arXiv subjects

Ricardo Suárez

Publications and source records attributed to Ricardo Suárez.

5 recordsLinked to original sources

arXiv2025

$\textrm{U}(n)$-structures and their induced minimal left ideals

In previous work, we associated to $\textrm{SU(3)}$, $\mathrm{G}_2$, and $\textrm{Spin(7)}$-structures minimal left ideals for the Clifford algebras $\mathbb{R}_{0,6},\mathbb{R}_{0,7}$, and $\mathbb{R}_{0,8}$, respectively. In this paper, we continue to analyze the link between Berger's classification theorem and the structure theorem of minimal left ideals for Clifford algebras of signature $(p,q)$ by identifying $\mathrm{U}(n)$-structures with minimal left ideals for Clifford algebras of various signatures via the induced Kahler polynomial $P(ω_{0})$ associated with the symplectic form $ω_{0}$ that defines the $\mathrm{U}(n)$-structure as a stabilizer subgroup of $\mathrm{O}(n)$.

math.DG↗

arXiv2025

The countable condensation on linear orders

The countable condensation on a linear order $L$ is the equivalence relation $\sim_ω$ defined by declaring $x \sim_ωy$ when the set of points between $x$ and $y$ is countable. We characterize the linear orders $L$ that condense to $1$ under the countable condensation by constructing a linear order $U$ that is universal for the order types $L$ such that $L/\!\!\sim_ω\, \cong 1$. We define a multiplication operation $\cdot_ω$ on the class of linear orders by setting $M \cdot_ωL$ to be the order type of $(ML)/\!\!\sim_ω$ (where $ML$ denotes the lexicographic product), and show that the right identities for $\cdot_ω$ are exactly the uncountable suborders of $U$. The order types of these uncountable suborders of $U$ form a left regular band under $\cdot_ω$, and the order types of all suborders of $U$ form a semigroup.

math.LO↗

arXiv2025

Algebraic structures arising from the finite condensation on linear orders

The finite condensation $\sim_F$ is an equivalence relation defined on a linear order $L$ by $x \sim_F y$ if and only if the set of points lying between $x$ and $y$ is finite. We define an operation $\cdot_F$ on linear orders $L$ and $M$ by $L \cdot_F M = \operatorname{o.t.}\left((LM)/\!\sim_F\right)$; that is, $L \cdot_F M$ is the order type of the lexicographic product of $L$ and $M$ modulo the finite condensation. The infinite order types $L$ such that $L / \! \sim_F\, \cong 1$ are $ω, ω^*,$ and $ζ$ (where $ω^*$ is the reverse ordering of $ω$, and $ζ$ is the order type of $\mathbb{Z}$). We show that under the operation $\cdot_F$, the set $R=\{1, ω, ω^*, ζ\}$ forms a left regular band. Further, each of the ordinal elements of $R$ defines, via left or right multiplication modulo the finite condensation, a weakly order-preserving map on the class of ordinals. We study these maps' effect on the ordinals of finite degree in Cantor normal form. In particular, we examine the extent to which one of these maps, sending $α$ to the order type of $α$ modulo the finite condensation, behaves similarly to a derivative operator on the ordinals of finite degree in Cantor normal form.

math.LO↗

arXiv2024

Group theoretic properties of Clifford multiplication on 2-torsion points on the Dirac Spinor Abelian Variety

In this manuscript we consider a special complex torus, denoted $S_{Δ_{2k}}$ (for each $k \in \mathbb{N},\, k \geq 1$) and called the Dirac spinor torus. It is an Abelian variety of complex dimension $2^{k}$ whose covering space is the space of Dirac spinors, $Δ_{2k}$, for the Clifford algebra $Cl(\mathbb{C}^{2k})$ associated with the vector space $\mathbb{C}^{2k}$. Fixing an isomorphism $ρ:Cl(\mathbb{C}^{2k})\rightarrow End (Δ_{2k})$, we define Clifford multiplication on $S_{Δ_{2k}}$ as the actions of those endomorphisms in the image of $ρ$ that preserve the full rank lattice. We analyze the properties of that Clifford multiplication on the 2-torsion points of the Dirac spinor torus. We identify the Clifford actions with permutation maps that represent all isomorphism classes of these actions on the group of 2-torsion points. We provide a structure theorem describing these isomorphism classes of Clifford actions in a way that is independent of the choice of representatives. We conclude by extending the scope of our analysis to the group of $n$-torsion points and analyzing the fixed points and translation constants of entry-permuting maps, a broader class of actions of which the Clifford actions on the 2-torsion points of $S_{Δ_{2k}}$ is a subset.

math.AG↗

arXiv2019

Generating sets for Clifford Algebras

The aim of this paper is to find generating sets of commuting involutions and use them to explicitly construct minimal representations of Clifford algebras $Cl(n)_{p,q}$. By results of [HL] and [LW], we know the dimension of such minimal representations, which is linked to the maximal number of commuting involutions in the algebra, dependent only on $p$ and $q$. We provide an algorithm to construct these generating sets of involutions explicitly for all Clifford algebras $Cl(n)_{p,q}$ and provide some examples. Involutions yield mutually non-annihilating idempotents whose product gives a projection map $P^+$ with image $Cl(n)P^+$ being a minimal left ideal, which is a spinor space. Using the projections, we find minimal representations of Clifford algebras by combining matrices of left multiplication endomorphisms in the spinor spaces. Finally, we provide examples showing calculations of minimal representations.

math.AG↗