TY - JOUR

T1 - Double axes and subalgebras of Monster type in Matsuo algebras

AU - Galt, Alexey

AU - Joshi, Vijay

AU - Mamontov, Andrey

AU - Shpectorov, Sergey

AU - Staroletov, Alexey

N1 - Publisher Copyright: © 2021 Taylor & Francis Group, LLC. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type (Formula presented.) Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

AB - Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type (Formula presented.) Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

KW - 3-transposition group

KW - Axial algebra

KW - non-associative algebra

UR - http://www.scopus.com/inward/record.url?scp=85108824609&partnerID=8YFLogxK

U2 - 10.1080/00927872.2021.1917589

DO - 10.1080/00927872.2021.1917589

M3 - Article

AN - SCOPUS:85108824609

VL - 49

SP - 4208

EP - 4248

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 10

ER -