We construct a new series of \(n^2\)-dimensional axial algebras of Monster type \((2\eta ,\eta )\). They arise as subalgebras A of the Matsuo algebras \(M_\eta (^-O_{n+1}^+(3))\), 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.

中文翻译：

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怪物类型$$（2 \ eta，\ eta）$$（2η，η）的一系列新的轴向代数

我们构造了一系列新的\（n ^ 2 \）维怪兽类型\（（（2 \ eta，\ eta）\））的轴向代数。它们作为由单轴和双轴生成的Matsuo代数\（M_ \ eta（^ -O_ {n + 1} ^ +（3））\）的子代数A出现。同样，我们解决了这些代数何时简单的问题。虽然我们没有解决这个问题，但我们根据计算机代数系统GAP计算了A上Frobenius形式的Gram矩阵的行列式的根，并针对n的几个小值，提出了三个具体的猜想。