Related papers: An Attempt of Construction for the Grassmann Numbe…
Phase space of General Relativity is extended to a Poisson manifold by inclusion of the determinant of the metric and conjugate momentum as additional independent variables. As a result, the action and the constraints take a polynomial…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.
This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.
Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure…
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…
In this paper a construction of affine exterior algebra of Grassmann, with a special attention to the revisitation of this subject operated by Peano and his School, is examined from a historical viewpoint. Even if the exterior algebra over…
The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…
The Lie algebra generated by $m\ $ $p$-dimensional Grassmannian Dirac operators and $m\ $ $p$-dimensional vector variables is identified as the orthogonal Lie algebra $\mathfrak{so}(2m+1)$. In this paper, we study the space $\mathcal{P}$ of…
We study structural limitations of purely algebraic reasoning in the analysis of arithmetic dynamical systems. Rather than addressing the truth of specific conjectures, we introduce a fragment - relative notion of algebraic refutability for…
The polysymplectic $(n+1)$-form is introduced as an analogue of the symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of field theory. The corresponding Poisson brackets on differential forms are constructed. The…
The main results on the theory of conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures…
On the Grassmann manifold G (m, n) of m-dimensional subspaces of an n-dimensional projective space P^n, a certain supplementary construction called the normalization is considered. By means of this normalization, one can construct the…
We report on experience with an investigation of the analytic structure of the solution of certain algebraic complex equations. In particular the behavior of their series expansions around the origin is discussed. The investigation imposes…
The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson…
In this essay, we explore the geometric structures involved in the Wolfram model of fundamental physics. Furthermore, we propose some directions of research aiming to get the bosons and fermions out of this framework.
We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. In particular, this enables us to define a notion of superalgebras that does…
In this note we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with Lagrangian boundary conditions, which in the case of the projective line are described in terms of real…
The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three…
Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. Operators of differentiation with respect to paragrassmann variables and a covariant…