By D. J. H. Garling

Clifford algebras, equipped up from quadratic areas have functions in lots of
areas of arithmetic, as normal generalizations of advanced numbers and the
quaternions they're famously utilized in proofs of the Atiyah-Singer index
theorem to supply double covers (spin teams) of the classical teams, and to
generalize the Hilbert rework. additionally they have .their position in physics atmosphere
the scene for Maxwell's equations in electromagnetic thought, for the spin of
elementary debris and for the Dirac equation.

This effortless creation to Clifford algebras makes the required
algebraic historical past - together with multilinear algebra, quadratic areas and
finite-dimensional actual algebras - simply obtainable to investigate scholars and
final-year undergraduates the writer additionally introduces many functions in
mathematics and physics, equipping the reader with Clifford algebras as a
working software in quite a few contexts.

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Extra resources for Clifford Algebras: An Introduction

Example text

When do we have a right B-module N such that N = M ⊗B A? Let D = A ⊗B A be the associated Sweedler canonical coring. Let M = (M, ρ) be a right Dcomodule. We will identify M ⊗A D ∼ = M ⊗B A using the natural isomorphism. The coassociativity and the counit property then take the form ρ(m[0] ) ⊗ m[1] = m[0] ⊗B 1A ⊗B m[1] and m[0] m[1] = m. 7) 1A ⊗B 1A is a grouplike element of D. 2, we havea pair of adjoint functors, between MB and MD , which we will denote by (K, R). The unit and counit of the adjunction will be denoted by η and ε.

Before answering this question, we make two observations that are valid in any Hopf algebra. 7 Let H be a Hopf algebra with invertible algebra ϕ ∈ h ∈ H, we have that ϕ, h(2) S −1 (h(1) ) = ϕ, h 1. l H∗ Moreover, t(2) ⊗ S −1 (t(1) ) ∈ WH . Proof. If ϕ ∈ l , H∗ then ϕ, h(2) h(1) = ϕ, h 1, for all h ∈ H, hence ϕ, h(2) S −1 (h(1) ) = ϕ, h S −1 (1) = ϕ, h 1. 52 and t ∈ l . 2 we know that t(1) ⊗ S(t(2) ) ∈ WH . Applying this property to the Hopf algebra H cop , and keeping in mind that the antipode of H cop is S −1 , we find that t(2) ⊗ S −1 (t(1) ) ∈ WH .

2) If f : C → D is a coalgebra morphism with I ⊂ Ker (f ), then f factors through C/I: there exists a unique coalgebra morphism f : C/I → D such that f ◦ p = f . 8 Let f : C → D be a surjective coalgebra map. Then we have a canonical coalgebra isomorphism C/Ker (f ) ∼ = D. 4 Comodules The definition of a comodule over a coalgebra is a dual version of the definition of module over an algebra. Let k be a commutative ring, and C a k-coalgebra. A right C-comodule (M, ρ) is a k-module M together with a k-linear map ρ : M → M ⊗ C such that the following diagrams commute: ρ ρ /M ⊗C /M ⊗C M M ρ  M ⊗C M ⊗∆ /  ρ⊗C M M ⊗C ⊗C  M rM /  M ⊗ε M ⊗k We will also say that C coacts on M , or that ρ defines a coaction on M .

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