Jordan algebras and automorphism groups of Matsuo algebras

Abstract: Primitive axial algebras of Jordan type half were introduced by Jonathan Hall, Felix Rehren and Sergey Shpectorov in 2015, generalizing Jordan algebras by requiring that their idempotents satsify the Peirce decomposition. More specifically, primitive axial algebras of Jordan type $\frac{1}{2}$ are commutative non-associative algebras generated by idempotents $a$ such that their multiplication operators $L_a$ are diagonalizable with eigenvalues $\{1,0,\frac{1}{2}\}$, such that the fusion laws $V_1 = \langle a\rangle$, $V_0^2 \subseteq V_0$, $V_0V_{\frac{1}{2}} \subseteq V_{\frac{1}{2}}$ and $V_{\frac{1}{2}}^2 \subseteq V_0 \oplus V_1$ hold, where $V_\lambda$ is the $\lambda$-eigenspace of $L_a$. The most well-known examples of this class of algebras are either Jordan algebras or Matsuo algebras, certain non-assocative algebras related to 3-transposition groups that Atsushi Matsuo discovered while studyin vertex operator algebras. In this talk, we will sketch how one can distinguish these two classes in terms of their automorphism groups. In particular, primitive axial algebras of Jordan type half with large automorphism groups are automatically Jordan while the automorphism groups of non-Jordan Matsuo algebras are usually finite, with one infinite family of exceptions.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar