Comtrans algebras, Thomas sums, and bilinear forms

Generic adjoints in comtrans algebras of bilinear spaces

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

ln a recent number of the Transactions of the Connecticut Acadelny, E. B. MRILSON obtains a necessary and sufficient condition that a linear transformation be factorable into two involutory transformations. If this condition is translated from the language of dyadics, which WILSON uses, into that of the algebra of matrices, its form suggests at once a theorem of FROBENIUS con. cerning transformations of a bilinear form into itself. A mere combination of these two theorems is sufficient to establish the following, which is simpler in statement than either: I. A necessary and sufScient condition that a lineat transforr)tatton be such as to carry some non-singulclr bilinear fornx into itself, iys that it be factorstble bnto two involutory transformations. It will be seen that the part of this theorem relating to the necessity of the conclition is a generalization of a theorem of P. F. SMITH t which VVrILsoN uses in proving his theorem, and which is substantially as follows: II. (SMITH). A necessary cond:tionthat a linear transformation be such as to carry some non-sing7xlar q?badfratic fortn into itself, iys that it be fcletorable into two involutory transformations. The converse of SMITH'S theorem is not true, as is remarked by WILSON; a substitution which is factorable into two involutory transformations is not necessarily capable of carrying a non-singular qt6adratic form into itself. In fact, the question of the possibility of factoring a transformation into two involutory transformations has no essential relatioll to the subject of symmetric bilinear forms, as distinguished from bilinear forlns in general. :§:

Equations in Hirota's bilinear form appear naturally whenever one needs to write integrable differential equations in terms of "nice" functions. In the case of soliton equations nice typically means polynomials of exponentials with lin- ear exponents, whereas in the context of Painleve equations nice often means entire functions. The natural or original dependent variables of the equation are usually not the best in this respect and a change of the dependent vari- ables is necessary. This dependent variable transformation can be somewhat involved. Since the solutions of Painleve equations are meromorphic by definition (the movable singularities can be at worst poles) it is natural to express them as ratios of entire functions, and this leads to homogeneous equations and often to equations in Hirota's bilinear form. Indeed, bilinear forms were already derived by Painleve himself (1). It should be noted, however, that the converse of the above is not necessarily true: a bilinear form does not by itself imply that the independent functions are regular in any sense, in fact bilinear forms exist even for non integrable equations. In the field of integrable partial differential equations, the so-called τ - functions play a major role. These functions actually provide a huge pool of such "nice" functions, in both of the aforementioned interpretations, and they also possess an extremely rich algebraic and algebro-geometric structure. This is the main theme of Sato-theory, one of the great unifying theories for the description of integrable systems. The relevance of τ -functions for the study of the Painleve equations has been recognized ever since the seminal work of Jimbo and Miwa on isomonodromy deformations (2) and in particular since the results obtained by Okamoto, regarding the algebro-geometric properties of the Painleve equations (3).

Given a bilinear form on , represented by a matrix , the problem of finding the largest dimension of a subspace of such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation is consistent (here denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices . To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A.

Algebraic formulation and topological structure of Boolean networks with state-dependent delay

Totally isotropic subspaces, complementary subspaces, and generalized inverses

We have found an easier and more effective method to describe the structure of meson algebras (also called Duffin-Kemmer algebras) by paying due respect to interior multiplications, to parity gradings and to the quadratic algebra Ω associated with the bilinear form under consideration. Some applications follow, in particular in the context of the meson wave equation, and in the treatment of orthogonal transformations (as restrictions of twisted inner automorphisms). Neutral meson algebras and mesonic interior multiplications lead to further applications, for instance mesonic Lipschitz monoids.

This paper is a contribution to the structure and representation theory of finitedimensional cylindric and polyadic equality algebras.The main result is that every singulary algebra is representable, where an algebra is singular y if it is generated by elements supported by singletons.In particular, all prime algebras are representable (a prime algebra has no proper subalgebra, and every cylindric or polyadic equality algebra has a prime subalgebra).Since every polyadic algebra can be embedded in a polyadic equality algebra the results also apply to them.Every infinite-dimensional singulary algebra is locally finite and hence is known to be representable, so we shall be concerned just with the finite-dimensional case, where previously very little was known about representation.It may be mentioned that for any dimension greater than one there are nonrepresentable cylindric and polyadic equality algebras(2).For some special kinds of singulary algebras we obtain a direct construction of the representation, which brings out the structure of the algebra clearly.The methods used essentially constitute an algebraic version of Behmann's solution of the decision problem for the singulary predicate calculus with equality(3).Since only finitely many variables are available, the algebraization has some novel features.The essential ideas can be seen in the case of prime algebras, for which no advanced results about cylindric or polyadic algebras are needed.In the general case, however, use is made of the fact that a cylindric algebra is representable if every finitely generated subalgebra of it is.In the last section of the paper we discuss the logical counterpart of the methods and results of the algebraic part of the paper.1. Introduction.A cylindric algebra of dimension a (a CAa) is a system 21 = <[A, +,-,-, cK,dKXyKtX<x such that (A, +,-, -> is a Boolean algebra, a is an ordinal, cK is a singulary Presented to the Society, November 17,1962 under the title On finite dimensional cylindric algebras; received by the editors April 2, 1963.0) The research herein reported on was supported by

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

This paper develops a novel framework for antisymmetric infinitesimal hom-bialgebras by introducing admissible bimodules of hom associative algebras, generalizing the foundational work of Hounkonnou et al. (2019) . We first construct admissible hom-associative bialgebras and rigorously establish their equivalence to matched pairs and Manin triples via a canonical bilinear form. Our approach systematically extends classical bialgebra theory to the hom-associative setting by imposing admissibility conditions on bimodules, which preserve the twisted multiplicativity of hom-structures. Key results include: (1) the characterization of admissible hom associative algebras through their regular bimodules, (2) the derivation of antisymmetric infinitesimal admissible hom-bialgebras as dual structures, and (3) the proof that these bialgebras are categorically equivalent to Manin triples when equipped with a standard invariant bilinear form on the direct sum A ⊕ A∗. This work unifies and generalizes prior studies on hom-Lie bialgebras and the hom Yang-Baxter equation, while providing new tools for deformation theory and non-commutative geometry. The constructions are contextualized within prominent examples, including q-deformations and σ-derivations, bridging connections to the Witt and Virasoro algebras. Our results demonstrate that the hom-associative admissible framework offers a robust algebraic foundation for studying twisted bialgebraic structures.

We study definable sets, groups, and fields in the theory of infinite-dimensional vector spaces over an algebraically closed field of any fixed characteristic different from 2 equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define a notion of dimension of a definable set, which enjoys many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all definable groups are (algebraic-by-abelian)-by-algebraic. We conclude that every definable field is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component.We also prove analogous results working over real closed fields.

Bilinear transformations and bilinear forms are introduced and studied. Their matrix representation, and especially the representation of symmetric bilinear forms, is presented. Orthogonality relative to a bilinear form is considered. These notions are then used to define the tensor product of vector spaces, and the properties of the tensor product are considered in detail.

A discrete group which admits a faithful, finite dimensional, linear representation over a field F of characteristic zero is called linear.This note combines the natural structure of semi-direct products with work of A. Lubotzky [13] on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear.Let G denote a discrete group which is a semi-direct product given by a split extensionThis note defines an additional type of structure for this semi-direct product called a stable extension below.The main results are as follows:(1) If π and Γ are linear, and the extension is stable, then G is also linear.Restrictions concerning this extension are necessary to guarantee that G is linear as seen from properties of the Formanek-Procesi "poison group" [7].(2) If the action of Γ on π has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over π" ( to be defined below ), then the associated extension is stable and thus G is linear.(3) The condition of a stable extension also implies that G admits filtration quotients which themselves give a natural structure of Lie algebra and