Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39842

A note on the ideal lattice of a weighted Leavitt path algebra

Raimund Preusser
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $K$ be a field and $L_K(E,w)$ the Leavitt path algebra of a row-finite weighted graph $(E,w)$. Suppose that $L_K(E,w)$ is not isomorphic to an ordinary Leavitt path algebra. We prove that the ideal lattice of every countably generated unital $K$-algebra embeds as a closed interval into the ideal lattice of $L_K(E,w)$. The embedding preserves multiplication of ideals modulo the lower endpoint of the interval. It follows that the ideal lattice of $L_K(E,w)$ is not distributive and that ideal multiplication in $L_K(E,w)$ is not commutative. Moreover, we show that if $(E,w)$ is countable, then $L_K(E,w)$ has exactly $|K|^{\aleph_0}$ ideals, prime ideals, left primitive ideals, and right primitive ideals.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Raimund Preusser. 2026-10-29. A note on the ideal lattice of a weighted Leavitt path algebra. https://arxiv.org/abs/2610.39842

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Structure of the Cayley-Dickson algebras

Viewing the Cayley-Dickson process as a graded construction provides a rigorous definition of associativity consisting of three classes, which divides the non-associative parts into four types. These simplify the Moufang loop identities and Malcev's identity and categorise non-associative loops. Analysing the non-associativity structure of sedenions and higher level Cayley-Dickson algebras uncovers octonion and three quasi-octonion, power-associative subalgebras. Power-associativity is proved to grow by multiples of seven that introduce zero divisors in factors of 84 for all Cayley-Dickson algebras. It is convenient to replace the terminology hypercomplex numbers with ultracomplex numbers to capture the quasi-algebras and since sedenions and higher level Cayley-Dickson algebras contain quasi-algebras these are ultracomplex. The non-associative types reduce the 84 zero divisors in sedenions via three cycles and four modes down to seven primary zero divisor pairs providing new techniques to analyse the seven quasi-octonions loops in sedenions. The graded notation allows the eight octonion and seven quasi-octonion subalgebras of sedenions to be uniquely derived, up to representation. The zero divisors for split sedenion algebras are analysed and mappings between three of these are provided showing 180 zero divisors and the other split algebra is analogous to sedenions with 84 zero divisors. These split algebras are shown to involve quasi-octonion algebras with mixed signature.

math.RA↗
arXiv2026

Automorphisms of Sedenions

This paper extends the octonion calibration in Clifford algebra to two related sedenion calibrations. Associative calibrations map quaternion rings in Clifford algebra to those in Cayley-Dickson algebras, with octonions consisting of seven rings and sedenions having 35 rings. A new non-associative calibration is found that is related to the sedenion associative calibration but is invertible, providing a classification of the ideals of Clifford algebra and hence of 8 x 8 matrices.This leads to new subalgebras of Clifford algebras and a thorough analysis of the possible automorphisms of sedenions is applied using the Spin and Pin groups. While the calibrations and automorphisms of octonions have minimal divergence, it is found sedenions extend Lie exceptional algebra, G2, to a Lie algebra of rank 49. This work clarifies and extends Schafer's and Brown's sedenion automorphisms. Brown introduced automorphisms that change the sedenion representation, unlike G2, but Clifford algebra analysis finds there are 92 such pure representations. The non-associative calibrations generate an infinite series of new Clifford subalgebras that match the finite geometry PG(N,2) series, discovered by Gino Fano, and connect the projections of simplices with those from hyper-cubes. The visual representation of sedenions as the PG(3,2) tetrahedron, representing the 15 Fano planes of sedenions, allows the sedenion power-associative loops to be visually distinguished from the octonion loops. The visual representation of sedenions may be important for particle physics and Clifford algebra provides analysis tools for the PG(N,2) series that uncovers the hidden structure of Cayley-Dickson algebras within an associative algebra.

math.RA↗
arXiv2026

Unitary-Invariant Decomposition of Reducible Total Least Squares Core Problems

The analysis of a total least squares problem can be reduced to that of an associated core problem, which typically has lower dimension and improved solubility properties. Nevertheless, even a core problem may remain reducible, admitting further decomposition into irreducible component sub- problems with simpler structure and better analytical properties. We seek the finest decomposition under unitary transformations over C; for real data this decomposition can be strictly finer than the finest one under real orthogonal transformations. We split the data matrix into row blocks associated with the singular subspaces of the system matrix and consider the graph whose ver- tices are the blocks and whose edges correspond to nonzero cross-products. After preliminary splittings that make every nonzero cross-product a scalar multiple of a unitary matrix, we show that the components can be obtained directly from these blocks if and only if the covariance matrices along all cycles of the graph are scalar matrices. Otherwise, there is a chordless cycle whose covariance matrix is not a scalar matrix, and splitting the blocks on this cy- cle along the eigenspaces of the covariance matrix removes the obstruction without changing the decomposition. Based on these results, we propose a recursive strategy that finds all irreducible components over the complex field. The strategy is exact and theoretical rather than an efficient numerical algo- rithm; a small floating-point computation illustrates it. The results depend only on the core problem and therefore apply to general linear approximation problems AX B.

math.RA↗