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A combinatorial construction of \(G_2\). (English) Zbl 1032.17015

Summary: We show how to construct the simple exceptional Lie algebra of type \(G_2\) by explicitly constructing its 7 dimensional representation. Technically no knowledge of Lie theory is required. The structure constants have a combinatorial meaning involving convex subsets of a partially ordered multiset of six elements. These arise from playing the Numbers and Mutation games on a certain directed multigraph.

MSC:

17B25 Exceptional (super)algebras