arXiv subjects

Antonio J. Calderón

Publications and source records attributed to Antonio J. Calderón.

12 recordsLinked to original sources

arXiv2024

On the structure of graded Lie superalgebras

We study the structure of graded Lie superalgebras with arbitrary dimension and over an arbitrary field ${\mathbb K}$. We show that any of such algebras ${\mathfrak L}$ with a symmetric $G$-support is of the form ${\mathfrak L} = U + \sum\limits_{j}I_{j}$ with $U$ a subspace of ${\mathfrak L}_1$ and any $I_{j}$ a well described graded ideal of ${\mathfrak L}$, satisfying $[I_j,I_k] = 0$ if $j\neq k$. Under certain conditions, it is shown that ${\mathfrak L} = (\bigoplus\limits_{k \in K} I_k) \oplus (\bigoplus\limits_{q \in Q} I_q),$ where any $I_k$ is a gr-simple graded ideal of ${\mathfrak L}$ and any $I_q$ a completely determined low dimensional non gr-simple graded ideal of ${\mathfrak L}$, satisfying $[I_q,I_{q'}] = 0$ for any $q'\in Q$ with $q \neq q'$.

math.RT↗

arXiv2024

On split regular Hom-Lie superalgebras

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra ${\frak L}$ is of the form ${\frak L} = U + \sum_j I_j$ with $U$ a linear subspace of a maximal abelian graded subalgebra $H$ and any $I_j$ a well described (split) ideal of ${\frak L}$ satisfying $[I_j,I_k] = 0$ if $j \neq k$. Under certain conditions, the simplicity of ${\frak L}$ is characterized and it is shown that ${\frak L}$ is the direct sum of the family of its simple ideals.

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arXiv2024

On split Leibniz superalgebras

We study the structure of arbitrary split Leibniz superalgebras. We show that any of such superalgebras ${\frak L}$ is of the form ${\frak L} = {\mathcal U} + \sum_jI_j$ with ${\mathcal U}$ a subspace of an abelian (graded) subalgebra $H$ and any $I_j$ a well described (graded) ideal of ${\frak L}$ satisfying $[I_j,I_k] = 0$ if $j \neq k$. In the case of ${\frak L}$ being of maximal length, the simplicity of ${\frak L}$ is also characterized in terms of connections of roots.

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arXiv2024

Modules over linear spaces admitting a multiplicative basis

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.

math.RT↗

arXiv2024

Leibniz algebras and graphs

We consider a Leibniz algebra ${\mathfrak L} = {\mathfrak I} \oplus {\mathfrak V}$ over an arbitrary base field $\mathbb{F}$, being ${\mathfrak I}$ the ideal generated by the products $[x,x], x \in {\mathfrak L}$. This ideal has a fundamental role in the study presented in our paper. A basis $\B=\{v_i\}_{i \in I}$ of ${\mathfrak L}$ is called multiplicative if for any $i,j \in I$ we have that $[v_i,v_j] \in {\mathbb F}v_k$ for some $k \in I$. We associate an adequate graph $Γ({\mathfrak L},\B)$ to ${\mathfrak L}$ relative to $\B$. By arguing on this graph we show that ${\mathfrak L}$ decomposes as a direct sum of ideals, each one being associated to one connected component of $Γ({\mathfrak L},\B)$. Also the minimality of ${\mathfrak L}$ and the division property of ${\mathfrak L}$ are characterized in terms of the weak symmetry of the defined subgraphs $Γ({\mathfrak L},\B_{\mathfrak I})$ and $Γ({\mathfrak L},\B_{\mathfrak V})$.

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arXiv2024

Weight modules over split Lie algebras

We study the structure of weight modules $V$ with restrictions neither on the dimension nor on the base field, over split Lie algebras $L$. We show that if $L$ is perfect and $V$ satisfies $LV=V$ and ${\mathcal Z}(V)=0$, then $$\hbox{$L =\bigoplus\limits_{i\in I} I_{i}$ and $V = \bigoplus\limits_{j \in J} V_{j}$}$$ with any $I_{i}$ an ideal of $L$ satisfying $[I_{i},I_{k}]=0$ if $i \neq k$, and any $V_{j}$ a (weight) submodule of $V$ in such a way that for any $j \in J$ there exists a unique $i \in I$ such that $I_iV_j \neq 0,$ being $V_j$ a weight module over $I_i$. Under certain conditions, it is shown that the above decomposition of $V$ is by means of the family of its minimal submodules, each one being a simple (weight) submodule.

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arXiv2024

Graded pseudo-H-rings

Consider a pseudo-$H$-space $E$ endowed with a separately continuous biadditive associative multiplication which induces a grading on $E$ with respect to an abelian group $G$. We call such a space a graded pseudo-$H$-ring and we show that it has the form $E = cl(U + \sum_j I_j)$ with $U$ a closed subspace of $E_1$ (the summand associated to the unit element in $G$), and any $I_j$ runs over a well described closed graded ideal of $E$, satisfying $I_jI_k = 0$ if $j \neq k$. We also give a context in which graded simplicity of $E$ is characterized. Moreover, the second Wedderburn-type theorem is given for certain graded pseudo-$H$-rings.

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arXiv2024

Decompositions of linear operators on pre-euclidean spaces by means of graphs

In this work we study a linear operator $f$ on a pre-euclidean space $\mathcal{V}$ by using properties of a corresponding graph. Given a basis $\B$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as an orthogonal direct sum of certain linear subspaces $\{U_i\}_{i \in I}$, each one admitting a basis inherited from $\B$, in such way that $f = \sum_{i \in I}f_i$, being each $f_i$ a linear operator satisfying certain conditions respect with $U_i$. Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to $f$ relative to two different bases. We also study the minimality of $\mathcal{V}$ by using the graph associated to $f$ relative to $\B$.

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arXiv2022

The algebraic and geometric classification of nilpotent Lie triple systems up to dimension four

In this paper we generalize the Skjelbred Sund method, used to classify nilpotent Lie algebras, in order to classify triple systems with non zero annihilator. We develop this method with the purpose of classifying nilpotent Lie triple systems, obtaining from it the algebraic classification of the nilpotent Lie triple systems up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Lie triple systems up to dimension four.

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arXiv2022

Graded Lie-Rinehart algebras

We introduce the class of graded Lie-Rinehart algebras as a natural generalization of the one of graded Lie algebras. For $G$ an abelian group, we show that if $L$ is a tight $G$-graded Lie-Rinehart algebra over an associative and commutative $G$-graded algebra $A$ then $L$ and $A$ decompose as the orthogonal direct sums $L = \bigoplus_{i \in I}I_i$ and $A = \bigoplus_{j \in J}A_j$, where any $I_i$ is a non-zero ideal of $L$, any $A_j$ is a non-zero ideal of $A$, and both decompositions satisfy that for any $i \in I$ there exists a unique $j \in J$ such that $A_jI_i \neq 0$. Furthermore, any $I_i$ is a graded Lie-Rinehart algebra over $A_j$. Also, under mild conditions, it is shown that the above decompositions of $L$ and $A$ are by means of the family of their, respective, gr-simple ideals.

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arXiv2020

Leibniz superalgebras with a set grading

Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case of ${\mathfrak L}$ being of maximal length, the simplicity of ${\mathfrak L}$ is also characterized in terms of connections.

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arXiv2017

Split Lie-Rinehart algebras

We introduce the class of split Lie-Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if $L$ is a tight split Lie-Rinehart algebra over an associative and commutative algebra $A,$ then $L$ and $A$ decompose as the orthogonal direct sums $L = \bigoplus_{i \in I}L_i$, $A = \bigoplus_{j \in J}A_j$, where any $L_i$ is a nonzero ideal of $L$, any $A_j$ is a nonzero ideal of $A$, and both decompositions satisfy that for any $i \in I$ there exists a unique $\tilde{i} \in J$ such that $A_{\tilde{i}}L_i \neq 0$. Furthermore any $L_i$ is a split Lie-Rinehart algebra over $A_{\tilde{i}}$. Also, under mild conditions, it is shown that the above decompositions of $L$ and $A$ are by means of the family of their, respective, simple ideals.

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