Infinite-dimensional quadratic forms admitting composition

New variables («new» as regards the space-time variables) are obtained starting from the requirement that their allowable values should leave the value of the Minkowski metric unchanged. To this purpose, the space-time variables are embedded into real and then complex nonassociative algebras (with an identity element) endowed with nondegenerate quadratic forms generalizing the Minkowski metric. The algebras are proved to fulfil the above-mentioned requirement if and only if they are composition algebras. Using the Hurwitz theorem, it is shown that in the real case there is only one allowable algebra, which is isomorphic to the eight-dimensional split real Cayley algebra, while, in the complex case, there are two allowable algebras: one is isomorphic to the four-dimensional complex quaternion algebra, the other is isomorphic to the eight-dimensional complex Cayley algebra. In the case of these three algebras, new variables and their allowable values are obtained. The essential role played in this context by the (multiplicative) noncommutativity and nonassociativity is pointed out in some final remarks.

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite-dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

Given a symmetric bilinear form F on a finite dimensional real vector space V we attempt to find ‘nice’ bases where the matrix representation of the form is as simple as possible, in the same sort of way as we did in the Gram-Schmidt orthogonalization process for inner products. The theorem which says that this is possible and which describes all the possible matrix representations one can get is called Sylvester’s law of inertia. (Sylvester’s law of inertia has nothing to do with inertia: the story goes that Sylvester said, ‘If Isaac Newton can have a law of inertia, then so can I’, and consequently gave this theorem the name it has been known by ever since.) This ‘law’ is exactly analogous to Theorem 2.33 which classified all finite dimensional vector spaces using dimension; it turns out that symmetric bilinear forms are classified by the dimension of the underlying vector space and two further numbers, called the rank and the signature of the form.

The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space X.A homomorphism is defined between them analogous to the generalized Hasse invariant.Upon evaluation, the Witt group is seen to be KO(X), the graded Brauer group 1+H1(X;ZZ) + H2(X; Z2) with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: I + m^ + h^.Introduction.The Hasse invariant of a quadratic form on a vector space over a field k is an element of the Brauer group of k.It can be generalized to a homomorphism from the Witt group of classes of quadratic forms to the graded Brauer group of classes of graded ^-algebras [3], [16].Curiously enough, in this guise it is quite analogous to the homomorphism given by the first two Stiefel-Whitney classes which maps real vector bundles over a space X to the group 1 + H\X; Z2) + H2(X;Z2), where multiplication is by truncated cup product.The purpose of this paper is to display this analogy.Delzant has defined Stiefel-Whitney classes of quadratic forms [4], [13] using the discriminant and Hasse invariant.It should add insight into his definition to see that the first two topological Stiefel-Whitney classes can also be defined in an algebraic way.A vector bundle L is known to be orientable for real L-theory if its first two Stiefel-Whitney classes vanish.In this case it can be read off from the diagram of Theorem 3.1 that the Clifford bundle of L is a bundle of graded endomorphism algebras of a graded module bundle, which is then used to construct the Thorn class of L. Donovan and Karoubi in studying this orientability question have done independently much of this same work, applying it further to define L-theory with local coefficients [6], [7].The connection between algebra and topology comes from considering bundles over X whose fibers are vector spaces with quadratic forms or graded algebras and subjecting them to the same analysis the fiber alone is accorded in algebra.This is described in some detail in 1.11 is devoted to the algebraic computations and 111 to a study of the topological problems.

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.

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

Let us consider the vector space E of all 2 × 2 matrices x provided with the quadratic form S(x) = -2 det x.

In this part of the book, we will study linear transformations in the same spirit as we studied inner products and quadratic forms. In this chapter we will see how a linear transformation can be represented by a matrix, with respect to a particular basis, and in later chapters we will discuss how to find the ‘best’ basis, so that the corresponding matrix is as simple as possible, with applications to the solution of many kinds of differential and difference equations, and to quadratic forms. Our vector spaces will be over a field F. For this chapter, F can be any field whatsoever. If it helps, it is possible to think of F as either ℝ. or ℂ without much being lost. In later chapters, we will need some further properties of the field, taking the form that certain polynomials have roots; these properties are always true of the field ℂ, so it is safe to think of F as ℂ throughout the rest of the book.

In this paper we consider the quadratic form $\phi [x] = \sum _{i=1}^{n} x_{i}^{2}$ over the vector space $\mathbb{Q}_{n}^{1}$. We take a $\mathbb{Z}$-maximal lattice $L$ in $\mathbb{Q}_{n}^{1}$ with respect to $\phi$. Let $\{L^{(i)}\}_{i=1}^{k(n)}$ be a complete set of representatives for the classes belonging to the genus of $L$. Applying Shimura's mass formula, we determine these representatives $L^{(i)}$ explicitly for $n = 11, 13$, and $14$. Consequently we obtain class numbers $k(11) = 3$, $k(13) = 4$, and $k(14) = 4$.

Bireflectionality of the weak orthogonal and the weak symplectic groups

Introduction. It is well-known that a group is uniquely determined by a system of generators, and a set of defining relations on those generators. Clearly it is of interest to find relations that are as simple as possible. In this paper, this question is dealt with for certain orthogonal groups of characteristic 2, which are generated by involutions.Let V be a vector space over a field K of characteristic 2 (we always exclude the prime field K = G F (2)). Let Q be a quadratic form over V, and let S be the set of orthogonal transformations of (V, Q) whose path is 1-dimensional and not contained in the radical of V. Letting O* be the group generated by S, we shall show that every relation among generators in S is a consequence of relations of length 2, 3, or 4.

Introduction. In [1] the authors studied pairs X, M of linear transformations of a complex vector space V into a complex vector space W (V, W possibly of infinite dimension). The original motivation for this study was its application to a perturbation theory of differential eigenvalue problems developed by Aronszajn (see the introduction to the cited paper). Our aim here is to make a small contribution to the algebraic problem of equivalence of pairs of linear transformations. The pair /,4: V-+ W is said to be equivalent (or isomorphic) to the pair a',&': V'-+ W' in case there exist linear isomorphisms 0 of V onto V' and il of W onto W' such that a?'+ = #a?, R'b = #X. Our considerations will be valid for vector spaces over any algebraically closed field; however we prefer to adhere to the terminology suitable for complex vector spaces. For finite-dimensional vector spaces the problem was solved by Kronecker who found canonical forms for pencils of rectangular matrices. In addition to [1] and the brief statement 1.7 below, one may find an exposition of Kronecker's result akin to our approach in [5] (which contains further references). In the so-called nonsingular case, in which an invertible linear combination of a? and R exists (and is known), the problem reduces to that of similarity of single operators. It well known that it may then be treated as the isomorphism problem for unitary modules over the complex polynomials, for which a vast literature exists. It appears that the infinite-dimensional general case was not considered before [1]. To treat the general case it is technically convenient to replace the structure (d1,4; V, W) by the following simple analog to a module. Let Y denote the 2-dimensional vector space of pairs of complex numbers. A (linear) system (V, W) is a pair of complex vector spaces V, W together with a system operation which assigns to every pair of elements A E Y, v E V an element Av of W so that (i) for every A e Y the map v -+ Av is a linear transformation of V into W; (ii) (ocA + /3B)v = x(Av) + ,B(Bv) for v E V, A, B E Y and a, , complex. Otherwise expressed, the map (A, v) -Av is a bilinear transformation of

In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of "equivalence"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K 0 , K 1 and K 2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshall's conjecture on signatures, which relies heavily on this theory.