On Generalized Clifford Algebras - a Survey of Applications

Abstract

The generalized Clifford numbers are defined [1] via a k-ubic form Q k replacing a quadratic one for ordinary construction of an appropriate ideal of tensor algebra. One of the epimorphic image of universal algebras k - C p,q ≅} T (V) /I(Q k ) is the algebra Cl p,q with p + q = dim V = n generators [2]. One enumerates and quotes here some applications of the Cl p,q algebras - mostly those to which the present author has had contributed. Generalized Clifford algebras Cl p,q possess inherent Z k ⊕ Z k ⊕ ... ⊕ Z k grading.This makes generalized Clifford numbers an efficient apparatus to deal with spin lattice systems [3]. For the same reason Cl p,q algebras served in providing the explicit cohomological classification and construction of e-Lie г graded algebras [1,4,5,6] and to prove the PBW theorem [7]. Another application [8] of generalized Clifford numbers originates from Herman Weyl’s example [9] of finite dimensional quantum mechanics. There, one degree of freedom is represented by a toroidal grid Z k × Z k i.e. classical phase space. The seeds of the “k-th order idea” may be traced back to Weierstrass [10] who considered possible commutative extensions of complex numbers to the case of arbitrary number of real dimensions. This possibility was afterwards realized [11, 12, 13] yielding quasi-number systems which form commutative subalgebras of generalized Clifford algebras Cl p,q . These “special” Clifford numbers are perfectly suited for the development of generalized “hyperbolon & ellipton” trigonometries as exposed in [11] and [13]. These quasinumber algebras [13] enable explicit construction of specific generalizations of Tchebyshev polynomials [14]. The generators of the generalized Pauli algebra and the corresponding group of automorphisms preserving k-ubic form are strictly related to the Last Fermat Theorem [15]. Recently the non-commuting matrix elements of matrices from the quantum group GL q (2; C) with q being the n-th root of unity were given a representation as operators in Hubert space with help of C 4 (n) generalized Clifford algebra generators [16]. This is also described here. Also recently [17] the quantum Torii Lie algebra and quantum universal enveloping algebra U q (sl (2)) where q = ω = exp\( q = \omega = \exp \left\{ {\frac{{2\pi i}} {n}} \right\} \) - were embedded in Generalized Clifford algebra C 4 (n) .