On the representability of actions of algebras with several bilinear multiplications
Abstract
We study the representability of actions in varieties of algebras with n ≥ 1 not necessarily associative bilinear multiplications. We first prove that, for homogeneous varieties, action accessibility is equivalent to being an Orzech category of interest. For an action accessible, operadic variety we then construct the external weak actor ℰ(X) of an algebra X, an algebra with n bilinear partial operations together with a natural monomorphism encoding the actions on X. We characterise the algebras admitting an actor in terms of ℰ(X), and we deduce an obstruction to action representability: if, for some algebra X with trivial multiplications, ℰ(X) does not satisfy a multilinear identity of the variety, then the variety is not action representable. Computing the external weak actor explicitly, we obtain that associative and matching dialgebras are weakly action representable but not action representable, and that Poisson algebras, left-right non-commutative Poisson algebras and matching Lie algebras are not action representable. Finally, we introduce the reduced type (p,q) of such a variety and prove that, over a field of characteristic different from $2$, an operadic variety of reduced type (p,0), (0,q) or (1,1) is action representable if and only if it is term equivalent to the variety of Lie algebras or to that of abelian algebras. For the reduced types (p,0) and (0,q), the arguments require no assumption on the cardinality of the field, whereas the case (1,1) requires the field to be infinite.
Related
- Weak central polynomials for algebras of multiplications of simple algebras
- Associativity of operator convolutions for group actions on von Neumann algebras
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- Selfless Reduced Crossed Product C*-Algebras Arising from Almost Periodic Actions
- Group actions on von Neumann algebras with compact open subgroups
BibTeX
@misc{mancini2026representability,
title = {{On the representability of actions of algebras with several bilinear multiplications}},
author = {Manuel Mancini},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02097},
eprint = {2611.02097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02097},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On the representability of actions of algebras with several bilinear multiplications
AU - Manuel Mancini
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02097
UR - https://arxiv.org/abs/2611.02097
AB - We study the representability of actions in varieties of algebras with n ≥ 1 not necessarily associative bilinear multiplications. We first prove that, for homogeneous varieties, action accessibility is equivalent to being an Orzech category of interest. For an action accessible, operadic variety we then construct the external weak actor ℰ(X) of an algebra X, an algebra with n bilinear partial operations together with a natural monomorphism encoding the actions on X. We characterise the algebras admitting an actor in terms of ℰ(X), and we deduce an obstruction to action representability: if, for some algebra X with trivial multiplications, ℰ(X) does not satisfy a multilinear identity of the variety, then the variety is not action representable. Computing the external weak actor explicitly, we obtain that associative and matching dialgebras are weakly action representable but not action representable, and that Poisson algebras, left-right non-commutative Poisson algebras and matching Lie algebras are not action representable. Finally, we introduce the reduced type (p,q) of such a variety and prove that, over a field of characteristic different from $2$, an operadic variety of reduced type (p,0), (0,q) or (1,1) is action representable if and only if it is term equivalent to the variety of Lie algebras or to that of abelian algebras. For the reduced types (p,0) and (0,q), the arguments require no assumption on the cardinality of the field, whereas the case (1,1) requires the field to be infinite.
ER -