允许乘法基的Leibniz三元系
Leibniz triple systems admitting a multiplicative basis
摘要：
设$(T,\langle \cdot, \cdot, \cdot \rangle)$是任意基域${\mathbb F}$上的任意维数的Leibniz三元系. $T$的基${\mathcal B} = \{e_{i}\}_{i \in I}$称为乘法的，如果对于任何$i,j,k \in I$，我们对于某些$r \in I$有该$\langle e_i,e_j,e_k\rangle\in {\mathbb F}e_r$。我们表明，如果$T$承认一个乘法的基础，那么它分解为正交直和$T= \bigoplus_k{\mathfrak I}_k$的良好描述的理想${\mathfrak I}_k$承认每一个乘法的基础。此外，极小的$T$的特点是在乘法基础，它表明，在一个温和的条件下，上述直和是通过家庭的极小理想。
英文摘要：
Let $(T,\langle \cdot, \cdot, \cdot \rangle)$ be a Leibniz triple system of arbitrary dimension, over an arbitrary base field ${\mathbb F}$. A basis ${\mathcal B} = \{e_{i}\}_{i \in I}$ of $T$ is called multiplicative if for any $i,j,k \in I$ we have that $\langle e_i,e_j,e_k\rangle\in {\mathbb F}e_r$ for some $r \in I$. We show that if $T$ admits a multiplicative basis then it decomposes as the orthogonal direct sum $T= \bigoplus_k{\mathfrak I}_k$ of well-described ideals ${\mathfrak I}_k$ admitting each one a multiplicative basis. Also the minimality of $T$ is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.
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