论文标题
格拉斯曼代数
Automorphisms and superalgebra structures on the Grassmann algebra
论文作者
论文摘要
让$ f $为特征零的字段，让$ e $为无限尺寸$ f $ - vector-vector space $ l $的格拉斯曼代数。在本文中，我们研究了代数$ e $允许的Superalgebra结构（即$ \ Mathbb {z} _ {2} $ - 等级）。通过使用二$ $ 2 $的超级二元组与自动形态之间的双重性，我们证明，在许多情况下，$ \ mathbb {z} _ {2} $ - 对此类结构的分级多项式身份与$ \ \ \ \ m athbb {z} $ a {2} $ eNcemed case case case case case case case case y ncise $ e ncistiation $ e_ {k^\ ast} $和$ e_ {k} $，其中vector space $ l $是同质的。回想一下，这些案例是由Di Vincenzo和Da Silva完全描述的\ cite {disil}。此外，我们展示了各种非同质$ \ mathbb {z} _ {2} $ - $ e $上的分级，即$ \ mathbb {z} _ {2} $ - isomorphic- isomorphic to $ e _ _ {\ infty} $，$ e _ {\ e_ _ {特别是，我们构造了一个$ \ mathbb {z} _ {2} $ - 在$ e $上进行分级，仅在$ l $中使用一个均质生成器，这是$ \ mathbb {z} _ {2} $ - iSomorphic- iSomorphic to y Mathbb {z} _ $ a {$ e {$ e {
Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$ admits. By using the duality between superalgebras and automorphisms of order $2$ we prove that in many cases the $\mathbb{Z}_{2}$-graded polynomial identities for such structures coincide with the $\mathbb{Z}_{2}$-graded polynomial identities of the "typical" cases $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$ where the vector space $L$ is homogeneous. Recall that these cases were completely described by Di Vincenzo and Da Silva in \cite{disil}. Moreover we exhibit a wide range of non-homogeneous $\mathbb{Z}_{2}$-gradings on $E$ that are $\mathbb{Z}_{2}$-isomorphic to $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$. In particular we construct a $\mathbb{Z}_{2}$-grading on $E$ with only one homogeneous generator in $L$ which is $\mathbb{Z}_{2}$-isomorphic to the natural $\mathbb{Z}_{2}$-grading on $E$, here denoted by $E_{can}$.
