 # Algebra Seminar Winter 2012

### Ottawa-Carleton Institute of Mathematics and Statistics

Organizers: Yuly Billig (Carleton) and Erhard Neher (Ottawa) Home page of the Algebra, Lie Theory, and Representation Theory Groups

#### Place and time:

• At the University of Ottawa: Wednesday, 4 - 5 pm, KED B015
• At Carleton University: Wednesday, 3:45 - 4:45pm, MacPhail room (HP 4351)

Talks in the Junior Algebra Seminar are aimed at graduate students in algebra.

#### January 18 (Carleton)

• Speaker: Yuly Billig (Carleton University)
• Title: Time-optimal decompositions in SU(2)
• Abstract: This work is motivated by applications to quantum mechanics and quantum computing. In quantum computing, an algorithm is an element of SU(N). An algorithm is realized using quantum controls, which are elements of some fixed 1-parametric subgroups. It is well-known that given a set of generators of the Lie algebra su(N), the corresponding 1-parametric subgroups generate the group SU(N). It is natural to ask how to find decompositions for the given set of generators that minimize the total time. We solve this problem for SU(2). The answer turns out to be non-trivial even in this simplest case.

#### January 25 (Ottawa) -- Junior Algebra Seminar

• Speaker: Caroline Junkins (University of Ottawa)
• Title: Characteristic classes: definitions, properties and examples
• Abstract: Characteristic classes are useful invariants which provide a great deal of information about a smooth manifold X. In particular, Stiefel-Whitney classes and Chern classes are associated to real and complex vector bundles over X, and can be defined axiomatically. Using these definitions, we may consider certain properties of these characteristic classes, and what type of information they can tell us about some familiar manifolds. We may then use this information to obtain a classic result: If V is a finite-dimensional real division algebra, then the dimension of V is a power of 2.

#### February 1 (Ottawa) --- Junior Algebra Seminar

• Speaker: Bea Schumann (University of Ottawa)
• Title: Classification of Quiver Representations
• Abstract: A quiver Q is a directed graph. If we attach to each vertex a vector space and to each arrow a linear map, then this is called a representation of the quiver Q. Quiver representations are linked to several other fields of mathematics. They not only give a natural generalization of some linear algebra problems but are also connected to the study of finite dimensional associative algebras, Kac-Moody algebras, quantum groups, etc. In this talk I will focus on the classification of quiver representations. For convenience, I will restrict myself mostly to finite representation type. I will discuss the Auslander-Reiten quiver, which encodes information about all indecomposable representation of a given quiver and the maps between them. Furthermore I will present the knitting algorithm, an inductive algorithm to compute this quiver.

#### February 15 (Carleton)

• Speaker: Vyacheslav Futorny (University of Sao Paulo)
• Title: q-difference Noether problem and quantum Gelfand-Kirillov Conjecture for gl(n)
• Abstract: The classical Gelfand-Kirillov Conjecture states that the skew field of fractions of the enveloping algebra of an algebraic Lie algebra is a Weyl skew field. The conjecture holds for gl(n) (hence sl(n)), nilpotent and solvable Lie algebras and for all Lie algebras of dimension less than 9. On the other hand there are counterexamples. In the quantum setting the conjecture was settled by J. Alev and F. Dumas (for sl(2) and sl(3)) and by F.Fauquant-Millet for sl(n). We will discuss the solution of the q-difference Noether problem and a new solution of the quantum Gelfand-Kirillov Conjecture for gl(n). The talk is based on joint work with J.Hartwig.

#### February 29 (Carleton)

• Speaker: Erwan Biland (University Paris 7 and Laval University)
• Title: An application of equivariant Morita theory
• Abstract: Consider two rings A and B ; in our context, A and B are block algebras of finite groups over a field k of positive characteristic. Morita's theorem asserts that the categories of modules over A and B are equivalent if, and only if, there exists an (A,B) -bimodule with nice properties. But what if A and B are both endowed with an action of a group G? In this talk, we will give a G-equivariant version of Morita's theorem. We will then define an induction functor which we use to turn G-equivariant Morita equivalences between simple algebras into equivalences between more complicated algebras. Finally, we will show how this technique allows us to simplify a proof of Robinson and Külshammer, initially based on Clifford theory. Our point of view easily brings functorial properties with interesting consequences, otherwise out of reach. This is based on our work for PhD, under the supervision of Michel Br oué.

#### March 7 (Ottawa)

• Speaker: Alistair Savage (University of Ottawa)
• Title: Equivariant map superalgebras
• Abstract: Suppose a finite group acts on a scheme (or algebraic variety) X and a "target'' Lie superalgebra g. Then the space of equivariant algebraic maps from X to g is a Lie superalgebra under pointwise multiplication. We call this an "equivariant map superalgebra". An important class of examples are the (twisted) loop superalgebras, where the variety X is the one-dimensional torus. In this talk we will present a classification of the irreducible finite-dimensional representations of an equivariant map superalgebra where the target is a basic classical Lie superalgebra and the group in question acts freely on X. It turns out that all irreducible finite-dimensional representations are generalized evaluation representations. In the case that the even part of g is semisimple, they are in fact all evaluation representations. As a corollary of our general result, we obtain the first classification of the twisted loop superalgebras. We will not assume any background knowledge of Lie superalgebras. Instead, the basic definitions will be given and several examples will be discussed.

#### March 14 (Ottawa)

• Speaker: Erhard Neher (University of Ottawa)
• Title: Rank one groups
• Abstract: In this talk we will consider abstract groups which look like invertible 2 x 2 matrices over division rings, i.e., split saturated Tits systems (= BN-pairs) of rank one. We will describe three different but equivalent axiomatic approaches to these groups, based on earlier work of Timmesfeld and Tits, and present examples which go beyond the case of invertible 2 x 2 matrices. We will present the necessary background, in particular we will not assume any prior knowledge of Tits systems. The talk is based on a part of my joint project with Ottmar Loos on Steinberg groups for Jordan pairs.

#### March 21 (Ottawa)

• Speaker: Vivien Ripoll (Université du Québec à Montréal)
• Title: Asymptotical behaviour of roots of infinite Coxeter groups
• Abstract: Let W be an infinite Coxeter group. We call limit roots the limit points of "projective" roots of W (representing the directions of the roots in a root system of W). It turns out that the set E of limit roots is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and we illustrate this behaviour with examples and pictures in rank 3 and 4. After defining a natural geometric action of W on E, we exhibit a finite subset of E (formed by the limit roots of some dihedral reflection subgroups of W), and prove that its orbit is dense in E. This talk is based on joint works with M. Dyer, Ch. Hohlweg and J.-P. Labbé.

#### March 28 (Ottawa)

• Speaker: Edward Richmond (UBC)
• Title: Littlewood Richardson coefficients for Kac-Moody flag varieties
• Abstract: Let G be a complex simple Lie group or Kac-Moody group and P a parabolic subgroup. One of the goals Schubert calculus is to understand the product structure of the cohomology ring H^*(G/P) with respect to its basis of Schubert classes. If G/P is the Grassmannian, then the structure constants corresponding to the Schubert basis are the classical Littlewood-Richardson coefficients which appear in various topics such as enumerative geometry, algebraic combinatorics and representation theory. In this talk, I will discuss joint work with A. Berenstein in which we give a combinatorial formula for these coefficients in terms of the Cartan matrix corresponding to G. In particular, our formula implies positivity of the "generalized" Littlewood-Richardson coefficients in the case where the corresponding Weyl group of G is a free Coxeter group (i.e. no braid relations). Moreover, this positivity result extends to the torus-equivariant coeffients of H^*_T(G/P) and does not rely on the geometry of the flag variety G/P.

#### April 4 (Ottawa) -- Junior Algebra Seminar

• Speaker: Khoa Pham (University of Ottawa)
• Title: Dynkin indices for simple Lie algebras
• Abstract: In his paper Semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of an index for representations of complex simple Lie algebras, now known as the Dynkin index. Indices of fundamental representations of simple Lie algebras were explicitly computed by Dynkin. It turns out that there is a fundamental representation whose Dynkin index divides the indices of other representations. In order to find a more conceptual proof to this result, we compute a formula for the index of the exterior powers of an arbitrary representation. We then apply the formula to prove the result for several types of simple Lie algebras. All concepts beyond those usually presented in a basic course on Lie algebras will be defined and explained.