SMS scnews item created by Alex Casella at Tue 24 Oct 2017 0909
Type: Seminar
Distribution: World
Expiry: 23 Jan 2018
Calendar1: 30 Oct 2017 1700-1800
CalLoc1: Carslaw 535A
CalTitle1: MaPSS: Mathematical Postgraduate Seminar Series
Auth: casella@10.17.25.116 (acas5565) in SMS-WASM

# MaPSS: Mathematical Postgraduate Seminar Series: Giulian Wiggins (Sydney University) -- Representation categories and reductive Lie algebras

Dear All,

We are delighted to present the MaPSS Seminar topic of Monday 30/10; please see the
abstract below.

**This Semester the Seminar will always run on Monday, at 5:00pm in 535A**

Following the talk, there will be pizza on offer.

Speaker: Giulian Wiggins (Sydney University)

Title: Representation categories and reductive Lie algebras

Abstract: We introduce some algebraic toys: Algebras with a partition of unity (APoU)
and their representations.  Given such an algebra $$A$$, we construct a category, CA, in
which the category of representations of $$A$$ is equivalent to the category of linear
functors from CA to the category of vector spaces over the ground field.  As an
application we take a reductive Lie algebra $$g$$, and construct an APoU, $$U*g$$
(Lusztigs idempotent form of $$g$$), whose representations are all the integral
representations of $$g$$.  Then applying the above theory, we are able to take a $$(g,A)$$-bimodule $$P$$ ( $$A$$ is any algebra) satisfying certain conditions, and
derive a presentation for the full subcategory of $$A$$ whose objects are direct sums of
the weight spaces of $$P$$.  If $$P$$ contains a copy of every irreducible $$A$$-module
then the Karoubi completion of this category is the whole category of representations of
$$A$$.  As an example, we give a presentation of the category of permutation modules of
$$S_n$$ and discuss how a presentation of the category of representations of $$S_n$$ may
be obtained from this.  This talk is accessible to anyone with a basic knowledge of
representations of finite dimensional algebras, and of the definitions of category and
functor.