arXiv subjects

Rodrigo Lucas Rodrigues

Publications and source records attributed to Rodrigo Lucas Rodrigues.

6 recordsLinked to original sources

arXiv2026

Isomorphism classes and geometric classification of three-dimensional Jordan algebras and the length function

We investigate the algebraic classification of Jordan algebras of dimensions at most three over arbitrary fields with more than two elements. Our approach combines the theory of nilalgebras and radicals with Peirce, Wedderburn decompositions and the length function. In characteristic two, we adapt the Peirce decomposition to organize the classification, including the field-dependent isomorphism conditions among parametrized families. We also study the varieties of Jordan algebra structures up to dimension three over an algebraically closed field of characteristic two. Besides explicit degenerations and standard arguments, we introduce the alicyclic and characteristic conditions as geometric obstructions to degeneration. We determine the rigid algebras and irreducible components in both dimensions. In dimension three, one component is the nine-dimensional affine space of square-zero products; unlike the other three components, it contains no open orbit.

math.RA↗

arXiv2026

Engel's Theorem for Alternative Superalgebras

In this paper, a nilpotency criterion is given for finite dimensional alternative superalgebras in the spirit of Engel's Theorem for Jordan superalgebras over infinite fields provided by Shestakov and Okunev. For alternative superalgebras, no restrictions on the cardinality of the ground field are required. Furthermore, we establish some connections between the concepts of graded-nil and nilpotent alternative superalgebras, and we also exhibit an example of an Engelian commutative power-associative superalgebra of dimension $4$ which is not nilpotent.

math.RA↗

arXiv2026

Varieties of nilpotent Jordan superalgebras of dimension five

The paper is devoted to the description of the varieties of complex 5-dimensional nilpotent Jordan superalgebras. We find all representatives for the isomorphism classes, using the Jordan normal form, results of simultaneous matrix triangularization, the Jordan-Kronecker theorem for a pair of skew-symmetric bilinear forms and similar arguments developed for $Δ$-modules by Burde and Grunewald. We also provide a complete geometric classification, determining the irreducible components of the corresponding varieties and describing all possible degenerations and non-degenerations between these superalgebras, in particular, applying some $\mathbb{Z}_2$-graded subspaces as invariants.

math.RA↗

arXiv2026

Construction of groups with triality and their corresponding code loops

We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality and $2n$ generators, based on embeddings of $G_n$ into direct products of copies of $G_3$. In the finite case, where $G_n$ is a group such that $|G_n| = 2^{4n+m}$ with $n \ge 3$ and $m = 3 {n \choose 2} + 2 {n \choose 3}$, we prove that the corresponding Moufang loop is the free loop $F_n$ with $n$ generators in the variety generated by code loops. The result depends on a construction similar to that of $G_n$, namely, embedding $F_n$ into direct products of copies of $F_3$, the free code loop associated with $G_3$.

math.GR↗

arXiv2025

The Variety of Jordan Superalgebras of dimension four and even part of dimension two

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $2$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$. We prove that the variety has $25$ irreducible components, $24$ of them correspond to the Zariski closure of the $GL_2(\mathbb{F})\times GL_2(\mathbb{F})$-orbits of rigid superalgebras and the other one is the Zariski closure of an union of orbits of an infinite family of superalgebras, none of them being rigid. Furthermore, it is known that the question of the existence of a rigid Jordan algebra whose second cohomology group does not vanish is still an open problem. We solve this problem in the context of superalgebras, showing a four-dimensional rigid Jordan superalgebra whose second cohomology group does not vanish.

math.RA↗

arXiv2025

Irreducible Components of the Varieties of Jordan Superalgebras of Types $(1,3)$ and $(3,1)$

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$. We prove that the variety is the union of Zariski closures of the orbits of $11$ and $21$ rigid superalgebras, respectively. In both cases, the irreducible components of the varieties are described. Furthermore, we exhibit a four-dimensional solvable rigid Jordan superalgebra, showing that an analogue to the Vergne conjecture for Jordan superalgebras does not hold.

math.RA↗