Generalized Clifford algebra

In, a Generalized Clifford algebra (GCA) is an that generalizes the , and goes back to the work of , who  utilized and formalized  these  operators introduced by  (1882), and organized by   (1898)  and.

Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces. The concept of a can further be linked to these algebras.

The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.

Abstract definition
The $n$-dimensional generalized Clifford algebra is defined as an associative algebra over a field $F$, generated by
 * $$e_j e_k = \omega_{jk} e_k e_j \,$$
 * $$\omega_{jk} e_l = e_l \omega_{jk} \,$$
 * $$\omega_{jk} \omega_{lm} = \omega_{lm} \omega_{jk} \,$$

and
 * $$e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \,$$

$∀ j,k,l,m = 1,...,n$.

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that
 * $$\omega_{jk} = \omega_{kj}^{-1} = e^{2\pi i \nu_{kj}/N_{kj}}$$

$∀ j,k = 1,...,n$,   and $$N_{kj} =$$$$ (N_j,N_k)$$. The field $F$ is usually taken to be the complex numbers C.

More specific definition
In the more common cases of GCA, the $n$-dimensional generalized Clifford algebra of order $p$ has the property $ω_{kj} = ω$, $$N_k=p$$   for all j,k, and $$\nu_{kj}=1$$. It follows that
 * $$e_j e_k = \omega \, e_k e_j \,$$
 * $$\omega e_l = e_l \omega \,$$

and
 * $$e_j^{p} = 1 = \omega^{p} \,$$

for all j,k,l = 1,...,n, and
 * $$\omega = \omega^{-1} = e^{2\pi i /p}$$

is the $p$th root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.

In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with $ω = −1, and p = 2$.
 * Clifford algebra

Matrix representation
The Clock and Shift matrices can be represented by $n×n$ matrices in Schwinger's canonical notation as

V = \begin{pmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ 0&0&\cdots&1&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&0&0&\cdots&0 \end{pmatrix} $$,   $$U = \begin{pmatrix} 1&0&0&\cdots&0\\ 0&\omega&0&\cdots&0\\ 0&0&\omega^2&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots&\omega^{(n-1)} \end{pmatrix} $$,   $$ W = \begin{pmatrix} 1&1&1&\cdots&1\\ 1&\omega&\omega^2&\cdots&\omega^{n-1}\\ 1&\omega^2&(\omega^2)^2&\cdots&\omega^{2(n-1)}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&\omega^{n-1}&\omega^{2(n-1)}&\cdots&\omega^{(n-1)^2} \end{pmatrix} $$.

Notably, $V^{n} = 1$, $VU = ωUV$ (the ), and $W^{−1}VW = U$ (the ). With $e_{1} = V, e_{2} = VU, and e_{3} = U$, one has three basis elements which, together with $ω$, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, $V$ and $U$, normally referred to as "", were introduced by   in the 1880s. (Note that the matrices  $V$ are cyclic  that perform a ; they are not to be confused with  which have ones only either above or below the diagonal, respectively).

Specific examples
In this case, we have  $ω$ =  −1, and
 * Case $n = p = 2$.

V = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} $$,   $$ U = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} $$,   $$ W = \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} $$ thus

e_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} $$,   $$ e_2 = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} $$,   $$ e_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} $$, which constitute the.

In this case we have $ω$ = $i$,  and
 * Case $n = p = 4$,

V = \begin{pmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0 \end{pmatrix} $$,    $$ U = \begin{pmatrix} 1&0&0&0\\ 0&i&0&0\\ 0&0&-1&0\\ 0&0&0&-i \end{pmatrix} $$,    $$ W = \begin{pmatrix} 1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i \end{pmatrix} $$ and $e_{1}, e_{2}, e_{3}$ may be determined accordingly.