Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004     

Quasisymmetric and quasiantisymmetric bilinear forms over a field

Translated from Matematicheskie Zametki, Vol. 52, No. 6, pp. 140–148, December, 1992.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

Existence of a time-like periodic trajectory for a time-dependent Lorentz metric

In this paper we deal with the problem of the existence ofT-periodic geodesics inR ^N × R equipped with a Lorentz metric g(x, t)[·, ·] which depends ontεR.     

Space of invariant bilinear forms

Let \({\mathbb {F}}\) be a field, V a vector space of dimension n over \({\mathbb {F}}\). Then the set of bilinear forms on V forms a vector space of dimension \(n^2\) over \({\mathbb {F}}\). For char \({\mathbb {F}}\ne 2\), if T is an invertible linear map from V onto V then the set of T-invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of T-invariant bilinear forms over \({\mathbb {F}}\). Also we investigate similar type of questions for the infinitesimally T-invariant bilinear forms (T-skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.     

Fox calculus, symplectic forms, and moduli spaces

An ``open pre-symplectic form' on surfaces with boundary and glueing formulae are provided to symplectically integrate the symplectic form on the deformation space of representations of the fundamental group of a Riemann surface into a reductive Lie group .

     

On a question of Sarkozy and Sos for bilinear forms

We prove that, if 2 k₁ k₂, then there is no infinite sequence of positive integers such that the representation functionr(n) = #{(a, a'): n = k₁a + k₂a', a, a' } is constant for nlarge enough. This result completes the previous work of Diracand Moser for the special case k₁ = 1 and answers a questionposed by Sárkozy and Sós.     

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for

(i)

bilinear forms over an algebraically closed or real closed field;

(ii)

sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and

(iii)

sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F; the canonical matrices are based on any given set of canonical matrices for similarity over F.

A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988) 481-501].     

Representations by bilinear forms (and.p a)

Summary An expression is derived for the number of representations of one bilinear form by another (mod p^a). From this, an explicit formula for the number of such representations is obtained in the case where both forms have square nonsingular matrices (mod p). A related bilinear analog of a lemma of Siegel on representations by quadratic forms (mod p^a) is also proved. In memory of guido Castelnuovo, in the recurrence of the first centenary of his birth. Research supported by National Science Foundation Grants GP-2542 and GP-4015.     

Additive regularization of singular bilinear forms

Conditions on the additive component of a singular bilinear form are found under which the sum is a regular form. As an application the question of regularization of -shaped forms is considered. The case of tensor products of spaces and forms is considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1199–1204, September, 1990.     

A counterexample to a conjecture on flat bilinear forms

We provide a counterexample to a conjecture on the dimension of the nullity of a flat symmetric bilinear form.