Related papers: Classification of sesquilinear forms with the firs…

Let $p$ be an odd prime, $k,\ell$ be positive integers, $q=p^k, Q=p^{\ell}$. In this paper we characterise planar functions of the form $f_{\underline{c}}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\mathbb{F}_{q^2}$ for any…

Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex…

This monograph starts with an upper triangular matrix with integer entries and 1's on the diagonal. It develops from this a spectrum of structures, which appear in different contexts, in algebraic geometry, representation theory and the…

Quaternions, split quaternions, and hybrid numbers are very well-known number systems. These number systems are used to make geometry in Euclidean and Lorentz spaces. These number systems can be obtained with the help of a quadratic form.…

We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly…

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort.…

Leibniz algebras ${\mathcal E}_n$ were introduced as algebraic structure underlying U-duality. Algebras ${\mathcal E}_3$ derived from Bianchi three-dimensional Lie algebras are classified here. Two types of algebras are obtained:…

We give two equivalent sets of invariants which classify pairs of coisotropic subspaces of finite-dimensional Poisson vector spaces. For this it is convenient to dualize; we work with pairs of isotropic subspaces of presymplectic vector…

We provide a systematic study of sesquilinear hermitian forms and a new proof of the calculus of some exponential sums defined with quadratic hermitian forms. The computation of the number of solutions of equations such as Tr(f(x)+v.x)=0 or…

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…

We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups…

We give an algorithm that uses only unitary transformations and for each square complex matrix constructs a *congruent matrix that is a direct sum of a nonsingular matrix and singular Jordan blocks.

The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in…

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on…

In this note, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the…

If two G-manifolds are G-cobordant then characteristic numbers corresponding to the fixed point sets (submanifolds) of subgroups of G and to normal bundles to these sets coincide. We construct two analogues of these characteristic numbers…

In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the…

In this paper, we first define a non-degenerate symmetric bilinear form on Fq^4. Then we get an incidence matrix G of Fq^4 by the bilinear form. By its corresponding quadratic form Q, the lines of Fq^4 are classified as isotropic and…

For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the…

Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spaces of bounded rank three were classified in 1983, and it has…