Type: Seminar

Distribution: World

Expiry: 23 Jan 2018

CalTitle1: MaPSS: Mathematical Postgraduate Seminar Series

Auth: casella@10.17.25.116 (acas5565) in SMS-WASM

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. Supervisors, please encourage your students to attend. Thanks, MaPSS Organisers

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