Highwater Algebra

• Highwater algebra is an infinite-dimensional, symmetric, two-generated primitive axial algebra defined over fields of characteristic not equal to 2,3, uniquely producing finite-dimensional quotients for every positive integer.
• It features a novel five-eigenvalue fusion law outside classical Monster-type, with a covering algebra that restores Monster type in characteristic 5.
• Its ideals are fully classified by explicit principal generators and bases, positioning Highwater algebra as the universal parent for non-Jordan Monster-type axial algebras.

Highwater algebra refers to the exceptional infinite-dimensional, symmetric, two-generated primitive axial algebra over a field of characteristic not equal to $2,3$, whose fusion structure falls outside the classical Monster type except in characteristic $5$. Discovered independently by Franchi–Mainardis–Shpectorov and Yabe, the Highwater algebra $\mathcal{H}$ is notable for providing the only example among two-generated Monster-type axial algebras with unbounded dimension and whose finite-dimensional quotients exhaust every positive integer. Its covering algebra $\widehat{\mathcal{H}}$ unifies characteristic behavior and reveals a previously unseen five-eigenvalue fusion law; in characteristic $5$, $\widehat{\mathcal{H}}$ itself is of Monster type $(2,2)$. Ideals and quotients within $\mathcal{H}$ and $\widehat{\mathcal{H}}$ admit explicit principal generators and bases, a distinctive feature enabling complete algebraic classification and corresponding to all known non-Jordan finite Monster-type axial algebras as quotients.

1. Construction and Basis

For a field $\mathbb{F}$ of characteristic $5$0, the Highwater algebra $5$1 is defined as a commutative, non-associative, infinite-dimensional $5$2-algebra generated by a countable family of axes $5$3, together with symmetric elements $5$4 ($5$5) and auxiliary elements $5$6 ($5$7, $5$8). The canonical $5$9-basis is

$\mathcal{H}$0

with multiplication specified by:

• $\mathcal{H}$1 $\mathcal{H}$2, $\mathcal{H}$3 such that $\mathcal{H}$4,
• $\mathcal{H}$5 $\mathcal{H}$6,
• $\mathcal{H}$7 $\mathcal{H}$8,
• $\mathcal{H}$9 $\widehat{\mathcal{H}}$0,
• $\widehat{\mathcal{H}}$1 $\widehat{\mathcal{H}}$2,
• $\widehat{\mathcal{H}}$3 $\widehat{\mathcal{H}}$4, where all $\widehat{\mathcal{H}}$5 are mod $\widehat{\mathcal{H}}$6, and $\widehat{\mathcal{H}}$7 if $\widehat{\mathcal{H}}$8 (Franchi et al., 2022).

2. Fusion Law and Eigenstructure

Each axis $\widehat{\mathcal{H}}$9 is a primitive idempotent and its adjoint action decomposes $5$0 into five eigenspaces with eigenvalues: $5$1 The corresponding fusion rules for the product of vectors from eigenvalue-$5$2 and eigenvalue-$5$3 spaces are determined by the table: $5$4 This non-classical five-eigenvalue fusion law is a hallmark of both $5$5 and its cover $5$6, and is novel within the axial algebra literature. Only in characteristic $5$7 does $5$8, $5$9, and $\widehat{\mathcal{H}}$0 all become $\widehat{\mathcal{H}}$1, returning to the classical Monster fusion law $\widehat{\mathcal{H}}$2 (Franchi et al., 2022).

3. Unified Cover and Characteristic Five Phenomenon

To unify behavior across characteristics, the algebra $\widehat{\mathcal{H}}$3 is constructed using the same generators and multiplication rules over $\widehat{\mathcal{H}}$4.

• For $\widehat{\mathcal{H}}$5, the quotient $\widehat{\mathcal{H}}$6, with $\widehat{\mathcal{H}}$7 the infinite-codimension ideal generated by all $\widehat{\mathcal{H}}$8.
• If $\widehat{\mathcal{H}}$9, $(2,2)$0 itself becomes Monster-type $(2,2)$1, and its quotients match the Franchi–Mainardis characteristic-$(2,2)$2 cover (Franchi et al., 2022). This explains the existence and uniqueness of nontrivial Monster-type covers precisely in characteristic $(2,2)$3—a direct reflection of the fusion constants’ collapse.

4. Ideals, Principal Generators, and Explicit Bases

The ideals of $(2,2)$4 and $(2,2)$5 form two types:

• Type (A): Ideals contained in $(2,2)$6 are principal and generated by

$(2,2)$7

with explicit basis

$(2,2)$8

• Type (B): Ideals not contained in $(2,2)$9 correspond to finite axial codimension $\mathcal{H}$0 and are principal, generated by elements

$\mathcal{H}$1

with an explicit basis involving $\mathcal{H}$2 parametrized by $\mathcal{H}$3 and determined by the ideal-type tuple $\mathcal{H}$4 (Franchi et al., 2022).

The rigidity result—every ideal is $\mathcal{H}$5-invariant and principal—fosters tractable classification and supports the explicit description of all quotients.

5. Quotients, Exceptional Isomorphisms, and Classification

Every ideal yields an explicit principal quotient, and $\mathcal{H}$6 (resp. $\mathcal{H}$7 in characteristic $\mathcal{H}$8) serves as the universal parent for all non-Jordan two-generated symmetric Monster-type axial algebras. In particular:

• For any $\mathcal{H}$9, the quotient $\widehat{\mathcal{H}}$0 has $\widehat{\mathcal{H}}$1 axes and dimension $\widehat{\mathcal{H}}$2, with automorphism group $\widehat{\mathcal{H}}$3.
• The quotients

$\widehat{\mathcal{H}}$4

correspond to finite non-Jordan Monster-type algebras such as $\widehat{\mathcal{H}}$5, $\widehat{\mathcal{H}}$6, and others, with explicit isomorphisms detailed by Yabe and others (Franchi et al., 2022).

The following table summarizes key quotient types derived from principal generators:

Quotient Type	Generator (Ideal)	Dimension, Automorphism Group
Axis-closure $\widehat{\mathcal{H}}$7	$\widehat{\mathcal{H}}$8	$\widehat{\mathcal{H}}$9, $\mathbb{F}$0
“1-type” $\mathbb{F}$1	$\mathbb{F}$2	$\mathbb{F}$3, Monster type $\mathbb{F}$4
Exceptional finite examples	$\mathbb{F}$5 (explicit)	$\mathbb{F}$6, etc.

6. Automorphism Group and Symmetry

A central symmetry is that

$\mathbb{F}$7

the infinite dihedral group acting naturally on the $\mathbb{F}$8. Each axis admits a Miyamoto involution $\mathbb{F}$9, and all ideals are $5$00-invariant, which crucially streamlines their classification and analysis (Franchi et al., 2022).

7. Significance and Unique Features

Highwater algebra represents the only infinite-dimensional two-generated axial Monster-type algebra, the only such algebra generating quotients of all finite dimensions, and the only one exhibiting a new five-eigenvalue fusion rule except in characteristic $5$01 where Monster type is restored. All non-Jordan two-generated symmetric primitive Monster-type algebras outside characteristic $5$02 are quotients of $5$03; in characteristic $5$04, the cover $5$05 realizes the Monster-dimension and fusion law. The complete principal ideal and explicit basis results are unprecedented in the axial algebra literature and essential for the modern classification program (Franchi et al., 2022, Franchi et al., 2021).

This suggests that the interplay between infinite-dimensional algebras, exceptional fusion laws, and characteristic phenomena in Highwater algebra will remain a focal point within axial algebra theory and its connections to finite simple groups.