Wargaming with Quadratic Forms and Brauer Configuration Algebras

Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa and Gabriel Bravo Rios
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Agustín Moreno Cañadas: Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogotá 11001000, Colombia
Pedro Fernando Fernández Espinosa: Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogotá 11001000, Colombia
Gabriel Bravo Rios: Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogotá 11001000, Colombia

Mathematics, 2022, vol. 10, issue 5, 1-19

Abstract: Recently, Postnikov introduced Bert Kostant’s game to build the maximal positive root associated with the quadratic form of a simple graph. This result, and some other games based on Cartan matrices, give a new version of Gabriel’s theorem regarding algebras classification. In this paper, as a variation of Bert Kostant’s game, we introduce a wargame based on a missile defense system (MDS). In this case, missile trajectories are interpreted as suitable paths of a quiver (directed graph). The MDS protects a region of the Euclidean plane by firing missiles from a ground-based interceptor (GBI) located at the point ( 0 , 0 ) . In this case, a missile success interception occurs if a suitable positive number associated with the launches of the enemy army can be written as a mixed sum of triangular and square numbers.

Keywords: Brauer configuration algebra; Dynkin graph; mixed sums of triangular and square numbers; path algebra; positive root; quadratic form; quiver representation; wargame (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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