Modules over linear spaces admitting a multiplicative basis
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911797466890240 |
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| author | Calderón, Antonio J. Izquierdo, Francisco J. Navarro Sánchez, José M. |
| author_facet | Calderón, Antonio J. Izquierdo, Francisco J. Navarro Sánchez, José M. |
| contents | We study the structure of certain modules $V$ over linear spaces $W$ with restrictions neither on the dimensions nor on the base field $\mathbb F$. A basis $\mathfrak B = \{v_i\}_{i\in I}$ of $V$ is called multiplicative respect to the basis $\mathfrak B' = \{w_j\}_{j \in J}$ of $W$ if for any $i \in I, j \in J$ we have either $v_iw_j = 0$ or $0 \neq v_iw_j \in \mathbb Fv_k$ for some $k \in I$. We show that if $V$ admits a multiplicative basis then it decomposes as the direct sum $V=\bigoplus_k V_k$ of well-described submodules admitting each one a multiplicative basis. Also the minimality of $V$ is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal submodules, admitting each one a multiplicative basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modules over linear spaces admitting a multiplicative basis Calderón, Antonio J. Izquierdo, Francisco J. Navarro Sánchez, José M. Representation Theory We study the structure of certain modules $V$ over linear spaces $W$ with restrictions neither on the dimensions nor on the base field $\mathbb F$. A basis $\mathfrak B = \{v_i\}_{i\in I}$ of $V$ is called multiplicative respect to the basis $\mathfrak B' = \{w_j\}_{j \in J}$ of $W$ if for any $i \in I, j \in J$ we have either $v_iw_j = 0$ or $0 \neq v_iw_j \in \mathbb Fv_k$ for some $k \in I$. We show that if $V$ admits a multiplicative basis then it decomposes as the direct sum $V=\bigoplus_k V_k$ of well-described submodules admitting each one a multiplicative basis. Also the minimality of $V$ is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal submodules, admitting each one a multiplicative basis. |
| title | Modules over linear spaces admitting a multiplicative basis |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2403.08779 |