论文标题
交叉产品$ 4 $ - 代数。申请
Crossed products of $4$-algebras. Applications
论文作者
论文摘要
$ 4 $ -Algebra是一个$ k $上的交换代数$ a $，因此$（a^2）^2 = 0 $，对于$ in $ a $ in a $。我们最近证明\ cite {mil} $ 4 $ - 代数在有限维度的伯恩斯坦代数的分类中起着重要作用。令$ a $为$ 4 $ -Algebra，$ e $ a vector Space和$π： $ e $上的所有$ 4 $ -Algebra结构，以使$π：e \ to a $是代数映射，并由全局共生对象$ {\ mathbb g} {\ Mathbb g} {\ Mathbb H}^{2} {2} \，（A，\，V）$。任何这样的$ 4 $ -Algebra都是跨产品$ v \＃a $和$ {\ Mathbb g} {\ Mathbb H}^{2} \，（A，A，\，V）$是共同的，在所有$ 4 $ -Algebras结构$ \ cdot_v $上，$ v $ nonnontonnony cohom， $ {\ mathbb h}^{2} _ {\ rm nab} \，\ bigl（a，\，（v，v，\，\ cdot_ {v}）\ bigl）$，这是所有分类对象的所有扩展对象，用于$ a $ a $ by $ v $。提供了几个应用程序和示例：尤其是，$ {\ Mathbb g} {\ Mathbb H}^{2} \，（A，A，\，K）$和$ {\ MathBB G} {\ MathBB g} {\ Mathbb H}^{2}^{2} \ {2} \，（K，\，V） （v \＃a/ v）扩展$ v \ hookrightarrow v \＃a $的$。
A $4$-algebra is a commutative algebra $A$ over a field $k$ such that $(a^2)^2 = 0$, for all $a \in A$. We have proved recently \cite{Mil} that $4$-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let $A$ be a $4$-algebra, $E$ a vector space and $π: E \to A$ a surjective linear map with $V = {\rm Ker} (π)$. All $4$-algebra structures on $E$ such that $π: E \to A$ is an algebra map are described and classified by a global cohomological object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$. Any such $4$-algebra is isomorphic to a crossed product $V \# A$ and ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is a coproduct, over all $4$-algebras structures $\cdot_V$ on $V$, of all non-abelian cohomologies ${\mathbb H}^{2}_{\rm nab} \, \bigl(A, \, (V, \, \cdot_{V} )\bigl)$, which are the classifying objects for all extensions of $A$ by $V$. Several applications and examples are provided: in particular, ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ and ${\mathbb G} {\mathbb H}^{2} \, (k, \, V)$ are explicitly computed and the Galois group ${\rm Gal} \, (V \# A/ V )$ of the extension $V \hookrightarrow V \# A$ is described.
