Research School 2017 "Associative and non-associative algebras"

Country: Russian Federation

City: Novosibirsk

Abstr. due: 23.04.2017

Dates: 07.08.17 — 19.08.17

Area Of Sciences: Physics and math;

Organizing comittee e-mail:

Organizers: Sobolev Institute of Mathematics, Cimpa, Novosibirsk State University, International Mathematical Center of Novosibirsk State University, and Russian Foundation for Basic Research (RFBR).


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.

1. From real to complex numbers, and the Cayley-Dickson 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)

to be announced

to be announced

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 PI-algebras

For an associative ring, satisfying a polynomial identity, i.e. being a PI-ring,
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 PI-algebra
is simple and finite dimensional over its center or Posner's theorem showing
that every prime PI-ring is Goldie. We also recall Amitsur's theorem on semiprime
PI-rings or the existence of nonzero central elements in semiprime PI-rings
as a result of Posner-Formanek-Rowen'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 PI-theory
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 non-vanishing 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 non-associative Lie theory in general.
The following topics will be addressed:

(1) Sabinin algebras and flat affine connections.
(2) Non-associative 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 co-authors 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 so-called 2-associative 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 2-compatible 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.

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