02/10 - Guidelines 2.8: Central simple algebras Alex Macedo [email protected]

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Motivation and goals • We will see that quaternion algebras are a special type of algebras called central simple algebras (CSA). • Most preliminary results on quaternion algebras are a consequence of classical results in the theory of CSA’s, namely Wedderburn’s theorem and the Skolem-Noether theorem. So it doesn’t make sense to study quaternion algebras (section 2.1) before talking about CSA’s (sections 2.8 and 2.9). • This week’s goal is to go over section 2.8, which is an introduction to the theory of CSA’s. • After reading 2.8 you should be familiar with: – centralizers; – the left regular representation of an algebra; – tensor products; – the definition of a CSA; – the definition of the Brauer group of a field;

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How to read section 2.8 • Read everything up to definition 2.8.3. Don’t worry about some fancy names. For example, although the left regular representation of an algebra A is an important construction, all you need to know is that it is the map λ : A → EndA (A) given by λ(a) = λa , where λa : A → A is the Aendomorphism given by λa (x) = ax. Here, we regard A as an algebra over itself (scalar multiplication A × A → A is the usual multiplication on A) and EndA (A) is the set of all Aendomorphisms of A. 1

• Before you proceed, you have to know a few things about tensor products. Click here and read sections 2 and 3. Don’t worry about the proofs. Instead focus on the examples, theorem 3.3, the questions (and answers) on page 12 and the remarks on page 13. In those notes, Conrad talks about tensor product between modules, which are “vector spaces over rings”. You can replace modules with algebras and the constructions/results are the same. To define a multiplication on the tensor product A ⊗ B we first define (a ⊗ b)(a0 ⊗ b0 ) = aa0 ⊗ bb0 and then extend this rule to the whole A ⊗ B by linearity. For example, (a1 ⊗ b1 + a2 ⊗ b2 )(a ⊗ b) = a1 a ⊗ b1 b + a2 a ⊗ b2 b. • Go back to the book and read the rest of section 2.8.

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Exercises

At the end of section 2.8, there’s a problem set. Try to solve or, at least, think about problems 1, 2 and 3. Ignore problems 4 and 5.

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