Abstract
We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. The decompositions of our algebras are now induced by nilpotent elements associated to lines in the corresponding Fischer space, rather than idempotent elements associated to points. For many 3-transposition groups, this still gives rise to a Z/2Z-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group Sym(4), the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group Ga2⋊Gm.
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