A new series of axial algebras of Monster type (2η,η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2\eta ,\eta )$$\end{document}

We construct a new series of n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^2$$\end{document}-dimensional axial algebras of Monster type (2η,η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2\eta ,\eta )$$\end{document}. They arise as subalgebras A of the Matsuo algebras Mη(-On+1+(3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_\eta (^-O_{n+1}^+(3))$$\end{document}, generated by single and double axes. Also, we address the question of when these algebras are simple. While we do not solve this question, we formulate three concrete conjectures based on the calculation, using the computer algebra system GAP, of the roots of the determinant of the Gram matrix of the Frobenius form on A for several small values of n.


Introduction
Axial algebras are a new class of commutative non-associative algebras introduced by Hall, Rehren, and Shpectorov [5,6]. Recently, Joshi [8] constructed a series of subalgebras 1 of dimension n 2 of the Matsuo algebras M η (S 2n ), generated by single and double axes. In this work, we deal with the case of Matsuo algebras M η ( − O + n+1 (3)) corresponding to a class of reflections of the isometry group of a nondegenerate orthogonal space over F 3 . In these Matsuo algebras, we construct a new series of subalgebras, also of dimension n 2 .
Let G = G O + n+1 (3) denote the group of orthogonal transformations of a vector space V of dimension n +1 over the finite field F 3 , endowed with a nondegenerate symmetric bilinear form admitting an orthonormal basis {e 0 , e 1 , . . . , e n }. That is, the vectors e i are pairwise orthogonal and (e 0 , e 0 ) = (e 1 , e 1 ) = · · · = (e n , e n ) = 1.
Let C be the set of reflections: where u is a vector with (u, u) = −1. Consider G = C ≤ G O + n+1 (3). Then, (G, C) is a 3-transposition group denoted − O + n+1 (3) (see [2]). We note that r u = r −u , so the elements of C are in a bijection with the set of 1-dimensional subspaces u of V , where (u, u) = −1, and not with the vectors u themselves. Hence, we can identify the elements of C with such subspaces u .
M. Alsaeedi (B) University of Birmingham, Birmingham, UK E-mail: mxa622@student.bham.ac.uk Let K be a field of characteristic not equal to 2 and let η ∈ K, η = 0, 1. Recall from [6] that the Matsuo algebra M η (G, C) over K corresponding to the 3-transposition group (G, C) has C as its basis, with multiplication of basis elements a, b ∈ C defined by: if a = b and ab = ba; The basis elements a ∈ C are the axes of the axial algebra M η (G, C), and we refer to them below as single axes. Double axes are sums a + b of two orthogonal (ab = 0) single axes. In the theorem below, we identify elements of C with the corresponding 1-spaces in the orthogonal space V (see the comment above). (3)) spanned by the set of single axes S = { e i + e j : 1 ≤ i < j ≤ n, = ±1} and the set of double axes D = { e 0 + e i + e 0 − e i | 1 ≤ i ≤ n}. Then, A is a primitive axial algebra of Monster type (2η, η) of dimension |S| + |D| = n(n − 1) + n = n(n − 1 + 1) = n 2 .

Theorem 1 Let A be the subspace of the Matsuo algebra M
In Sect. 2, we provide the necessary background on axial algebras. It is divided into five subsections. First, we introduce the basics of axial algebras, and define the Matsuo algebra and the Frobenius form on it. Then, we discuss double axes and the fusion rules M(2η, η) they satisfy. We conclude this section by defining the subalgebras of dimension n 2 constructed in [8].
In Sect. 3, we prove our main result, Theorem 1, and show that the subalgebra A of dimension n 2 in the Matsuo algebra M η ( − O + n+1 (3)) is not isomorphic to the subalgebra of M η (S 2n ) constructed by Joshi [8]. The goal of Sect. 4 is to study the conditions under which A is simple. In Proposition 4.3, we show that no proper ideal of A can contain one of the generating axes in S ∪ D. By [9], this means that A is simple exactly when the Frobenius form on A has zero radical (equivalently the Gram matrix of the Frobenius form on A has non-zero determinant). We calculate, in GAP [4], the determinant as a polynomial in η and we find its roots for n ≤ 14. Based on this, we put forward several precise conjectures describing the roots and their multiplicities for arbitrary n.
The author would like to express her gratitude to Sergey Shpectorov who provided help in preparation of this paper.

Background
In this paper, we consider non-associative algebras, which means algebras that are not necessarily associative.

Axial algebras
Suppose that A is a commutative algebra over a field F. For arbitrary a ∈ A, we write ad a for the adjoint map in End(A) that is given by ad a : b → ab. The eigenvalues, eigenvectors, and eigenspaces of a are the eigenvalues, eigenvectors, and eigenspaces of ad a , respectively. The element a is said to be diagonalisable if there exists a basis of A consisting of eigenvectors for ad a . For λ ∈ F, we write: for the λ-eigenspace of a. This is trivial when λ is not an eigenvalue of a. For F ⊆ F: Note that A ∅ (a) = 0.

Definition 2.1
A fusion law is a pair (F, * ) where F ⊆ F and * is a symmetric map * : F × F → P(F). Now, we give the definition of an F-axial algebra.
for all λ, μ ∈ F. That is, every product bc of a λ-eigenvector b with a μ-eigenvector c is a sum of some ν-eigenvectors, for ν ∈ λ * μ.
Let A be an algebra over F. Notice that if a is an idempotent, then 1 is an eigenvalue of ad a . Hence, we always assume that 1 ∈ F. Definition 2. 3 The algebra A is an F-axial algebra if it is generated by a set of F-axes.

Definition 2.4
An axis a ∈ A is primitive if A 1 (a) = Fa, (i.e., it is 1-dimensional). The algebra A is called primitive if it is generated by primitive axes.
The following is a well-known example.
Example 2. 5 The Griess algebra over R is of dimension 196, 884. This algebra is primitive axial and the fusion rules are given in the table in Fig. 1.

Matsuo algebras
A point-line geometry is a pair (P, L) consisting of a set of points, P, and a set of lines, L. We will assume that every line is a set of points, that is, L ⊆ 2 P , and that every line has size at least 2. A partial linear space is a point-line geometry where any two distinct lines intersect in at most one point. A partial triple system is a partial linear space (P, L) where every line consists of exactly three points. For any two collinear points a and b in a partial triple system, there exists a unique line through a and b. This line consists of a, b, and a third point c = a ∧ b. (We borrow this notation from [3]. ) We will write a ∼ b to indicate that a and b are collinear. If a and b are non-collinear, that is, there is no line through a and b, then we write a b. We denote by a ∼ the set of all points collinear to a (this excludes a). The complement of {a} ∪ a ∼ is the set a of all points that are not collinear with a.
A non-empty set of points P ⊆ P is a subspace of (P, L) if any line L ∈ L containing two points from P is fully contained in it. The subspace P can be viewed as a geometry if we endow it with the set of lines L = {L ∈ L : L ⊆ P }. Clearly, any non-empty intersection of subspaces is again a subspace. This allows us to define the subspace generated by a set of points X , X . This is the unique smallest subspace containing X . Subspaces generated by three points not contained in single line are called planes. It is easy to see that a subspace generated by a pair of intersecting lines is a plane.

Definition 2.6
A Fisher space is a partial triple system where any plane generated by two intersecting lines is isomorphic to the dual affine plane of order 2, denoted by P ∨ 2 , (see Fig. 2) or to the affine plane of order 3, denoted by P 3 .
Given a 3-transposition group (G, C) (see [1]), we define a geometry G = (P, L) by setting P = C and: This geometry G is a Fisher space and, conversely, every Fisher space can be obtained in this way from a 3-transposition group. Let F be a field with char F = 2 and η ∈ F, η = 0, 1. For a Fisher space G = (P, L), define the Matsuo algebra M = M η (G) as the algebra over F whose basis is the set of points P and multiplication on the basis is given by: Note that the elements a ∈ P are idempotent and, in fact, they are the axes of this algebra; namely, it is shown in [6] that a satisfies the fusion law of Jordan type η, as in Fig. 3.
That is, Matsuo algebras belong to the class of axial algebras of Jordan type η introduced by Hall, Rehren, and Shpectorov in [6]. In addition to Matsuo algebras, the class of algebras of Jordan type contains all Jordan algebras generated by idempotents. This is because each idempotent in a Jordan algebras satisfies the Peirce decomposition, which exactly represents the fusion law of Jordan type 1 2 . Hence, Jordan algebras are algebras of Jordan type 1 2 .

Frobenius form
Definition 2.8 A Frobenius form on an axial algebra A is a non-zero symmetric bilinear form that associates with the algebra product: According to [6], the Matsuo algebra M η (G) admits the Frobenius form given by: . We note that any axial subalgebra A of M η (G) inherits a Frobenius form as long as the form is not zero on A.

Double axes
Here, we are focussing on the case η = 1 2 , so that 2η = 1. We define double axes as follows. Definition 2.9 Consider a Matsuo algebra M = M η (G, C), where (G, C) is a group of 3-transpositions. Let a, b be any two Matsuo axes, such that a · b = 0. Then, x = a + b will be called a double axis.
Note that axes a and b satisfying a · b = 0 are called orthogonal. It is easy to see that a double axis is an idempotent: Note that x = a + b acts on M αβ (a, b) as the scalar α + β.
The fusion table in this case is given in Fig. 4. ' Remarks • Note that the fusion rules J (η) satisfied by every single axis a ∈ M is obtained by dropping a row and a column from M(2η, η). This corresponds simply to the 2η-eigenspace being zero. Here, we introduce the subalgebra of dimension n 2 in the Matsuo algebra M η (S 2n ), which was constructed in [8]. Recall that the generating axes of M η (S 2n ) are the transpositions of S 2n . The single axes are: (2i − 1, 2i), i = 1, . . . , n, and the double axes are: In this paper, we construct a similar subalgebra of dimension n 2 in the Matsuo algebra M η ( − O + n+1 (3)). However, our subalgebra contains n(n − 1) single axes and n double axes.

The new n 2 -algebra
We will see below that (G, C) is a 3-transposition group, and it is denoted − O + n+1 (3). Let M = M η (G, C) be the corresponding Matsuo algebra. In this section, we construct a subalgebra of M of dimension n 2 , generated by single and double axes.
Recall that a reflection in a nonsingular vector u (i.e., u satisfies (u, u) = 0) is given by: Remarks Since (u, u) = −1 and 2 = −1 in F 3 , in this case,

Lemma 3.1 For every
Hence, we obtain that r α u = r u α . Note that r u = r v if and only if v = ±u. Indeed, it is easy to see that r u = r αu for 0 = α ∈ F 3 . Conversely, (u,u) u, which immediately implies that u is a multiple of v. ) and suppose that u and v are independent; that is, u = ±v. Then, Proof If (u, v) = 0, then u r v = u, and so, by Lemma 3.0.1, r r v u = r u . This means that (r u r v ) 2 = 1, and so, |r u r v | = 2. Now, suppose that (u, v) = 0. Substituting −v for v if necessary, we may assume that (u, v) = −1.
In particular, r r v u = r r u v , which means that (r u r v ) 3 = 1, and so, |r u r v | = 3.
This proposition shows that the class C of reflections r u with (u, u) = −1 is a class of 3-transpositions and so − O + n+1 (3) = (G, C), where G = C , is a 3-transposition group, as claimed in the introduction and at the beginning of this section. We recall from the introduction that we identify the element r u ∈ C with the one-dimensional subspace u as both u and −u define the same element r u = r −u of C. .
We will now prove the main result of the paper, Theorem 1. Recall that η ∈ F and η / ∈ {1, 0, 1 2 }. Proof To show that A is a subalgebra, we need to check that A is closed under multiplication. We establish this by looking through the possible cases of pairs of axes a, b ∈ S ∪ D and showing in each case that ab ∈ A. Note that every axis, single or double, is an idempotent, so we just need to consider pairs of distinct axes: a = b.
Let us start with two single axes: a = e i + e j and b = e i + e j . Then, |{i, j} ∩ {i , j }| is 0, 1 or 2. If {i, j} and {i , j } are disjoint, then, clearly, ab = 0 since (e i + e j , e i + e j ) = 0. If |{i, j} ∩ {i , j }| = 1, then, without loss of generality, i = i , that is, b = e i + e j . In this case, ab = η 2 (a + b − c), where c = b r a = − e j + e j . Manifestly, c ∈ S, and so, ab ∈ A. Finally, suppose that |{i, j} ∩ {i , j }| = 2. Then, without loss of generality, a = e i + e j and b = e i − e j . Here, (e i + e j , e i − e j ) = 0, and so again, as in the first case, ab = 0.
Next, assume that a = e i + e j is a single axis and b = e 0 +e k + e 0 −e k is a double axis. Here, we have two options: either k / ∈ {i, j} or k ∈ {i, j} (say, k = i). In the first case, ab = 0, since (e i ± e j , e 0 ± e k ) = 0. If k = i, then: Clearly, e 0 + e j + e 0 − e j ∈ D, and so, ab ∈ A.
Clearly, all summand here are in A, so in this final case, ab ∈ A.
We have shown that A is a subalgebra. Manifestly, the vectors in S ∪ D are linearly independent, and so, they form a basis of A. This yields the claim concerning the dimension of A.
It remains to show that the double axes x = e 0 + e i + e 0 − e j are primitive in A. Consider σ = r e 0 . This involution fixes all single axes in S, and it switches two single axes a = e 0 + e i and b = e 0 − e i in every x = a + b ∈ D. Hence, S ∪ D is contained in the fixed subalgebra M σ , which means that A is contained in M σ . Recall from Theorem 2.10 that M 1 (x) = a, b . Within M 1 (x), σ fixes a + b = x and inverts a − b. Hence, A 1 (x) = A ∩ M 1 (x) = x , and so, x is indeed primitive in A.
Clearly, this algebra is different as axial algebra from the algebra constructed by Joshi. This is because the number of single and double axes do not match.

Frobenius form and simplicity of A
In this section, A is the n 2 (3)) that we constructed in the previous section. Here, we use the theory developed in [9] to investigate the following question: for which values of η is A a simple algebra?

Ideals in A
According to [9], the ideals of A containing axes from S ∪ D are controlled by the projection graph on the set S ∪ D of axes of A.  Proof According to [9], it suffices to show that the projection graph is connected. By Proposition 4.1, we see that the single axis e i + e j is connected by edges to both double axes e 0 + e i + e 0 − e i and e 0 + e j + e 0 − e j . Thus, all double axes and all single axes are contained in the same connected component of the projection graph.

Radical
We turn now to ideals of A that contain no axes from S ∪ D. All such ideals are contained in the radical of A, which is defined in [9] as the largest ideal not containing any of the generating axes of A. It is also shown in [9] that, in the presence of a Frobenius form having non-zero values (a, a) on all generating axes a, the radical of A coincides with the radical: of the Frobenius form on A. Clearly, this radical is non-zero if and only if the determinant of the Gram matrix of the Frobenius form is non-zero. Clearly, the determinant of the Gram matrix (written with respect to the basis S ∪ D of A) is a polynomial in η of degree depending on n. In the next (and final) subsection, we compute this polynomial for n ≤ 14, and based on this, we put forward exact conjectures concerning the values of η for the radical is non-zero (and, hence, A is not simple).

Critical values of η
Here, we use GAP [4] to compute and factorize the determinant of the Gram matrix of the Frobenius form on A for small values of n. We conclude this section with some conjectures. Hence, A ⊥ is non-zero and A is not simple if and only if η = 1 2 or η = − 1 4 . Similarly, we do calculations for n ≤ 14. The results are summarized in Table 1. The data in the table are very suggestive and they allow us to formulate several conjectures. Conjecture 4. 6 The determinant of the Gram matrix G is a polynomial of degree n(n+1) 2 , unless n = 3.