Baric Axial Algebras of Jordan type

Shi, Yunxi (2025). Baric Axial Algebras of Jordan type. University of Birmingham. Ph.D.

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Abstract

An axial algebra is a non-associative commutative algebra and generated by special idempotents (axes) satisfying a prescribed fusion law. An axial algebra is baric if there exists a surjective homomorphism from the algebra onto the ground field, which does not send axes to zero. In this text, we investigate baric algebras within the class of algebras of Jordan type η. We show that such algebras only exist when η=2 or η=1/2. We completely classify the case η=2 by showing that baric algebras of Jordan type 2 are exactly the Matsuo algebras of Moufang type and their factors. The case of η=1/2 is more complicated as it includes Jordan algebras. We demonstrate the existence of the universal k-generated baric algebra of Jordan type 1/2 and in doing so we also establish a few interesting facts about more general axial algebras in terms of magma algebras. We also explicitly construct the 4-generated baric axial algebras of Jordan type 1/2, which turns out to have dimension 54, which is quite below the known upper bound of 81. As a consequence of this calculation, we also deduce that the universal k-generated baric algebra of Jordan type 1/2 is Jordan for all k.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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