September 13-21, 2008

the University of Ottawa

ABSTRACTS

**Bruce Allison** (Alberta/Victoria)

*Some 3x3 matrix algebras and cubic structures in Lie theory*

It is classical fact that the space H(n,A) of nxn hermitian matrices over an associative algebra A with involution forms a Jordan algebra. In addition, there is one exceptional coordinate algebra, the octonion algebra O, for which H(3,O) is a Jordan algebra. The exceptional Jordan algebra H(3,O) plays a crucial role in Jordan theory, and it can also be constructed from a cubic form N satisfying the adjoint identity. In this setting the trace form of N is a non-degenerate symmetric bilinear form.

In this talk we look at some analogs for Lie algebras of these facts. It turns out that, suitably interpreted, the space K(3,A) of 3x3 skew-hermitian matrices has the structure of a Lie algebra for a large class of nonassociative coordinate algebras with involution. Moreover, some of the most important examples of these coordinate algebras, which are called structurable algebras, can be constructed from a suitable generalization of a cubic form N satisfying the adjoint identity. Here the analog of the trace form of N is a non-degenerate hermitian form.

We will discuss these topics and, time permitting, describe some applications in the theory of finite dimensional and infinite dimensional Lie algebras. Much of this talk will be based on joint work with John Faulkner and Yoji Yoshii.

**Jose Anquela** (Oviedo)

*More on Minimal Ideals of Jordan Systems*

In 1968 Zhelvlakov posed the question of whether minimal ideals of linear Jordan algebras are always simple or trivial (with zero product). In 1983, Nam and McCrimmon extended Zhevlakov's question to quadratic Jordan algebras, for which the notion of triviality is understood as having zero cube. Zhevlakov's question was settled in the affirmative by Skosyrskii in 1988, while the answer to Nam-McCrimmon's question appeared in 2007 in a joint paper with Cortés. Moreover, analogues for pairs and triple systems were also obtained establishing that minimal ideals of Jordan systems are either simple or have zero cube. The natural question in the Jordan algebra setting about whether trivial minimal ideals have also zero square has been answered affirmatively by McCrimmon. We will discuss McCrimmon's beautiful proof and the possibility of extending this result to pairs and triple systems (this is joint work with Cortés and McCrimmon).

**V. Drensky** (Bulgarian Academy of Science, Sofia)

*Locally Nilpotent Derivations of Polynomial and Free Associative Algebras
*

A derivation d of an algebra R is called locally nilpotent if for any r in R there exists a positive integer d such that d^d (r)=0. Locally nilpotent derivations of the polynomial algebra C[x_1,…,x_n], C the complex numbers, are related with many important problems on automorphisms and invariant theory, including the Jacobian conjecture, the 14th Hilbert problem, etc.

We present an elementary proof of the conjecture of Nowicki (recently confirmed
in the Ph.D. thesis of Khoury, and also by Derksen) that the algebra of constants
of the locally nilpotent derivation d of C[x_1,…,x_n,y_1,…,y_n] defined
by

d(y_i)=x_i, d(x_i)=0, i=1,…,n, is generated by x_1,…,x_n and x_iy_j-x_jy_i,
1=i<j=n.

Although the theory of locally nilpotent derivations of the free associative algebra C<x_1,…,x_n> repeats the main steps of the case of C[x_1,…,x_n], there are a lot of specific differences, due to the noncommutativity. We describe the constants of locally nilpotent derivations of C<x,y> and show that, up to a multiplicative constant, the derivations are determined by their constants.

These results are obtained jointly with Leonid Makar-Limanov.

**John R. Faulkner** (University of Virginia, Charlottesville):

*Structurable superalgebras of classical type *

The Grassmann envelope is a method of converting superalgebras into ordinary
algebras. Its usual use is to give a quick description of the defining identities
of a superalgebra; e.g., A is a stucturable superalgebra if and only if the
Grassmann envelope G[A] is a structurable algebra. To attack the classification
of simple structurable superalgebras over algebraically closed fields of characteristic
0, we develop methods using the Grassmann envelope to transfer constructions
and results from the algebra setting to the superalgebra setting. This allows
us to generally work with structurable algebras over rings rather than with
structurable superalgebras over fields.

Given an associative algebra A with involution and a symmetric idempotent e,
we define a structurable product on eA. This construction is equivalent to the
construction of the structurable algebra of a hermitian form, but is convenient
in the classification of simple structurable superalgebras whose Kantor algebra
is a Lie superalgebra of type A,B,C,D, P and Q. There are two structurable superalgebras
constructed from cubic forms satisfying the adjoint identity giving Kantor algebras
of types F(4) and G(3), thereby completing the classification for superalgebras
of classical type.

**Osamu Iyama**, Paris

*n-cluster tilting and n-APR tilting *

**Bernhard Keller**, (Paris)

*Quiver mutation and derived equivalence*

Quiver mutation is the central ingredient of Fomin-Zelevinsky's recent theory of cluster algebras. In this talk, we will interpret quiver mutation as the shadow of a derived equivalence between suitable differential graded algebras. We will rely on Derksen-Weyman-Zelevinsky's recent work on mutations of quivers with potentials and on an important construction due to Ginzburg. This is joint work with Dong Yang.

**Ottmar Loos** (Innsbruck)

*The scheme of quaternionic algebras*

We study associative unital algebras of degree 2 and rank 4 over a commutative ring and show that they form a non-smooth scheme with a singularity of normal crossing type. The usual quaternion algebras (Azumaya algebras of rank 4) form an open but not dense subscheme.

**Kevin McCrimmon** (University of Virgina, Charlottesville)

*Odd automorphism dreams*

**Holger P. Petersson** (Fernuniversität Hagen)

*Composition algebras over local fields revisited*

Composition division algebras over a local field k with residue field ? are reasonably well understood provided one works under the additional assumption that their residue algebras be separable over ?. This lecture will be concerned with the opposite case of what I call residually inseparable composition algebras, which have a chance of survival only if ? is an imperfect field of characteristic 2. We attach invariants to residually inseparable composition algebras and relate them to the classification problem. In stark contrast to the unramified case, it will follow, for example, that there are an infinite number of octonion division algebras over appropriate local fields of characteristic zero whose residue algebras are all isomorphic to the same pre-assigned purely inseparable field extension of exponent 1 and degree 8 over ?.

**Claus Michael Ringel**, Bielefeld

*The structure of length categories*

Abstract: Length categories are abelian categories with all objects being of finite length. We will consider length categories A with only finitely many isomorphism classes of simple objects, but infinitely many isomorphism classes of indecomposable objects, and we will assume that A is a k-category with all Hom and Ext spaces being of finite dimension. Such categories arise in many different parts of mathematics and we will discuss the global structure of such categories, in particular the existence of "take-off" subcategories.

**Claus Michael Ringel**, Bielefeld

**Colloquium**

*The torsionless modules for an artin algebras*

Let A be an artin algebra. An A-module M is said to be torsionless provided
it can be embedded into a projective A-module. The algebra A is said to be orsionless-finite
provided there are only finitely many isomorphism classes of indecomposable
torsionless modules. The aim of the lecture will be to survey properties of
the torsionless modules. In particular, We will consider the question which
algebras are torsionless-finite. Note that the representation dimension of a
torsionless-finite algebra is always bounded by 3.

**Ivan Shestakov** (Sao Paulo)

*The Chevalley and Kostant theorems for Malcev algebras*

Abstract: Centers of universal envelopes for Mal'tsev algebras are studied. It is proved that the center of the universal envelope U(M) for a finite-dimensional semisimple Mal'tsev algebra M over a field of characteristic 0 is isomorphic to a polynomial ring in a number of variables equal to the dimension of a Cartan subalgebra of M, and that the algebra U(M) is a free module over its center. Centers of universal enveloping algebras are computed for some Mal'tsev algebras of small dimensions.

This is joint work with V.N.Zhelyabin.

**Paul Smith**, Washington

*An equivalence of categories involving the graded Weyl algebra and an algebraic
quotient stack*

Abstract: Let A denote the first Weyl algebra over the complex field. It is
isomorphic to the ring of differential operators with polynomial coefficients
on the complex line. It is generated by elements x and y subject to the relation
xy-yx=1. We consider it as a Z-graded ring with the degrees of x and y being
+1 and -1 respectively. Let S denote the commutative algebra over the complex
numbers generated by indeterminates x_n, n \in Z, modulo the relations x_n^2+n=x_m^2+m.
Let Z_{fin} be the abelian group whose elements are the finite subsets of Z
(the integers) with group operation "exclusive or". We give S a grading
by Z_{fin} by declaring degree(x_n)={n}. Let G be the affine group scheme Spec(CZ_{fin})
where CZ_{fin} is the complex group algebra for Z_{fin} endowed with its standard
Hopf algebra structure. The grading on C corresponds to and action of G as automorphisms
of S or, equivalently, as automorphisms of Spec(S).

Let X denote the stack-theoretic quotient [Spec(S)/G].

Theorem: The following three categories are equivalent:

(1) Gr(A,Z)

(2) Gr(S,Z_{fin})

(3) Qcoh X

(4) G-equivariant S-modules

where Gr(-,-) denotes the category of graded modules with degree preserving
homomorphisms, and Qcoh X is the category of quasi-coherent sheaves on X.

The equivalence of the categories in (2), (3), and (4) is well-known. The equivalence
with the category of graded modules over the Weyl algebra is new

and is the focus of this talk.

**Efim Zelmanov** (San Diego)

* Jordan Superalgebras*

We will survey basic examples and representations of Jordan superalgebras and their links to superconformal algebras.

**Efim Zelmanov** (San Diego)

**Colloquium **

*Asymptotic properties of finite groups and finite dimensional algebras. *

We will discuss the limits of finite groups and finite dimensional algebras
and their connections with number theory, low dimensional topology, combinatorics
etc.