Inverse semialgebras and partial actions of Lie algebras
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arXiv
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| Format: | Preprint |
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2025
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| author | Dokuchaev, Mikhailo Johari, Farangis Vilca-Rodríguez, José L. |
| author_facet | Dokuchaev, Mikhailo Johari, Farangis Vilca-Rodríguez, José L. |
| contents | We introduce the concept of a non-associative (i.e. non-necessarily associtive) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a non-associative algebra, whereas the associative case is based on such examples as the set of all partially defined linear maps of a vector space, the set of all sections of the structural sheaf of a scheme, the set of all regular functions defined on open subsets of an algebraic variety and the set of all smooth real valued functions defined on open subsets of a smooth manifold. Given a Lie algebra $L$ we define the notion of a partial action of $L$ on a non-associative algebra $A$ as an appropriate premorphism and introduce a Lie inverse semialgebra $E(L),$ which is a Lie analogue of R. Exel's inverse semigroup $S(G)$ that governs the partial actions of a group $G.$ We discuss how $E(L)$ controls the premorphisms from $L$ to $A,$ obtaining results on its total control. We define the concept of an $F$-inverse Lie semialgebra and obtain Lie theoretic analogues of some classical results of the theory of inverse semigroups, namely, we show that the category of partial representations of $L$ in meet semilattices is equivalent to the category ${\mathcal F}$ of $F$-inverse Lie semialgebras with morphisms that preserve the greatest elements of $σ$-classes. In addition, we establish an adjunction between the category of Lie algebras and the category ${\mathcal F}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03081 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse semialgebras and partial actions of Lie algebras Dokuchaev, Mikhailo Johari, Farangis Vilca-Rodríguez, José L. Rings and Algebras 20M18, 17B99, 16W25, 06A12, 17A9 We introduce the concept of a non-associative (i.e. non-necessarily associtive) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a non-associative algebra, whereas the associative case is based on such examples as the set of all partially defined linear maps of a vector space, the set of all sections of the structural sheaf of a scheme, the set of all regular functions defined on open subsets of an algebraic variety and the set of all smooth real valued functions defined on open subsets of a smooth manifold. Given a Lie algebra $L$ we define the notion of a partial action of $L$ on a non-associative algebra $A$ as an appropriate premorphism and introduce a Lie inverse semialgebra $E(L),$ which is a Lie analogue of R. Exel's inverse semigroup $S(G)$ that governs the partial actions of a group $G.$ We discuss how $E(L)$ controls the premorphisms from $L$ to $A,$ obtaining results on its total control. We define the concept of an $F$-inverse Lie semialgebra and obtain Lie theoretic analogues of some classical results of the theory of inverse semigroups, namely, we show that the category of partial representations of $L$ in meet semilattices is equivalent to the category ${\mathcal F}$ of $F$-inverse Lie semialgebras with morphisms that preserve the greatest elements of $σ$-classes. In addition, we establish an adjunction between the category of Lie algebras and the category ${\mathcal F}.$ |
| title | Inverse semialgebras and partial actions of Lie algebras |
| topic | Rings and Algebras 20M18, 17B99, 16W25, 06A12, 17A9 |
| url | https://arxiv.org/abs/2505.03081 |