MHB You're welcome, Peter! I'm glad it was helpful.

[Math Amateur]

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series)

In Chapter 2: Linear Algebras and Artinian Rings, Cohn introduces representations of k-algebras as follows:
View attachment 3152So, essentially Cohn considers a right multiplication:

$$\rho_a \ : \ x \mapsto xa$$ where $$x \in A$$

and then declares the representation to be the matrix $$( \rho_a )_{ij}$$

BUT … what is the point here … … ?

… … and why take a right multiplication anyway …Can anyone help me to see the motivation for introducing the notion of representations of k-algebras?

Peter

 

[Deveno]

A ($k$-)algebra is, essentially an algebraic structure. As with many algebraic structures, information about the "internal workings" of a particular example of this structure is often revealed by the behavior of the (algebra) homomorphisms to and from our particular example. This is a highly "conceptual" point of view, and often properties of a given algebra are deduced by the properties of various mappings, without ever looking at "a single element".

On the other hand, matrices are fairly "concrete" things, with operations we can manipulate mechanically, through arithmetic. There is an analogy with groups here: an "abstract group" can be realized (faithfully) as a "concrete" group of permutations of a set. That is, we can transfer "abstract" characterizations, such as normality, to specific shuffling operations on a set.

For example, the dihedral group of order 6, can be realized as "the game of 3-card monte". Conjugation (of an abstract group) corresponds to "replacement" of a shuffling sequence "with different cards".

Moreover, the theory of linear algebra is quite extensively developed, with many useful results on inverting matrices, useful decompositions, and various "canonical" or "normal" forms. These results can be "pulled back" to abstract statements involving $k$-algebras (since this representation is FAITHFUL).

There are two parallel benefits, here: the first is that the abstract theory allows us to "save computation" with specific examples, by applying high-level theorems to "skip steps". The second benefit is that by studying the PARTICULAR $k$-algebra $\text{End}(k^n)$, we can learn many things about how $k$-algebras work IN GENERAL, allowing us to develop a sense of what feels "intuitive" (we gain INSIGHT).

This kind of trade-off is at the border between "pure" and "applied" math-chemists, for example, will work with the representation (images) themselves in analyzing molecular symmetry, whereas a group theorist is more likely to look at the associated $F[G]$-module. Going in a more abstract direction is "why", and in a more concrete direction is "how".

 

[Math Amateur]

A ($k$-)algebra is, essentially an algebraic structure. As with many algebraic structures, information about the "internal workings" of a particular example of this structure is often revealed by the behavior of the (algebra) homomorphisms to and from our particular example. This is a highly "conceptual" point of view, and often properties of a given algebra are deduced by the properties of various mappings, without ever looking at "a single element".

On the other hand, matrices are fairly "concrete" things, with operations we can manipulate mechanically, through arithmetic. There is an analogy with groups here: an "abstract group" can be realized (faithfully) as a "concrete" group of permutations of a set. That is, we can transfer "abstract" characterizations, such as normality, to specific shuffling operations on a set.

For example, the dihedral group of order 6, can be realized as "the game of 3-card monte". Conjugation (of an abstract group) corresponds to "replacement" of a shuffling sequence "with different cards".

Moreover, the theory of linear algebra is quite extensively developed, with many useful results on inverting matrices, useful decompositions, and various "canonical" or "normal" forms. These results can be "pulled back" to abstract statements involving $k$-algebras (since this representation is FAITHFUL).

There are two parallel benefits, here: the first is that the abstract theory allows us to "save computation" with specific examples, by applying high-level theorems to "skip steps". The second benefit is that by studying the PARTICULAR $k$-algebra $\text{End}(k^n)$, we can learn many things about how $k$-algebras work IN GENERAL, allowing us to develop a sense of what feels "intuitive" (we gain INSIGHT).

This kind of trade-off is at the border between "pure" and "applied" math-chemists, for example, will work with the representation (images) themselves in analyzing molecular symmetry, whereas a group theorist is more likely to look at the associated $F[G]$-module. Going in a more abstract direction is "why", and in a more concrete direction is "how".

Thanks for a very insightful and informative post ...

Peter