Generalized Clifford Algebra and related mathematical branches [uts-001Z]
A Clifford algebra is a \(Z_2\)-graded algebra, and a Filtered algebra, the associated graded algebra is the exterior algebra.
It may be thought of as quantizations (cf. quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Specifically, for \(V\) an inner product space, the symbol map constitutes an isomorphism of the underlying super vector spaces of the Clifford algebra with the exterior algebra on \(V\), and one may understand the Clifford algebra as the quantization Grassmann algebra induced from the inner product regarded as an odd symplectic form.
Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.
A Clifford module is a representation of a Clifford algebra.
A Generalized Clifford algebra (GCA) can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms, e.g.
> For \(q_1, q_2, \ldots , q_m \in \mathbb {k}^*\), the generalized Clifford algebra \(C^{(n)}\left (q_1, q_2, \ldots , q_m\right )\) is a unital associative algebra generated by \(e_1, e_2, \ldots , e_m\) subject to the relations
> \[ e_i^n=q_i \mathbf {1}, \quad e_i e_j=\omega e_j e_i, \quad \forall j \lt i . \]
>
> It is easy to see that \(C^{(n)}\left (q_1, q_2, \ldots , q_m\right )\) is \(\mathbb {Z}_n\)-graded where the degree of \(e_i\) is \(\overline {1}\), the generator of \(\mathbb {Z}_n\). [cheng2019new]
In [cheng2019new], note also that "Clifford algebras are weak Hopf algebras in some symmetric tensor categories." while "generalized Clifford algebras are weak Hopf algebras in some suitable braided linear categories of graded vector spaces." as well as that "the Clifford process is a powerful technique to construct larger dimensional Clifford algebras from known ones."
TODO: add papers linking Hopf algebra and Clifford algebra together learned from the adjoint discord here.
Clifford algebras can be obtained by twisting of group algebras [albuquerque2002clifford], where twisted group algebras are studied in [conlon1964twisted], [edwards1969twisted], [edwards1969twisted2].
There exists isomorphisms between certain Clifford algebras and NDAs (Normed Division Algebras) over \(\mathbb {R}\).
Variants of Clifford algebras whose generators are idempotent or nilpotent can be considered. Zeon algebras ("nil-Clifford algebras") have proven to be useful in enumeration problems on graphs where certain configurations are forbidden, such as in the enumeration of matchings and self-avoiding walks. The idempotent property of the generators of idem-Clifford algebras can be used to avoid redundant information when enumerating certain graph and hypergraph structures. See [ewing2022zeon].
It's also closely related to universal enveloping algebra (see [figueroa2010spin] and "The universal enveloping algebra of a Lie algebra is the analogue of the usual group algebra of a group." from group algebra on nlab).
Great discussions about the limitations and generalizations of Clifford algebras can be found in John C. Baez's [baez2002octonions]. Particularly, note Cayley-Dickson construction, Bott periodicity, matrix algebra, triality, and \(\mathbb {R}\) as a real commutative associative nicely normed ∗-algebra. Also see Anthony Lasenby's work on the embedding of octonions in the Clifford geometric algebra for space-time STA ( \(\mathop {\mathcal {C}\ell }(1, 3)\) ) [lasenby2024some].
Note also Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form [depies2024octonions].