Noncommutative Vieta theorem in Clifford geometric algebras

Noncommutative Vieta Theorem and Symmetric Functions

There are two ways to generalize basic constructions of commutative algebra for a noncommutative case. More traditional way is to define commutative functions like trace or determinant over noncommuting variables. Beginning with [6] this approach was widely used by different authors, see for example [5], [15], [14], [12], [11], [7]. However, there is another possibility to work with purely nonc...

Geometric Models for Noncommutative Algebras

Hecke algebras as subalgebras of Clifford geometric algebras of multivectors

Clifford geometric algebras of multivectors are introduced which exhibit a bilinear form which is not necessarily symmetric. Looking at a subset of bi-vectors in Cl(IK, B), we proof that theses elements generate the Hecke algebra HIK(n + 1, q) if the bilinear form B is chosen appropriately. This shows, that q-quantization can be generated by Clifford multivector objects which describe usually c...

Geometric tri-product of the spin domain and Clifford algebras

We show that the triple product defined by the spin domain (Bounded Symmetric Domain of type 4 in Cartan’s classification) is closely related to the geometric product in Clifford algebras. We present the properties of this tri-product and compare it with the geometric product. The spin domain can be used to construct a model in which spin 1 and spin1/2 particles coexist. Using the geometric tri...

Generalized Clifford Algebras and the Last Fermat Theorem

Abstract One shows that the Last Fermat Theorem is equivalent to the statement that all rational solutions x + y = 1 of equation (k ≥ 2) are provided by an orbit of rationally parametrized subgroup of a group preserving k–ubic form. This very group naturally arrises in the generalized Clifford algebras setting [1]. I. The stroboscopic motion of the independent oscilatory degree of freedom is gi...