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Louis H. Rowen

Publications and source records attributed to Louis H. Rowen.

13 recordsLinked to original sources
arXiv2025

Makar-Limanov's problem on values of polynomials on matrices

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th roots, for any $n \ge m'$ there are matrices $A_1,\dots,A_m$ in~$M_n(F)$ such that $f(A_1,\dots,A_m)$ is upper triangular with $n-m'$ prescribed diagonal entries. When f is homogeneous, $f(A_1,\dots,A_m)$ is diagonal with $n-m'$ prescribed diagonal entries. When f is multilinear, we can take $d=1$ and $m' = [\frac{m-1}{2}]$, and the upper left $(n-m')\times (n-m')$ piece of $f(A_1,\dots,A_m)$ can be taken to be $diag(\beta_1,\dots, \beta_{n-m'})$, for indeterminates $\beta_i$. Furthermore, if $f$ is not a polynomial identity of $ k \times k $ matrices, then at least $ n - k $ characteristic values of $ f(A_1,\dots,A_m) $ may be taken to be algebraically independent.

math.RA↗
arXiv2024

The spectrum of prime congruences of a pair

Continuing the study of the structure of semirings, we turn to the spectrum of prime congruences. Joo and Mincheva developed an elegant theory in the special case of idempotent semirings, which is generalized here to ``semiring pairs,'' which include supertropical semirings and various classes of hyperrings. Our main result is that the Joo-Mincheva spectrum can be embedded into the prime spectrum of a semiring pair, and the mapping is onto when the pair satisfies a mild restriction which we call ``positive $e$-type.''

math.RA↗
arXiv2024

Tensor products of bimodules and bimodule pairs over monoids

We modify the well-known tensor product of modules over a semiring, to provide a general theory encompassing the classical theory of tensor products over algebras, tropical structures, and hyperfields. We discuss tensor products from three viewpoints: tensor extensions, tensor products of modules and bimodules over monoids, and tensor products of semi-algebras. The tensor product of residue hypermodules is functorial with respect to this construction. Special attention is paid to different kinds of morphisms and the connection to the work of Nakamura and Reyes.

math.RA↗
arXiv2024

Residue Structures

We consider residue structures $R/G$ where $(G,+)$ is an additive subgroup of a ring $(R,+,\cdot)$, not necessarily an ideal. Special instances include Krasner's construction of quotient hyperfields, and Pumpluen's construction of nonassociative algebras. The residue construction, treated formally, satisfies the Noether isomorphism theorems, and also is cast in a broader categorical setting which includes categorical products, sums, and tensor products.

math.RA↗
arXiv2016

Bimodule Structure of Central Simple Algebras

For a maximal separable subfield $K$ of a central simple algebra $A$, we provide a semiring isomorphism between $K$-$K$-bimodules $A$ and $H$-$H$ bisets of $G = \Gal(L/F)$, where $F = \operatorname{Z}(A)$, $L$ is the Galois closure of $K/F$, and $H = \Gal(L/K)$. This leads to a combinatorial interpretation of the growth of $\dim_K((KaK)^i)$, for fixed $a \in A$, especially in terms of Kummer sets.

math.RA↗
arXiv2015

Unions of Chains of Primes

The union of an ascending chain of prime ideals is not always prime. We show that this property is independent of the parallel property for semiprimes. We also show that the PI-class is a tight bound on the number of non-prime unions of subchains in a chain of primes in a PI-algebra.

math.RA↗
arXiv2015

Kummer Spaces in Cyclic Algebras of Prime Degree

We classify the monomial Kummer subspaces of division cyclic algebras of prime degree $p$, showing that every such space is standard, and in particular the dimension is no greater than $p+1$. It follows that in a generic cyclic algebra, the dimension of any Kummer subspace is at most $p+1$.

math.RA↗
arXiv2011

Structure of Zariski-closed algebras

The objective of this paper is to describe the structure of Zariski closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a version of Wedderburn's principal theorem, as well as a more explicit description using representations, in terms of "gluing" in Wedderburn components. Finally, we construct "generic" Zariski closed algebras, whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras. Special attention is given to infinite dimensional algebras over finite fields.

math.RA↗
arXiv2011

Full quivers of representations of algebras

We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to \Zcd\ algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: {\it Frobenius} (along the diagonal) and, more generally {\it proportional} Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.

math.RA↗
arXiv2004

Coxeter covers of the symmetric groups

We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set $T$ of transpositions. These quotients, denoted here by C_Y(T), are a special type of the generalized Coxeter groups defined in \cite{CST}, and also arise in the computation of certain invariants of surfaces. We use a surprising action of $S_n$ on the kernel of the surjection $C_Y(T) \ra S_n$ to show that this kernel embeds in the direct product of $n$ copies of the free group $π_1(T)$ (with the exception of $T$ being the full set of transpositions in $S_4$). As a result, we show that the groups $C_Y(T)$ are either virtually Abelian or contain a non-Abelian free subgroup.

math.GR↗
arXiv2003

Fields of definition for division algebras

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if $A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that: (1) In many cases $A$ can be defined over a rational extension of $k$. (2) If $A$ has odd degree $n \ge 5$, then $A$ is defined over a field $F_0$ of transcendence degree at most $(n-1)(n-2)/2$ over $k$. (3) If $A$ is a $Z/m \times Z/2$-crossed product for some $m \ge 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, $M_m(A)$ can be defined over a field $F_0$ of transcendence degree at most 4 over $k$. (4) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree at most 4. (In (1), (3), and (4) we assume that the center of $A$ contains certain roots of unity.)

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