arXiv subjects

Louis Halle Rowen

Publications and source records attributed to Louis Halle Rowen.

6 recordsLinked to original sources

arXiv2026

Axial identities

Axial algebras are algebras generated by semisimple idempotents whose eigenspace decompositions satisfy prescribed fusion rules, such as Jordan fusion rules, which hold in Jordan algebras and Matsuo algebras. Axial algebras are studied here in terms of identities involving idempotents and/or axes, and at times involving Frobenius forms. After laying out the general theory, organized as a PI-type theory of marked varieties, we turn to Jordan type, providing insight into a major theorem of I.~Gorshkov, S.~Shpectorov, and A.~Staroletov about solid subalgebras. This approach also leads to generic constructions of primitive $N$-generated axial algebras of Jordan type $\gl$, for $N\le 4$, and for arbitrary $N$ we obtain corresponding ``$M$-primitive'' constructions. We discuss the relevance to the problem of finding a primitive nonsingular axial algebra of Jordan type~$\half$, which is neither Jordan nor a homomorphic image of a Matsuo algebra, although we cannot provide an answer at this stage.

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arXiv2026

Roots of polynomials over semirings and hyperfields

We continue our investigation of roots of polynomials over semirings and hyperfields, employing a property on semiring and hyperfield ``pairs'' with a surpassing relation $\preceq,$ which we call $\preceq$-reversibility. There are two kinds of roots generalizing the classical algebraic theory, ``null roots,'' and $\preceq$-roots. The theory works best when all null roots are also $\preceq$-roots. Ensuing results include the fundamental theorem of algebra for pairs, that tangible polynomials with enough roots ``$\preceq$-split,'' at times uniquely, into linear factors. We also see that polynomials that agree on ``almost'' all null roots are ``almost'' equal. Finally, we obtain roots of integral polynomials over extension pairs, providing a construction of integrally closed pairs over hyperfields and over zero sum free semirings.

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arXiv2026

Semirings

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main features are a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, and a ``surpassing relation,'' taking the place of equality, which permit generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' $(\mcA,\mcA_0)$ is studied along the lines of universal algebra.

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arXiv2024

Weakly primitive axial algebras

In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and other commutative and noncommutative examples. Also we broaden our investigation and describe 2-generated algebras in which only one of the generating axes is weakly primitive and satisfies the fusion rules, on condition that its zero-eigenspace is one dimensional. We also characterize when both axes satisfy the fusion rules (weak PAJ's), and describe precisely the 2-dimensional axial algebras. In contrast to the previous situation, there are weak PAJ's of dimension~$> 3$ generated by two axes.

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arXiv2022

Structure of primitive axial algebras

"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren, and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and vertex operator algebras. In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite dimensional noncommutative algebras, and commutative axial algebras generated by primitive axes of the same type. In particular,all primitive axial algebras are flexible. They also have Frobenius forms. We give a precise description of all the primitive axes of axial algebras generated by two primitive axes.

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arXiv2021

Algebras with a negation map

Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras, one often is challenged by the lack of negation. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called \textbf{systems}, in the context of universal algebra, showing how these unify the more viable (super)tropical versions, as well as hypergroup theory and fuzzy rings, thereby "explaining" similarities in their theories. Special attention is paid to \textbf{meta-tangible} $\mathcal T$-systems, whose algebraic theory includes all the main tropical examples and many others, but is rich enough to facilitate computations and provide a host of structural results. Basic results also are obtained in linear algebra, linking determinants to linear independence. Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby further explaining the mysterious link between classical algebraic results and their tropical analogs, as well as with hyperfields. We utilize the tropicalization functor to propose tropical analogs of classical algebraic notions. The systems studied here might be called "fundamental," since they are the underlying structure which can be studied via other "module" systems, which is to be the third stage of this project, involving a theory of sheaves and schemes and derived categories with a negation map.

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