Abstract
We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field.
Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam.
Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object E(X) for the actions on (or split extensions of) an object X. We actually obtain a partial algebra E(X), which we call external weak actor of X, together with a monomorphism of functors SplExt(−,X)↣Hom(U(−),E(X)), which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor USGA(X) and that of E(X) are described in detail. We end with some open questions.
Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam.
Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object E(X) for the actions on (or split extensions of) an object X. We actually obtain a partial algebra E(X), which we call external weak actor of X, together with a monomorphism of functors SplExt(−,X)↣Hom(U(−),E(X)), which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor USGA(X) and that of E(X) are described in detail. We end with some open questions.
| Original language | English |
|---|---|
| Pages (from-to) | 401-444 |
| Number of pages | 44 |
| Journal | Journal of Algebra |
| Volume | 669 |
| Issue number | N/A |
| DOIs | |
| Publication status | Published - 1 May 2025 |
Bibliographical note
Funding Information:The first author is supported by a postdoctoral fellowship \u201CConvocatoria 2021\u201D funded by Universidad de Valladolid, and partially supported by grant PID2022-137283NB-C22 funded by MCIN/AEI/10.13039/501100011033 and ERDF \u201CA way of making Europe\u201D. The second author is supported by Ministerio de Econom\u00EDa y Competitividad (Spain) with grant number PID2021-127075NA-I00. The third author is supported by the University of Palermo, by the \u201CNational Group for Algebraic and Geometric Structures, and their Applications\u201D (GNSAGA \u2013 INdAM), by the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 1409 published on 14/09/2022 by the Italian Ministry of University and Research (MUR), funded by the European Union \u2013 NextGenerationEU \u2013 Project Title Quantum Models for Logic, Computation and Natural Processes (QM4NP) \u2013 CUP B53D23030160001 \u2013 Grant Assignment Decree No. 1371 adopted on 01/09/2023 by the Italian Ministry of Ministry of University and Research (MUR), by the Sustainability Decision Framework (SDF) Research Project \u2013 CUP B79J23000540005 \u2013 Grant Assignment Decree No. 5486 adopted on 04/08/2023, and he is a postdoctoral researcher of the Fonds de la Recherche Scientifique\u2013FNRS. The fourth author is a Senior Research Associate of the Fonds de la Recherche Scientifique\u2013FNRS. The fifth author is supported by the Fonds Thelam of the Fondation Roi Baudouin.
Funding Information:
The first author is supported by a postdoctoral fellowship \u201CConvocatoria 2021\u201D funded by Universidad de Valladolid, and partially supported by grant PID2022-137283NB-C22 funded by MCIN/AEI/10.13039/501100011033 and ERDF \u201CA way of making Europe\u201D. The second author is supported by Ministerio de Econom\u00EDa y Competitividad (Spain) with grant number PID2021-127075NA-I00. The third author is supported by the University of Palermo; by the \u201CNational Group for Algebraic and Geometric Structures, and their Applications\u201D (GNSAGA \u2013 INdAM); by the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 1409 published on 14/09/2022 by the Italian Ministry of University and Research (MUR), funded by the European Union NextGenerationEU \u2013 Project Title Quantum Models for Logic, Computation and Natural Processes (QM4NP) \u2013 CUP B53D23030160001 \u2013 Grant Assignment Decree No. 1371 adopted on 01/09/2023 by the Italian Ministry of Ministry of University and Research (MUR); by the Sustainability Decision Framework (SDF) Research Project \u2013 MISE decree of 31/12/2021 (MIMIT Dipartimento per le politiche per le imprese \u2013 Direzione generale per gli incentivi alle imprese) \u2013 CUP: B79J23000530005, COR: 14019279, Lead Partner: TD Group Italia Srl, Partner: University of Palermo; and he is a postdoctoral researcher of the Fonds de la Recherche Scientifique\u2013FNRS. The fourth author is a Senior Research Associate of the Fonds de la Recherche Scientifique\u2013FNRS. The fifth author is supported by the Fonds Thelam of the Fondation Roi Baudouin.
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