论文标题
$ g_2 $和$ f_4 $的非求和Frobenius代数
Non-associative Frobenius algebras of type $G_2$ and $F_4$
论文作者
论文摘要
最近，Maurice Chayet和Skip Garibaldi引入了一类交换性非缔合代数，对于任意领域的每个简单的线性代数群（对特征的限制有一些较小的限制）。 我们对这些代数的明确描述分别针对$ g_2 $和$ f_4 $的组分别在八月代数和阿尔伯特代数方面。作为副产品，我们确定代表上的所有可能不变的交换代数产品，重量最高$2Ω_1$，$ g_2 $，以及最高权重$2Ω_4$的代表性，对于$ f_4 $。 Chayet和Garibaldi已经观察到，代数$ f_4 $的代数的汽车组等于$ f_4 $本身的类型。使用我们的新描述，我们能够证明类型$ G_2 $的结果相同。
Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). We give an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight $2ω_1$ for $G_2$ and on the representation with highest weight $2ω_4$ for $F_4$. It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type $F_4$ is equal to the group of type $F_4$ itself. Using our new description, we are able to show that the same result holds for type $G_2$.
