The Research School will be dedicated to Prof. Ivan Shestakov's 70th Birthday.
The organizers of the meeting are
Sobolev Institute of Mathematics,
Cimpa,
Novosibirsk State University,
International Mathematical Center of Novosibirsk State University.
The programme of the school will consist of lectures (see the abstracts below).
The conference will take place at the Sobolev Institute of Mathematics
of the Siberian Branch of the Russian Academy of Sciences, Akademgorodok, Novosibirsk, Russia.
Scientific Committee:
Ivan Shestakov (Brazil)
Alberto Elduque (Spain)
Consuelo Martinez (Spain)
Vladislav Kharchenko (Mexico)
Pavel Kolesnikov (Russia)
Jacob Mostovoy (Mexico)
Local organizing committee:
Irene Paniello (Spain)
Aleksandr Pozhidaev (Russia)
Yulia Mikhalchishina (Russia)
Main data for participants:
Research School will be held on August 719, 2017
The deadline for registrations at CIMPA of all participants is
23/04/2017
Registration:
Participants from outside the country of the Research School
must register at CIMPA,
filling out a
form that may be browsed here.
Lecturers do not need to register.
Registration of participants from Russia will be entirely managed
by local organizers.
For registration send an email to
ana2017@math.nsc.ru
with the following information:
First Name, Last Name, University/Institution,
Scientific interests, City, Invitation (Y/N), Hotel (Y/N).
Registration fee:
Participants registered at CIMPA which have enough funding
will need to pay
a registration fee to CIMPA through the CIMPA web site,
in order to help CIMPA work.
This fee is not mandatory. Those are the
only registration fees required.
The registration will be on August 7, Institute of Mathematics, pr. Koptyuga 4, 8.00  9.00 Am.
Hotels: Goldvalley , Student hostel.
Important local information: is here
Slides of the lectures and Conference photo
The courses:
(will be given in English)
Ivan Shestakov (Brazil, Russia)
Jordan algebras and superalgebras
Jordan algebras were introduced in 1933 by von Neumann, Jordan and Wigner
as an algebraic formalism
of quantum mechanics. Since that time, they have
found fundamental applications in Mathematics and Physics and now form an
intrinsic part of modern algebraic methods. In our course we plan to give
an introduction to the structure theory and representations
of Jordan
algebras and superalgebras. The following topics will be addressed:
 Finite dimensional Jordan algebras,

Representations of Jordan algebras,
 Special and exceptional Jordan algebras,
 Jordan superalgebras.
Alberto Elduque (Spain)
Composition algebras
After reviewing the process of the construction of the complex numbers
from the real numbers, this
process will be iterated to produce the
algebra of quaternions. Applications of this algebra to the study of
rotations in the
Euclidean spaces of dimension 3 and 4 will be
considered. A further iteration provides the algebra of octonions.
These algebras
will be generalized over arbitrary fields.
Symmetric composition algebras will be introduced too, and
several connections of
composition algebras with exceptional
simple Lie and Jordan algebras will be given.
Content:
1. From real to complex numbers, and the CayleyDickson doubling process.
2. Hamilton's quaternions. Rotations.
3.
Octonions.
4. Symmetric composition algebras.
5. Composition algebras and simple exceptional Lie and Jordan algebras.
Vladislav Kharchenko (Mexico)
Quantum Lie theory
The numerous attempts over the previous 1520 years to define
a quantum Lie algebra as an elegant algebraic object with a binary
``quantum" Lie bracket have not been widely accepted. In the lectures
we discuss an alternative approach that includes multivariable operations.
There are many fields in which multivariable operations replace the Lie bracket,
such as investigations of skew derivations in ring theory,
local analytic loop theory, and theoretical research
on generalizations of Nambu mechanics.
Among the problems discussed in the lectures are the following:
multilinear quantum Lie operations,
the principle generic quantum Lie operation,
the basis of symmetric generic operations,
ShestakovUmirbaev operations for the Lie theory
of nonassociative products.
Consuelo Martinez (Spain)
Jordan Superalgebras
The aim of the course is to give an introduction to the theory of Jordan superalgebras.
It will
include connections with Lie superalgebras, examples, classification results,
both in zero characteristic and in prime characteristic,
and representation results in zero characteristic.
Pavel Kolesnikov (Russia)
Conformal algebras
Conformal algebras (also called vertex Lie algebras) appeared as
"infinitesimal" version of vertex operator algebras in
mathematical
physics have also found applications in representation theory, ring theory,
and combinatorics. The structure theory
of associative conformal
algebras with finite faithful representation has been developed in a
series of works by A.D'Andrea, V.
Kac, E. Zelmanov, A. Retakh, and
P.Kolesnikov. In particular, it was shown that the second cohomology group
of a simple
associative conformal algebra with finite faithful
representation may not be trivial. Structure of this group is
responsible
for splitting of the nilpotent radical in a conformal
algebra with finite faithful representation and thus it is an important
feature of structure theory in this class. We are planning to compute
precisely a series of cohomology group for different simple
conformal algebras.
Irene Paniello (Spain)
Nonassociative PIalgebras
For an associative ring, satisfying a polynomial identity, i.e. being a PIring,
can be understood as a kind of finiteness condition. Then, as it happens for other
such conditions, like finite dimensionality or descending chain conditions,
general structure theories and regularity conditions, as those related to primitive
or prime algebras, specialize providing more concrete descriptions of the rings involved.
Consider, for example, Kaplansky's theorem stating that every primitive PIalgebra
is simple and finite dimensional over its center or Posner's theorem showing
that every prime PIring is Goldie. We also recall Amitsur's theorem on semiprime
PIrings or the existence of nonzero central elements in semiprime PIrings
as a result of PosnerFormanekRowen's theorem.
An extension of the notion of polynomial identity for associative algebras is given
by generalized polynomial identities (GPI) consisting on polynomial identities admitting
not only scalar coefficients but involving also coefficients from the ring itself.
The main structural results are due to Amitsur and Martindale for primitive and prime
GPI algebras respectively.
The situation in the nonassociative case is slightly different since here
(consider, for instance, Jordan or alternative algebras) the PItheory
is an integral part of the structure theory. One can consider, for example,
strongly prime Jordan algebras, whose classification theorem, due to Zelmanov,
depends on the existence of particular classes of nonvanishing identities
(hermitian polynomials) and polynomial identities (e.g. Clifford identities).
The GPI counterpart for Jordan algebras (and in general for Jordan systems)
corresponds to homotope polynomial identities (HPI), that is, polynomial identities
that hold in all homotope algebras. Similar results to those mentioned above
for associative algebras hold for Jordan algebras.
Jacob Mostovoy (Mexico)
Sabinin algebras
In a certain sense, Sabinin algebras are a relative version of Lie algebras
and the techniques of the theory fall into the scope of the
classical Lie theory.
In these lectures I will give an overview of the theory of Sabinin algebras
and nonassociative Lie theory in
general.
The following topics will be addressed:
(1) Sabinin algebras and flat affine connections.
(2) Nonassociative Hopf
algebras and the integration.
(3) Representations and cohomology.
(4) Particular cases: Malcev and Bol algebras,
Lie triple
systems, nilpotent Sabinin algebras.
(5) Applications to discrete loops.
Murray Bremner (Canada)
Associative and Nonassociative Structures Arising from Algebraic Operads
The classical theories of associative and nonassociative algebras deal almost
exclusively with structures having a single binary operation. The recent rapid
development of the theory of algebraic operads and the closely related topic
of higher categories has made clear the importance of studying structures with two
or more operations. In particular, for two associative binary operations
a . b and a # b, the work of Loday and his coauthors shows that there are
many different ways in which one can define associativities between the two operations.
If one assumes no further relations, one obtains the socalled 2associative algebras;
if one assumes also that ( a ◦ b ) • c ~ a ◦ ( b • c ) then one obtains
duplicial algebras;
if one assumes instead that every linear combination of the two operations is associative,
then one obtains 2compatible algebras; and finally, if one assumes that all four combinations
of the operations are associative, ( a * b ) *' c ~ a * ( b *' c ) for all *, *' in { ◦, • },
then one obtains totally associative algebras.
In every case the identities defining the algebras can be expressed in terms of the vanishing
of certain associators, and weakening this vanishing to the corresponding alternating property
leads to notions of alternative algebras in the setting of two operations.
Similarly, there are analogues of the Lie bracket and the Jordan product in each case,
and the identities satisfied by these operations lead to notions of Lie and Jordan algebras
in the setting of two operations.
This short course will summarize the necessary background in the theory of
algebraic operads, recall known results on structures with two operations,
and conclude with an overview of current research and open problems.
Additional lectures
will be given by Leonid Bokut, Vladimir Levchuk, Yuriy Malcev, Alexandr Pozhidaev, and Uziel Vishne.
Sobolev Institute of Mathematics
4 Acad. Koptyug avenue,
630090, Novosibirsk, Russia
Phone: (383) 3297630
Fax: (383) 3332598
Email: app@math.nsc.ru