相关论文: A Minimalist Construction of the Geometric Algebra
We construct real and complex matrices in terms of Kronecker products of a Witt basis of 2n null vectors in the geometric algebra over the real and complex numbers. In this basis, every matrix is represented by a unique sum of products of…
The algebraic geometry of a universal algebra $\mathbf{A}$ is defined as the collection of solution sets of term equations. Two algebras $\mathbf{A}_1$ and $\mathbf{A}_2$ are called algebraically equivalent if they have the same algebraic…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
This is a survey paper on Alegbraic Geometry over Lie Algebras
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.
It is pointed out that despite of the non-linearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.
We construct a geometric system from which the Hall algebra can be recovered. This system inherently satisfies higher associativity conditions and thus leads to a categorification of the Hall algebra. We then suggest how to use this…
We obtain a complete classification of minimal simple unitary $W$-algebras.
This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric…
A connected algebraic group in characteristic 0 is uniquely determined by its Lie algebra. In this paper an algorithm is given for constructing an algebraic group in characteristic 0, given its Lie algebra. Using this an algorithm is…
We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
We study the Frattini subalgebra of Leibniz algebras generated by one element. We also investigate Leibniz algebras all of whose proper subalgebras are elementary.
A new simplified approach for teaching electromagnetism is presented using the formalism of geometric algebra (GA) which does not require vector calculus or tensor index notation, thus producing a much more accessible presentation for…
We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories.The algebraic tools…
Does there exist for any $\sigma$-algebra a minimal (with respect to inclusion) generating set? We formulate this problem and answer it in the very special instance of partition generated and standard measurable spaces, the general case…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic…
We give an algorithm for constructing the algebraic hull of a given matrix Lie algebra in characteristic zero. It is based on an algorithm for finding integral linear dependencies of the roots of a polynomial, that is probably of…