Punctual Spectra of Algebraic Structures and Isomorphisms

We study the isomorphism problem of two “natural” algebraic structures – $\mathbb{F}$ -algebras and cubic forms. We prove that the $\mathbb{F}$ -algebra isomorphism problem reduces in polynomial time to the cubic forms equivalence problem. This answers a question asked in [AS05]. For finite fields of the form $3 \Lambda(\#\mathbb{F} - 1)$ , this result implies that the two problems are infact equivalent. This result also has the following interesting consequence: Graph Isomorphism ${\leq}^P_m$ $\mathbb{F}$ -algebra Isomorphism ${\leq}^P_m$ Cubic Form Equivalence.

Consider the ideal \((x_{1} , \dotsc , x_{n})^{d} \subseteq k[x_{1} , \dotsc , x_{n}]\), where k is any field. This ideal can be resolved by both the L-complexes of Buchsbaum and Eisenbud, and the Eliahou-Kervaire resolution. Both of these complexes admit the structure of an associative DG algebra, and it is a question of Peeva as to whether these DG structures coincide in general. In this paper, we construct an isomorphism of complexes between the aforementioned complexes that is also an isomorphism of algebras with their respective products, thus giving an affirmative answer to Peeva’s question.

We study from an algebraic point of view the question of extending an action of a group Γ on a commutative domain R to a formal pseudodifferential operator ring B = R((x ; d)) with coefficients in R, as well as to some canonical quadratic extension C = R((x 1/2 ; 1 2 d))2 of B. We give a necessary and sufficient condition of compatibility between the action and the derivation d of R for such an extension to exist, and we determine all possible extensions of the action to B and C. We describe under suitable assumptions the invariant subalgebras B Γ and C Γ as Laurent series rings with coefficients in R Γ. The main results of this general study are applied in a numbertheoretical context to the case where Γ is a subgroup of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. Denoting by Mj the vector space of algebraic modular forms in R of weight j (even or odd), we build for any nonnegative integer k a linear isomorphism between the subspace C Γ k of invariant operators of order ≥ k in C Γ and the product space M k = j≥k Mj, which can be identified with a space of algebraic Jacobi forms of weight k. It results in particular a structure of noncommutative algebra on M0 and an algebra isomorphism Ψ : M0 → C Γ 0 , whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.

With this paper we begin a study of the structure of the group algebra RG of a finite group G over the ring of algebraic integers R in an algebraic number field k. The basic question is whether non-isomorphic groups can have isomorphic algebras over R. We shall show that this is impossible if G is (a) abelian,(b) Hamiltonian,(c) one of a special class of p-groups.

Crossed products for weak Hopf algebras with coalgebra splitting

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M_1 \to M_2 be a homotopy equivalence of closed, oriented n -manifolds. Then the induced equivalence, Lf : LM_1 \to LM_2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h_* that supports an orientation of the M_i 's.

The module isomorphism problem reconsidered

The structure of orthodox semigroups with a normal band of idempotents has been described by Yamada. Since, for a naturally ordered strong Dubreil-Jacotin onhodox semigroup, ir can be shown that the band of idempotents is normal, it is of interest to investigate the structure of these ordered onhodox semigroups via the Yamada decomposition. What is hard in such matters, and no exception here, is how to marry together the order theoretic structure with the algebraic structure theory. Basically, the problem is how to define orders on the building bricks of the structure theory in such a way tha~ the algebraic isomorphisms become order isomorphisms. This we are able to do for a variety of different types, obtaining structure theorems which concern not only cartesian orders but also several new types of lexicographic orders. Examples are given to illustrate the hierarchy of orders and the corresponding algebraic conditions required for the order isomorphisms.

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

The study, by the last-named author, of the ring of all real-valued continuous functions ~(X, IR) on a tpopological space X, was begun some 33 years ago in the article [1]. The principal notions of this paper are realcompact spaces (originally named Q-spaces for no very good reason) and hyperreal fields (which we call here H-fields). These concepts have found applications in a number of areas. First, topologists have found realcompact spaces to be of interest: see for example [16] and [19]. Second, ultraproducts and ultrapowers, first considered by Log [31], have proved to be a powerful model-theoretic tool for investigation from a unified point of view of varied problems in set theory, logic, model theory, certain branches of algebra, and the theory of numbers (Artin's hypothesis and p-adic fields): see [40] and [8]. The method of ultraproducts and ultrapowers has been worked out in detail, and one may say that a more or less complete theory exists. Its traditional aspects are set forth in the two monographs of Chang and Keisler ([7] and [8]), in the monograph of Comfort and Negrepontis [14], and in the monograph of A.I. Mal'cev [33]. In particular, much progress has been made in such classical problems as the structure of ultrafilters and the cardinal numbers of ultraproducts. The key to the results obtained up to now lies in axiomatic set theory, generic models, and the theory of large cardinal numbers. Studies in recent years have tended to neglect the theory of H-fields. One should not forget, however, that ultrapowers of 1R are exactly the H-fields obtained from discrete spaces. Hence the study of arbitrary H-fields, their algebraic structure, classification, and isomorphism types, should and does lead to more complicated topological and set-theoretical problems. Little progress has been made up to now with

Rabin automata have applications in many areas of mathematics and computer science. In this chapter our goal is to show how results about Rabin automata can be applied to prove that some theories of well-studied mathematical structures are decidable. This chapter consists of ten sections. The first four sections introduce the notions of algebraic structure, the monadic second order logic, the truth of formulas in algebraic structures, isomorphisms, and theories of classes of structures. Section 5 is devoted to proving that the monadic second order theory of two successor functions, known as S2S, is decidable. In particular, the section shows the relationship between definable relations in the monadic second order logic of two successors and languages accepted by Rabin automata.

The minimality theorem states, in particular, that on cohomology H(A) of a dg algebra there exists sequence of operations mi : H(A)⊗i → H(A), i = 2, 3, . . . , which form a minimal A ∞ -algebra (H(A), {m i }). This structure defines on the bar construction BH(A) a correct differential dm so that the bar constructions (BH(A), d m ) and BA have isomorphic homology modules. It is known that if A is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on BA there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations Ep,q : H(A) ⊗p ⊗H(A) ⊗q → H(A), p, q = 0, 1, 2, . . ., which turn (H(A), {m i }, {E p,q }) into a B ∞ -algebra. These operations determine on BH(A) correct multiplication, so that (BH(A), d m ) and BA have isomorphic homology algebras.

On when a separable field extension in characteristic p > 0 is determined by its Frobenius structure, I

(One-value) graph magma algebras are algebras having a basis such that, for all Such bases induce graphs and, conversely, certain types of graphs induce graph magma algebras. The equivalence relation on graphs that induce isomorphic magma algebras is fully characterized for the class of associative graphs having only finitely many non-null connected components. In the process, the ring-theoretic structure of the magma algebras induced by those graphs is given as it is shown that they are precisely those graph magma algebras that are semiperfect as rings. A complete description of the semiperfect rings that arise in this fashion, in ring theoretic and linear algebra terms, is also given. In particular, the precise number of isomorphism classes of one-value magma algebras of dimension n is shown to be where, for any p(i) is the number of partitions of i. While it is unknown whether uncountable dimensional algebras always have amenable bases, it is shown here that graph magma algebras do.

The paper is devoted to the classification of finite-dimensional complex Lie algebras of analytic vector fields on the complex plane and the corresponding actions of Lie groups on complex two-dimensional manifolds. These Lie algebras were specified by Sophus Lie. He specified vector fields which form bases of the Lie algebras. However the structure of the Lie algebras was not clarified, and isomorphic Lie algebras among listed were not established. Thus, the classification was far from complete, and the situation has not been essentially changed until now. This paper is devoted to the completion of the above mentioned classification. We consider the part of this classification which concerns transitive actions of Lie groups.