Last update: 12 August 2013
Definition 1.1 An algebra is a vector space $A$ over the complex numbers $\u2102$ with multiplication such that $A$ is an associative ring with identity and
$$\left(c{a}_{1}\right){a}_{2}=c\left({a}_{1}{a}_{2}\right)={a}_{1}\left(c{a}_{2}\right)\phantom{\rule{1em}{0ex}}\forall c\in \u2102,\hspace{0.17em}{a}_{1},{a}_{2}\in A$$The last condition merely says that the subalgebra $\u21021\subseteq A$ is central, i.e. the scalars are scalars.
Homework Problem 1.2
1. | Develop the following theory for algebras over other fields. Identify places where prime characteristic or lack of algebraic closure causes problems. |
2. | Develop this theory for Lie algebras, Jordan algebras, or alternative algebras, etc. |
3. | Develop a specific base theoretic cohomology for semisimple algebras. |
Example 1.3 An important example of an associative algebra with identity is the set ${M}_{d}\left(\u2102\right)$ of all $d\times d$ matrices over the complex numbers with the usual matrix multiplication. We will write
$${I}_{d}=\left(\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ & \vdots & \ddots & \\ 0& 0& \cdots & 1\end{array}\right)$$for the identity element. Then the scalars $\u2102$ are isomorphic to the subalgebra of scalar matrices $\u2102{I}_{d}\text{.}$
We will often work with particular bases of an algebra.
Definition 1.4 Let $A$ be a finite dimensional algebra and choose a basis ${a}_{1},{a}_{2},\dots ,{a}_{d}\text{.}$ The structure constants of $A$ are the scalars ${c}_{i,j}^{k}\in \u2102$ defined by the multiplication rules
$${a}_{i}{a}_{j}=\sum _{k=1}^{d}{c}_{i,j}^{k}{a}_{k}$$Alternatively, one may define an algebra structure on a vector space by specifying a set of structure constants. Of course, the structure constants depend on the particular choice of basis.
For example, in ${M}_{d}\left(\u2102\right),$ one might choose the basis of matrix units
$$\left\{{E}_{i,j}\hspace{0.17em}\right|\hspace{0.17em}1\le i,j\le d\}$$where ${E}_{i,j}$ is the matrix with a 1 in the $i\text{th}$ row and $j\text{th}$ column, and zeros elsewhere. If $T=\left({t}_{i,j}\right)$ is a matrix, then we express $T=\sum _{i,j}{t}_{i,j}{E}_{i,j}$ in this basis. Matrix multiplication is merely the linear extension of the multiplication rule
$${E}_{i,j}{E}_{r,s}={\delta}_{j,r}{E}_{i,s}$$and so the structure constants are given by
$${c}_{\{i,j\}\{r,s\}}^{\{k,\ell \}}=\{\begin{array}{cc}1& j=r,\hspace{0.17em}k=i,\hspace{0.17em}\ell =s\\ 0& \text{otherwise}\end{array}$$We turn next to defining our notions of representation and module.
Definition 1.5 An algebra homomorphism is a map $f:A\u27f6B$ such that
1. | $f({a}_{1}+{a}_{2})=f\left({a}_{1}\right)+f\left({a}_{2}\right)\phantom{\rule{1em}{0ex}}\text{(}f$ is an additive group homomorphism) |
2. | $f\left(c{a}_{1}\right)=cf\left({a}_{1}\right)$ |
3. | $f\left({a}_{1}{a}_{2}\right)=f\left({a}_{1}\right)f\left({a}_{2}\right)$ |
4. | $f\left({1}_{A}\right)={1}_{B}$ |
Definition 1.6 An algebra homomorphism $\varphi :A\u27f6{M}_{d}\left(\u2102\right)$ is called a representation. The number $d\in \mathbb{N}$ is called the dimension of the representation.
Group representations fit into this scheme as follows. One notion is that a representation of a group $G$ is a group homomorphism $\varphi :G\u27f6\text{GL}(d,\u2102)\text{.}$ Equivalently, we may consider group representations as a special case of algebra representations as follows:
Definition 1.7 The group algebra $\u2102G$ (often written $\u2102\left[G\right]\text{)}$ is the vector space of finite $\u2102\text{\xe2\u20ac\u201clinear}$ combinations of elements of $G$ with multiplication given as the linear extension of the group multiplication.
Example 1.8 If $G=\{{g}_{1},{g}_{2},\dots ,{g}_{5}\}$ then
$$\u2102G=\{{c}_{1}{g}_{1}+{c}_{2}{g}_{2}+\dots {c}_{5}{g}_{5}\hspace{0.17em}|\hspace{0.17em}{c}_{i}\in \u2102\}$$with multiplication defined as in the following example:
$$(5{g}_{1}+3{g}_{2})(2{g}_{3}+4{g}_{4})=10\left({g}_{1}{g}_{3}\right)+20\left({g}_{1}{g}_{4}\right)+6\left({g}_{2}{g}_{3}\right)+12\left({g}_{2}{g}_{4}\right)$$where the products ${g}_{i}{g}_{j}$ are taken in the group $G\text{.}$
By definition, the elements of $G$ form a basis for $\u2102G\text{.}$
Definition 1.9 A group representation of a group $G$ is an algebra representation of the group algebra $\u2102G\text{.}$
Definition 1.10 A module for an algebra $A$ is a vector space $V$ over $\u2102$ together with an $A\text{-action,}$ that is a map $A\times V\u27f6V$ denoted $\u27e8a,v\u27e9\mapsto a\xb7v$ such that
1. | $({c}_{1}{a}_{1}+{c}_{2}{a}_{2})\xb7v={c}_{1}\left({a}_{1}v\right)+{c}_{2}\left({a}_{2}v\right)$ |
2. | ${a}_{1}\xb7({c}_{1}{v}_{1}+{c}_{2}{v}_{2})={c}_{1}\left({a}_{1}v\right)+{c}_{2}\left({a}_{2}v\right)$ |
3. | ${a}_{1}\xb7({a}_{2}\xb7{v}_{1})=\left({a}_{1}{a}_{2}\right)\xb7{v}_{1}$ |
4. | $1\xb7{v}_{1}={v}_{1}$ |
We next explore the relationship between modules and representations. Let $V$ be a vector space of dimension $d$ and let $\{{v}_{1},{v}_{2},\dots ,{v}_{d}\}$ be a basis of $V\text{.}$ If $T\in \text{End}\left(V\right)$ is a linear transformation then in terms of this basis we may express $T{v}_{i}=\sum _{j}{t}_{j,i}{v}_{j}$ for some scalars ${t}_{j,i}\in \u2102\text{.}$ The map $\text{End}\left(V\right)\u27f6{M}_{d}\left(\u2102\right)$ given by $T\mapsto \left({t}_{j,i}\right)$ is an algebra isomorphism.
Now suppose that $V$ is an $A\text{-module.}$ If $a\in A$ then $a{v}_{i}=\sum _{j}{a}_{j,i}{v}_{j}$ for some scalars ${a}_{j,i}\text{.}$ We claim that $\varphi :A\u27f6{M}_{d}\left(\u2102\right)$ defined by $\varphi \left(a\right)=\left({a}_{j,i}\right)$ is a representation of $A\text{.}$ The bilinearity of the $A\text{-module}$ action implies $\varphi $ is a linear transformation. Clearly $\varphi \left({1}_{A}\right)={I}_{d}$ and it remains to check that $\varphi $ is multiplicative. Let $a,b\in A$ and write $\varphi \left(a\right)=\left({a}_{j,i}\right)$ and $\varphi \left(b\right)=\left({b}_{j,i}\right)\text{.}$ Then
$$ab\xb7{v}_{i}=a\xb7\left(\sum _{j=1}^{d}{b}_{j,i}{v}_{j}\right)=\sum _{j,k=1}^{d}{a}_{k,j}{b}_{j,i}{v}_{k}=\sum _{k=1}^{d}\left(\sum _{j=1}^{d}{a}_{k,j}{b}_{j,i}\right){v}_{k}=\left(\varphi \left(a\right)\varphi \left(b\right)\right){v}_{i}\text{.}$$By definition, $\varphi \left(ab\right)=\varphi \left(a\right)\varphi \left(b\right),$ hence $\varphi $ is a representation of $A\text{.}$
Thus, given an $A\text{-module}$ and a choice of basis, we can produce a representation of $A\text{.}$ What happens if we choose a different basis of $V\text{?}$
Let $\{{w}_{1},{w}_{2},\dots ,{w}_{d}\}$ be another basis of $V,$ and let $\psi :A\u27f6{M}_{d}\left(\u2102\right)$ be the representation defined as above using this new basis.
Definition 1.11 Two representations $\varphi :A\u27f6{M}_{d}\left(\u2102\right)$ and $\psi :A\u27f6{M}_{d}\left(\u2102\right)$ are equivalent provided there exists a matrix $P\in \text{GL}(d,\u2102)$ such that
$$\psi \left(a\right)={P}^{-1}\varphi \left(a\right)P$$for all $a\in A\text{.}$
If we take $P$ to be the change of basis matrix such that $P{w}_{i}={v}_{i}$ then
$${P}^{-1}\varphi \left(a\right)P{w}_{i}={P}^{-1}\varphi \left(a\right){v}_{i}={P}^{-1}\sum _{j}{a}_{j,i}{v}_{j}=\sum _{j}{a}_{j,i}{w}_{j}=\psi \left(a\right){w}_{i},$$hence ${P}^{-1}\varphi \left(a\right)P=\psi \left(a\right),$ i.e. the representations are equivalent. We have shown that up to equivalence, a module corresponds to a unique representation.
On the other hand, if we have a representation $\varphi :A\u27f6{M}_{d}\left(\u2102\right),$ then letting $W={\u2102}^{d}$ be the space of $d\times 1$ column vectors produces an $A\text{-module}$ by $a\xb7w=\varphi \left(a\right)w\text{.}$
Definition 1.12 Given modules $V$ and $W,$ a linear transformation $f:V\u27f6W$ is a module homomorphism provided $f\left(av\right)=af\left(v\right)$ for all $a\in A$ and $v\in V\text{.}$ We say $V$ and $W$ are isomorphic (and write $V\cong W\text{)}$ provided $f$ is invertible.
Homework Problem 1.13
1. | Show that two $A\text{-modules}$ $V$ and $W$ are isomorphic if and only if their associated representations are equivalent. |
2. | Given two equivalent representations $V:A\u27f6{M}_{d}\left(\u2102\right)$ and $W:A\u27f6{M}_{d}\left(\u2102\right)\text{.}$ Show that $V$ and $W$ induce isomorphic $A\text{-module}$ structures on the space of column vectors ${\u2102}^{d}\text{.}$ |
Thus module and representation are equivalent notions. In the future we will not separate the two halves of the brain, but instead use these concepts simultaneously. Often, we will denote a module and its corresponding representation by a single letter, i.e. the module $V$ affords the representation $V:A\u27f6{M}_{d}\left(\u2102\right)$ defined by $a\xb7v=V\left(a\right)v\text{.}$
Definition 1.14 A submodule $V\prime \subseteq V$ is a vector subspace of $V$ that is invariant under the action of $A$ that is $a\xb7v\prime \in V\prime $ for all $A\in A$ and $v\prime \in V\prime \text{.}$
A representation $\psi :A\u27f6{M}_{n}\left(\u2102\right)$ is a subrepresentation of $\varphi :A\u27f6{M}_{d}\left(\u2102\right)$ provided $d\ge n$ and $\varphi $ is equivalent to a representation of the form
$$a\mapsto \left(\begin{array}{cc}\psi \left(a\right)& *\\ {\scriptscriptstyle 0}& *\end{array}\right)$$Definition 1.15 A module $V$ is simple provided $V$ has no nonzero proper submodules.
A representation $\varphi :A\u27f6{M}_{d}\left(\u2102\right)$ irreducible provided $\varphi $ has no nonzero proper subrepresentations.
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.