of Brauer groups under certain secial circumstances will be studied in order to get further infornation about anisotropic phenomena.
Classical anisotropy invariants like the u-invariant will be studied for general fields. There is a subproject on cohomology of right symmetric algebras: An extension theory of Leibniz algebras will be developed, polynomial identities for Witt algebras, and Casimir operators will be studied. Also, topological Hochschild cohomology and MacLane cohomology together with relations to K-theory is investigated. Moreover, questions about motivic cohomology and K-theory of low dimensional fields, the Gersten conjecture in K-theory, Suslin's rigidity theorem and the Weyl tranfer functor with applications to classical varieties will be investigated.
It is expected that the research in this project will bring fruitful results for all fields and disciplines involved, like linear algebraic groups, Azumaya algebras, quadratic forms, Brauer groups, Lie algebras and associated (co-)homology theories.