Rocky Mountain Algebraic Combinatorics Seminar

New theorems of Skolem-Noether and Morita type for nilpotent products

James B. Wilson
Colorado State University

Elementary Representation Theory relies on the structure of matrix rings over fields. Foundational results include the theorems of Skolem and Noether, Morita, and Lemmas of Schur and Nakayama. Each of those results assume some form of simplicity. This limits their use in the study unipotent and nilpotent representations.

In this talk we adapt these famous claims to nilpotent products. We introduce a module theory based ideally suited for nilpotent representations. Then we link the irreducible representations to semifields and other geometrically significant creatures.


A new proof of the Artin-Zorn theorem

Tim Penttila
Colorado State University

Emil Artin first proved the theorem (first published by his student Max Zorn in 1931) that a finite alternative ring is a field. In 1942, Friedrich Levi was the first to state the geometric equivalent that a finite projective plane satisfying the little Desargues' theorem satisfies Pappus' theorem. In 1972, George Glauberman gave a characterization of finite fields of odd order using results of Adrian A. Albert and Kevin McCrimmon on finite Jordan algebras. In 1984, Gunter Heimbeck extended Glauberman's characterization to include fields of even order, and gave an elementary proof of the combined result. In this talk, the Artin-Zorn theorem is deduced from the Glauberman-Heimbeck theorem. The method of relating the two topics is one familiar to loop theorists: the study of the set of left multiplications. A brief survey will follow of extensions of the Artin-Zorn theorem, still in the finite setting, by various authors led by Michael Kallaher over the period 1968-2006, culminating in a characterisation of finite nearfield planes by a weakening of the alternative condition (due to Gerrit Bol in 1937).


Weber 223
4–6 pm
Friday, September 11, 2015
(Refreshments in Weber 117, 3:30–4 pm)
Colorado State University

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.

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Previous Seminars:

May 8, 2015
Amanda Schaeffer Fry, Peter Brooksbank
April 24, 2015
Heide Gluesing-Luerssen, Phil DeOrsey
March 6, 2015
Felice Manganiello, Eric Moorhouse
February 20, 2015
Anton Dzhamay, Anton Betten
February 6, 2015
Alexander Hulpke, Morgan Rodgers
December 5, 2014
Stefaan De Winter, Gretchen Matthews
November 14, 2014
Greg Coxson, Tom Dorsey
October 31, 2014
Octavio Paez Osuna, Sylvia Hobart
October 10, 2014
Takunari Miyazaki, Eric Moorhouse
September 26, 2014
Elissa Ross, Anton Betten
September 12, 2014
Petr Vojtěchovský, Alexander Hulpke
May 9, 2014
Philip DeOrsey, Tim Penttila
April 25, 2014
William J Martin, Jason Williford
April 11, 2014
Victor Pambuccian, George Shakan
March 7, 2014
Nathan Lindzey, Jens Harlander
February 21, 2014
Ross McConnell, Anton Betten
November 22, 2013
Justin Hughes, Josh Maglione

Department of Mathematics
Fort Collins, Colorado 80523