Talk:Classification of Clifford algebras

A comment by User:Krivonosov_Leonid

This was left in the article text, but I've put it here till someone can integrate it into the article structure properly. 4pq1injbok (talk) 23:26, 10 September 2011 (UTC)

All parities of symmetry and also full classification of Clifford Algebras follow from three initial isomorphisms

$C\ell _{p,q}(\mathbb {R} )\otimes \mathbb {H} \cong C\ell _{q,p+2}(\mathbb {R} )$,

$C\ell _{p,q}(\mathbb {R} )\otimes \mathbb {R} (2)\cong C\ell _{p+1,q+1}(\mathbb {R} )\cong C\ell _{q+2,p}(\mathbb {R} )$

These isomorphisms are not mentioned here for unknown reasons. Their proofs are precisely the same, as for the isomorphism

$C\ell _{n,0}\otimes \mathbb {H} \cong C\ell _{0,n+2}$

proved, for example, in Clifford Algebras, Hugh Griffiths, May 2007. It is necessary to mean, that there are three anticommutative basis vectors in $\mathbb {R} (2)$:

$i={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}$, $p={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$, $q={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$,

$i^{2}=-1\$, $p^{2}=1\$, $q^{2}=1\$.

The algebra with such basis is pseudoquaternion algebra $\mathbb {H} \_$, so $\mathbb {R} (2)\cong \mathbb {H} \_$. Replacement in the proof of the first isomorphism of imaginary units $i,\ j,\ k$ of quaternion algebra $\mathbb {H}$ with $i,\ p,\ q$ of pseudoquaternion algebra gives the proof of the two others isomorphisms. Just these three isomorphisms underlie 8-fold periodicity in isomorphism distribution between $C\ell _{p,q}(\mathbb {R} )$.

Use of undefined structure

The article explicitly assumes only the ring structure of the Clifford algebras. It would be appropriate to exclude reference to any structure that is not defined in this context: pseudoscalars when n is even, or the generating vector space V. The sections Classification of Clifford algebras#Classification of quadratic form and Classification of Clifford algebras#Unit pseudoscalar are rife with this sort of problem. The first should be reworked, and the second could possibly be deleted. A few other minor instances should also be dealt with. Comments? — Quondum 10:47, 3 September 2012 (UTC)

Charge conjugation

So, this article does discuss a party operator, which in physics I can interpret as P-symmetry. However, there is also the idea of charge conjugation and an explicit charge conjugation operator is constructed in Weyl–Brauer matrices. There is no mention of charge conjugation in this article; why? Surely some representations are charge-conjugate, others are not. There's a complicated table in Higher-dimensional gamma matrices. 67.198.37.16 (talk) 22:11, 17 November 2020 (UTC)