Report
The length and depth of associative algebras
| Τίτλος: | The length and depth of associative algebras |
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| Συγγραφείς: | Sercombe, Damian, Shalev, Aner |
| Πηγή: | Pacific J. Math. 311 (2021) 197-220 |
| Έτος έκδοσης: | 2020 |
| Συλλογή: | Mathematics |
| Θεματικοί όροι: | Mathematics - Rings and Algebras, Mathematics - Group Theory, Primary 20P70, Secondary 20P10 |
| Περιγραφή: | Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an associative, not necessarily unital, algebra over $k$. An unrefinable chain of $A$ is a chain of subalgebras $A=A_0>A_1>...>A_t=0$ for some integer $t$ where each $A_i$ is a maximal subalgebra of $A_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $A$. It turns out that finite length, finite depth and finite dimension are equivalent properties for $A$. For $A$ finite dimensional, we give a formula for the length of $A$, we bound the depth of $A$, and we study when the length of $A$ equals its dimension and its depth respectively. Finally, we investigate under what circumstances the dimension of $A$ is bounded above by a function of its length, or its depth, or its length minus its depth. Comment: 18 pages |
| Τύπος εγγράφου: | Working Paper |
| DOI: | 10.48550/arXiv.2004.06920 |
| Σύνδεσμος πρόσβασης: | http://arxiv.org/abs/2004.06920 |
| Αριθμός Καταχώρησης: | edsarx.2004.06920 |
| Βάση Δεδομένων: | arXiv |
| DOI: | 10.48550/arXiv.2004.06920 |
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