arXiv 2009.040552020-09math.RA

理想扩张与直无限代数

Ideal Extensions and Directly Infinite Algebras

Daniel P. Bossaller

备注：: 19 Pages. Significant revision of the previous version. To appear in Pure and Applied Algebra

摘要：

直接无限代数，那些有一对元素$x$和$y$的代数，$E$，其中$1 = xy \neq yx$，众所周知有一个子代数同构于$M_\infty(K)$，无限$\zplus \times \zplus$ -索引矩阵的集合，其中只有100个非零元素。当这个子代数实际上是一个理想时，我们可以根据某个代数$A$通过$M_\infty(K)$的扩张来分析这个代数，即$K$ -代数$0 \to M_\infty(K) \to E \to A \to 0$的短正合序列。本文通过考察$K[x,x^{-1}]$的平凡扩张作为无限矩阵代数的子代数，刻画了由$M_\infty(K)$构成的$K[x,x^{-1}]$的所有平凡扩张.此外，我们构造了一个无限族的两两非同构扩张$\{\mathcal T_i : i \geq 0\}$，所有这些都可以写为一个扩张$0 \to M_\infty(K) \to \mathcal T_i \to K[x,x^{-1}] \to 0$。

英文摘要：

Directly infinite algebras, those algebras, $E$ which have a pair of elements $x$ and $y$ where $1 = xy \neq yx$, are well known to have a sub-algebra isomorphic to $M_\infty(K)$, the set of infinite $\zplus \times \zplus$-indexed matrices which have only finitely many nonzero entries. When this sub-algebra is actually an ideal, we may analyze the algebra in terms of an extension of some algebra $A$ by $M_\infty(K)$, that is, a short exact sequence of $K$-algebras $0 \to M_\infty(K) \to E \to A \to 0$. The present article characterizes all trivial (split) extensions of $K[x,x^{-1}]$ by $M_\infty(K)$ by examining the extensions as sub-algebras of infinite matrix algebras. Furthermore, we construct an infinite family of pairwise non-isomorphic extensions $\{\mathcal T_i : i \geq 0\}$, all of which can be written as an extension $0 \to M_\infty(K) \to \mathcal T_i \to K[x,x^{-1}] \to 0$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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