I Multiplication Maps on Algebras ... Bresar, Lemma 1.25 ...

We know that ##M(A) = \langle L_a , R_b \,\vert \, a,b \in A \rangle ## is generated by left- and right-multiplications by definition.
Lemma 1.24 guarantees us that ##\{L_{u_i} \cdot R_{u_j} = L_{u_i} \circ R_{u_j}\,\vert \, 1 \leq i,j \leq d \}## are linear independent ##^{*})##. These are ##d^2## many, so the dimension of ##M(A)## can only be greater than ##d^2##, because we already have found ##d^2## linear independent vectors, which can be extended to a basis.

##^{*}) \; 0 = \sum \lambda_{ij}L_{u_i}R_{u_j} = \sum L_{u_i} R_{b_i} \Longrightarrow ## (Lemma 1.24) ## b_i = \sum \lambda_{ij}u_j = 0 \Longrightarrow \lambda_{ij}=0## because ##\{u_k\}## is a basis, hence ##L_{u_i}R_{u_j} = L_{u_i} \cdot R_{u_j} = L_{u_i} \circ R_{u_j}## are linear independent.

Well, ##L_a## as well as ##R_b## are endomorphisms of ##A##, i.e. ##\mathbb{F}-##linear mappings ##A \rightarrow A##.
Therefore ##M(A) \subseteq End_\mathbb{F}(A)## and we have a subspace ##M(A)## which has at least dimension ##d^2##. On the other hand ##End_\mathbb{F}(A)## has exactly the dimension ##d^2##, so there is no room left between ##M(A)## and ##End_\mathbb{F}(A)##.
That ## \dim End_\mathbb{F}(A)=d^2 ## can fastest be seen, if we think of matrices: Since ##\{u_1,\ldots, u_d\}## is a basis of ##A##, all linear functions, i.e. every element of ##End_\mathbb{F}(A)## can be written as a ##(d \times d)-##matrix with respect to this basis.

 

I simply wanted to indicate, that multiplication ##"\cdot"## here is the successive application of mappings ##"\circ"##, no matter how it is written even if without multiplication sign. In the end it is ##(L_{u_i}R_{u_j})(x) = L_{u_i}(R_{u_j}(x))=L_{u_i}(x\cdot u_j) = u_i \cdot x \cdot u_j##.

 

A ##\mathbb{K}-##algebra ##\mathcal{A}## is central simple, if the center ##\mathcal{C}(\mathcal{A})=\{c\in \mathbb{A}\,\vert \,ca=ac \,\forall \,a\in\mathcal{A}\}## of ##\mathcal{A}## equals ##\mathbb{K}## and ##\mathcal{A}## as a ring is simple, i.e. without ideals.

The Brauer group is a long time no see. So I've read the definition again. A funny thing, that you're talking about. How does it come?

According to the theorem of Artin-Wedderburn, every central simple algebra is isomorphic to a matrix algebra ##\mathbb{M}(n,\mathcal{D})## over a division ring ##\mathcal{D}##, here ##\mathcal{D}=\mathbb{K}##. Now all ##\mathbb{M}(n,\mathbb{K})## are considered equivalent, i.e. ##\mathbb{M}(n,\mathbb{K}) \text{ ~ } \mathbb{M}(m,\mathbb{K})## and the elements of the Brauer group (of ##\mathbb{K}##) are the equivalence classes. E.g. ##[1] = [\mathbb{M}(1,\mathbb{K})]=[\mathbb{K}]## and the inverse element is the opposite algebra ##\mathcal{A}^{op}## with the multiplication ##(a,b) \mapsto ba##.

However, the really interesting question here is: Do all Scottish mathematicians (Hamilton, Wedderburn, ...) have a special relationship to strange algebras and why is it so?

 

I wasn't quite satisfied with this lapidary description of an equivalence relation here. Unfortunately the English and German Wikipedia page are a one-to-one translation. But the French has been a little bit better. Starting with a central simple algebra ##\mathcal{A}## over a field ##\mathbb{K}##, we have ##\mathcal{A} \otimes_{\mathbb{K}} \mathbb{L} \cong \mathbb{M}(n,\mathbb{L})## for a finite field extension ##\mathbb{L} \supseteq \mathbb{K}##

Now ##\mathcal{A} \text{ ~ } \mathcal{B}## are considered equivalent, if there are natural numbers ##n,m## and an isomorphism such that ##\mathcal{A} \otimes_{\mathbb{K}} \mathbb{M}(n,\mathbb{K}) \cong \mathcal{B} \otimes_{\mathbb{K}} \mathbb{M}(m,\mathbb{K})##.
The (Abelian) Brauer group are now the equivalence classes with multiplication ##\otimes##.

(At least as far as my bad French allowed me to translate it.)

 

@fresh_42 - Re: Scots & maths - try the Jack polynomial.

 

Hmmm ... I wonder whether they spoke Gaelic ...

 

That's correct. In the lines ahead of Definition 1.22 Bresar mentions the rules by which ##\{L_{a_1}R_{b_1}+\ldots +L_{a_n}R_{b_n}\}## becomes an algebra. Without them, it would simply be a set of some endomorphisms.