---
title: Highwater Algebra
url: https://www.emergentmind.com/topics/highwater-algebra
type: topic
---

# Highwater Algebra

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 $\ne 2,3$, the Highwater algebra $\mathcal{H}$ is defined as a commutative, non-associative, infinite-dimensional $\mathbb{F}$-algebra generated by a countable family of axes $\{a_i : i \in \mathbb{Z}\}$, together with symmetric elements $s_j$ ($j \geq 1$) and auxiliary elements $p_{r,3j}$ ($j \geq 1$, $r = 1,2$). The canonical $\mathbb{F}$-basis is
\[
\mathcal{H} = \bigoplus_{i \in \mathbb{Z}} \langle a_i \rangle \oplus \bigoplus_{j \geq 1} \langle s_j \rangle \oplus \bigoplus_{\substack{j \geq 1 \\ j \equiv 0 \, (3)}} (\langle p_{1,j} \rangle \oplus \langle p_{2,j} \rangle)
\]
with multiplication specified by:
- $(H1)$ $a_i a_\ell = (a_i + a_\ell) + s_{|i-\ell|} + p_{r,|i-\ell|}$, $r$ such that $|i-\ell| \equiv r \pmod{3}$,
- $(H2)$ $a_i s_j = -[a_i + 2(a_{i-j} + a_{i+j}) + 2 s_j] - p_{1,j} + p_{2,j}$,
- $(H3)$ $a_i p_{r,3j} = 2 p_{r,3j} - p_{r+1, 3j}$,
- $(H4)$ $s_j s_k = (s_j + s_k) - \sum_{\delta \in \{\pm 1\}} (s_{|j-k|} + s_{j+k})$,
- $(H5)$ $s_j p_{r,3k} = 2(p_{r,3j} + p_{r,3k}) - (p_{r,3|j-k|} + p_{r,3(j+k)})$,
- $(H6)$ $p_{r,3j} p_{s,3k} = (p_{r+s,3j} + p_{r+s,3k}) - (p_{r+s,3|j-k|} + p_{r+s,3(j+k)})$,
where all $r+s$ are mod $2$, and $p_{r,3m} = 0$ if $m < 1$ [2205.02200].

## 2. Fusion Law and Eigenstructure

Each axis $a_i$ is a primitive idempotent and its adjoint action decomposes $\mathcal{H}$ into five eigenspaces with eigenvalues:
\[
\{1,\,0,\,\alpha,\,\beta,\,\gamma\};\qquad \alpha = 2,\ \beta = 2,\ \gamma\ \text{arbitrary}
\]
The corresponding fusion rules for the product of vectors from eigenvalue-$\lambda$ and eigenvalue-$\mu$ spaces are determined by the table:
\[
\begin{array}{c|ccccc}
* & 1 & 0 & \alpha & \beta & \gamma \\
\hline
1 & \{1\} & \emptyset & \{\alpha\} & \{\beta\} & \{\gamma\} \\
0 & \emptyset & \{0\} & \emptyset & \emptyset & \emptyset \\
\alpha & \{\alpha\} & \emptyset & \{1,0\} & \{\gamma\} & \{\beta\} \\
\beta & \{\beta\} & \emptyset & \{\gamma\} & \{1,0\} & \{\alpha\} \\
\gamma & \{\gamma\} & \emptyset & \{\beta\} & \{\alpha\} & \{1,0\}
\end{array}
\]
This non-classical five-eigenvalue fusion law is a hallmark of both $\mathcal{H}$ and its cover $\widehat{\mathcal{H}}$, and is novel within the axial algebra literature. Only in characteristic $5$ does $\alpha$, $\beta$, and $\gamma$ all become $0$, returning to the classical Monster fusion law $\mathrm{M}(2,2)$ [2205.02200].

## 3. Unified Cover and Characteristic Five Phenomenon

To unify behavior across characteristics, the algebra $\widehat{\mathcal{H}}$ is constructed using the same generators and multiplication rules over $\mathrm{char}(\mathbb{F}) \ne 2$. 
- For $\mathrm{char}(\mathbb{F}) \neq 5$, the quotient $\widehat{\mathcal{H}} / J \cong \mathcal{H}$, with $J$ the infinite-codimension ideal generated by all $p_{r,3j}$.
- If $\mathrm{char}(\mathbb{F}) = 5$, $\widehat{\mathcal{H}}$ itself becomes Monster-type $(2,2)$, and its quotients match the Franchi–Mainardis characteristic-$5$ cover [2205.02200].
This explains the existence and uniqueness of nontrivial Monster-type covers precisely in characteristic $5$—a direct reflection of the fusion constants’ collapse.

## 4. Ideals, Principal Generators, and Explicit Bases

The ideals of $\widehat{\mathcal{H}}$ and $\mathcal{H}$ form two types:
- **Type (A):** Ideals contained in $J$ are principal and generated by
  \[
  x = \sum_{j=1}^k \beta_{3j}(p_{1,3j} + p_{2,3j})
  \]
  with explicit basis
  \[
  \{x,\, x^{T_0},\, s_{3i}x,\, (s_{3i}x)^{T_0} \mid i \geq 1 \}
  \]
- **Type (B):** Ideals not contained in $J$ correspond to finite axial codimension $D$ and are principal, generated by elements
  \[
  x = \sum_{i=0}^D \alpha_i a_i,\,\,\, \alpha_0, \alpha_D \neq 0,\,\,\, \sum_{i=0}^D \alpha_i = 0,\,\,\, \exists \epsilon \in \{\pm 1\} : \alpha_i = \epsilon \alpha_{D-i}
  \]
  with an explicit basis involving $(x_k, y_k, p_k^{(1)}, p_k^{(2)})$ parametrized by $k$ and determined by the ideal-type tuple $(\alpha_0, ..., \alpha_D)$ [2205.02200].

The rigidity result—every ideal is $\Aut$-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 $\mathcal{H}$ (resp. $\widehat{\mathcal{H}}$ in characteristic $5$) serves as the universal parent for all non-Jordan two-generated symmetric Monster-type axial algebras. In particular:
- For any $n \ge 1$, the quotient $\mathcal{H}/(a_0 - a_n)$ has $n$ axes and dimension $n + \lfloor n/2 \rfloor$, with automorphism group $D_{2n}$.
- The quotients
  \[
  \mathcal{H}/(2a_0 - (a_{-n} + a_n)),\quad \mathcal{H}/(a_0 - a_2),\quad \mathcal{H}/(a_0 + a_1 - a_2 - a_3)
  \]
  correspond to finite non-Jordan Monster-type algebras such as $3C(2)$, $S(2)^\circ$, and others, with explicit isomorphisms detailed by Yabe and others [2205.02200].

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

| Quotient Type                 | Generator (Ideal)                   | Dimension, Automorphism Group         |
|-------------------------------|-------------------------------------|---------------------------------------|
| Axis-closure $(n)$            | $a_0 - a_n$                         | $n + \lfloor n/2 \rfloor$, $D_{2n}$  |
| “1-type” $(n)$                | $2a_0 - (a_{-n} + a_n)$             | $3n - 1$, Monster type $(2,2)$        |
| Exceptional finite examples    | $\sum \alpha_i a_i$ (explicit)      | $3C(2), S(2)^\circ$, etc.             |

## 6. Automorphism Group and Symmetry

A central symmetry is that
\[
\Aut(\mathcal{H}) \cong D_\infty
\]
the infinite dihedral group acting naturally on the $a_i$. Each axis admits a Miyamoto involution $T_i$, and all ideals are $\Aut$-invariant, which crucially streamlines their classification and analysis [2205.02200].

## 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 $5$ where Monster type is restored. All non-Jordan two-generated symmetric primitive Monster-type algebras outside characteristic $5$ are quotients of $\mathcal{H}$; in characteristic $5$, the cover $\widehat{\mathcal{H}}$ 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 [2205.02200, 2101.10315].

*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.

Source: https://www.emergentmind.com/topics/highwater-algebra