论文标题
高水代数及其封面的商
Quotients of the Highwater algebra and its cover
论文作者
论文摘要
轴向代数是一类非缔合代数，与有限（尤其是简单）组有很强的联系，这些组最近受到了很多关注。主要感兴趣的是Monster Type $（α，β）$的轴向代数，其中Griess代数（以怪物为其自动形态组）是一个重要的激励示例。在本文中，我们完成了对称$ 2 $ 2 $的原始轴向代数（α，β）$的分类。 根据Yabe和Franchi和Mainardis的先前工作，任何此类代数都是明确已知的，或者是Infinite高水量高水代数$ \ MATHCAL {H} $的商，或其特征$ 5 $ cover $ \ hat $ \ hat {\ Mathcal {h h}} $。在本文中，我们将$ \ Mathcal {h} $和$ \ hat {\ Mathcal {h}} $的理想分类。此外，我们为理想提供明确的基础。实际上，我们通过定义$ \ hat {\ Mathcal {h}} $ $ \ Mathcal {h} $在所有特征中的$ \ hat {\ Mathcal {h}} $中以统一的方式进行。我们的新代数$ \ hat {\ Mathcal {h}} $具有以前看不见的融合法，并提供了一个见解，说明为什么高水代数为何具有仅具有特征性$ 5 $的怪物类型的封面。
Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(α, β)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(α, β)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra $\mathcal{H}$, or its characteristic $5$ cover $\hat{\mathcal{H}}$. In this paper, we classify the ideals of $\mathcal{H}$ and $\hat{\mathcal{H}}$ and thus their quotients. Moreover, we give explicit bases for the ideals. In fact, we proceed in a unified way, by defining a cover $\hat{\mathcal{H}}$ of $\mathcal{H}$ in all characteristics and classifying its ideals. Our new algebra $\hat{\mathcal{H}}$ has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic $5$.
