Can I read On the Bundle of Clifford Algebras Over the Space of Quadratic Forms on EtoBox?

On the Bundle of Clifford Algebras Over the Space of Quadratic Forms by Arkadiusz Jadczyk is a Mathematics article available to read on EtoBox.

What is On the Bundle of Clifford Algebras Over the Space of Quadratic Forms about?

For each quadratic form Q ∈ Quad(V ) on a vector space over a field K, we can define the Clifford algebra Cl(V, Q) as the quotient T(V )/I(Q) of the tensor algebra T(V ) by the two-sided ideal generated by expressions of the form x ⊗ x -Q(x), x ∈ V. In the present paper we consider the whole family {Cl(V, Q) : Q ∈ Quad(V )} in a geometric way as a Z2-graded vector bundle over the base manifold Quad(V ). Bilinear forms F ∈ Bil(V ) act on this bundle providing natural bijective linear mappings λF between different Clifford algebras Cl(V, Q). Alternating (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as 'gauge transformations'. We develop here the formalism of Bourbaki, which generalizes the well known Chevalley's isomorphism Cl(V, Q) → End( (V )) → (V ). In particular we realize the Clifford algebra twisting gauge transformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of 2 (V \* ) representing these forms. Throughout all this paper we intentionally avoid using the so far accepted term "Clifford algebra of a bilinear form" (known otherwise as "Quantum Clifford algebra"), which we consider as possibly

Who reads On the Bundle of Clifford Algebras Over the Space of Quadratic Forms?

Author

Arkadiusz Jadczyk

Publisher

Springer Science and Business Media LLC

Published

2023

Language

EN

Field

Mathematics (Physical Sciences)