English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce on the abstract level in real Clifford algebras \cl_{p,q} of anon-degenerate quadratic space (V,Q), where Q has signature \epsilon=(p,q), atransposition anti-involution \tp. In a spinor representation, theanti-involution \tp gives transposition, complex Hermitian conjugation orquaternionic Hermitian conjugation when the spinor space \check{S} is viewed asa \cl_{p,q}-left and \check{K}-right module with \check{K} isomorphic to R orR^2, C, or, H or H^2.  \tp is a lifting to \cl_{p,q} of an orthogonal involution \tve: V \rightarrowV which depends on the signature of Q. The involution is a symmetric correlatio\tve: V \rightarrow V^{*} \cong V and it allows one to define a reciprocalbasis for the dual space (V^{*},Q). The anti-involution \tp acts as reversionon \cl_{p,0} and as conjugation on \cl_{0,q}. Using the concept of a transposeof a linear mapping one can show that if [L_u] is a matrix in the left regularrepresentation of the operator L_u: \cl_{p,q} \rightarrow \cl_{p,q} relative toa Grassmann basis B in \cl_{p,q}, then matrix [L_{\tp(u)}] is the matrixtranspose of [L_u].  Of particular importance is the action of \tp on the algebraic spinor spaceS, generated by a primitive idempotent f, or a sum f+\hat{f} in simple orsemisimple algebras. \tp allows us to define a new spinor scalar product S\times S \rightarrow \check{K}, where K=f\cl_{p,q}f and \check{K}=K or K \oplus\hat{K} in the simple or semisimple case. Our scalar product reduces to wellknown ones in Euclidean and anti-Euclidean signatures. \tp acts as identity,complex conjugation, or quaternionic conjugation on \check{K}. The action of\tp on spinors results in matrix transposition, complex Hermitian conjugation,or quaternionic ermitian conjugation. We classify the automorphism groups ofthe new product as O(N), U(N), Sp(N), O(N)^2, or Sp(N)^2.

Transposition anti-involution in Clifford algebras and invariance groups of scalar products on spinor spaces