Generators and relations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 20 March 2012

Generators and relations

Let $S$ be a set. Let $A$ be an algebra.

A free $A-$module on $S$ is a pair $(A^S,\iota )$ where

1. $A^S$ is an $A-$module
2. $\iota :S\to A^S$ is a function

such that,

1. if $M$ is an $A-$module with a function $ȷ:S\to M$ then there exists a unique morphism
2. $\tilde{ȷ}:A^S\to M$ such that $\tilde{ȷ}\circ \iota =ȷ.$

This is a definition by a universal property.

• A morphism in the category of sets is a function.
• A morphism in the category of categories is a functor.

The forgetful functor is $$\begin{array}{rrcl}F:& A-modules& \to & \mathrm{Sets}\\ & M& \mapsto & M\end{array}$$

The free module functor is $$\begin{array}{rcl}\mathrm{Sets}& \to & A-modules\\ S& \mapsto & A^S\end{array}$$

The universal property for free modules tells us $$\begin{array}{rcl}{\mathrm{Hom}}_A(A^S,M)& \simeq & {\mathrm{Hom}}_{\mathrm{Sets}}(S,M)\\ \tilde{ȷ}& ↤& ȷ\end{array}$$ The free module functor is the left adjoint to the forgetful functor.

Let $M$ be an $A-$module.

A presentation of $M$ is an exact sequence $$A^T\stackrel{\rho }{\to }{A}^S\stackrel{\gamma }{\to }M\to 0$$ where $A^T$ and $A^S$ are free modules.

The term exact sequence means that

1. $\mathrm{im}\rho =\ker \gamma$
2. $\mathrm{im}\gamma =\ker 0.$

Let $X$ be a group.

A presentation of $X$ is an exact sequence $$R\to G\to X\left\{1\right\}$$ where $R$ and $G$ are free groups.

Examples.

1. $\frac{\mathbb{Z}}{m\mathbb{Z}}$ is generated by $1$ with relation $${\displaystyle \underset{m \; times}{\underbrace{1+1+\cdots +1}}}=0.$$ The cyclic group of order m $C_m$ is generated by $g$ with relation $g^m=1.$
   
   Alternatively: $\frac{\mathbb{Z}}{m\mathbb{Z}}$ is presented by $$\begin{array}{ccccccc}\mathbb{Z}& \to & \mathbb{Z}& \to & \frac{\mathbb{Z}}{m\mathbb{Z}}& \to & 0\\ & & 1& \mapsto & 1& & \\ 1& \mapsto & m& & & & \end{array}$$ and $C_m$ is presented by $$\begin{array}{ccccccc}{G}^{\left\{r\right\}}& \to & G^{\left\{g\right\}}& \to & C_m& \to & \left\{1\right\}\\ & & g& \mapsto & g& & \\ r& \mapsto & g^m& & & & \end{array}$$ where $G^S$ denotes the free group on the set $S.$
2. The dihedral group of order $2m,$ $G_{m,m,2}$ is generated by $x$ and $y$ with relations $$x^2=1, y^m=1 \; and\; xy=y^{-1}x.$$ Alternately: $$\begin{array}{ccccccc}{F}_3& \to & F_2& \to & G_{m,m,2}& \to & \left\{1\right\}\\ & & g_1& \mapsto & x& & \\ & & g_2& \mapsto & y& & \\ r_1& \mapsto & g_1^2& & & & \\ r_2& \mapsto & g_2^m& & & & \\ r_3& \mapsto & g_1{g}_2{g}_1^{-1}{g}_2& & & & \end{array}$$ where $F_k$ denotes a free group on a set with $k$ elements.

Presentations by exact sequences

A presentation of $E$ is an exact sequence $$F\left(Y\right)\to F\left(X\right)\to E\to 0$$ where $F\left(Y\right)$ is the free object on the set $Y$ and $F\left(X\right)$ is the free object on the set $X.$

An exact sequence is a sequence of morphisms $$\cdots \to E_i\stackrel{{\gamma }_i}{\to }{E}_{i+1}\stackrel{{\gamma }_{i+1}}{\to }{E}_{i+2}\stackrel{{\gamma }_{i+2}}{\to }\cdots$$ such that if $i\in \mathbb{Z}$ then $\mathrm{im}{\gamma }_i=\ker {\gamma }_{i+1}.$

Let $G$ be a group and let $g_1,...,g_n \in G$ and $r_1,r_2,...,r_m \in F\{x_1,...,x_n\}.$

The group $G$ is presented by generators $g_1,...,g_n$ and relations $r_1,r_2,...,r_m$ if $$\begin{array}{ccccccc}F\{y_1,...,y_m\}& \to & F\{x_1,...,x_n\}& \to & G& \to & \left\{1\right\}\\ & & x_i& \mapsto & g_i& & \\ y_j& \mapsto & r_j& & & & \end{array}$$ is an exact sequence.

Let $V$ be a vector space and let $b_1,...,b_n\in V.$

The vector space $V$ has basis $\{b_1,...,b_n\}$ if $$\begin{array}{ccccccc}\left\{0\right\}& \to & 𝔽\{x_1,...,x_n\}& \to & V& \to & \left\{0\right\}\\ & & x_i& \mapsto & b_i& & \end{array}$$ is an exact sequence.

1. A cyclic group is a group generated by one element.
2. A dihedral group is a group generated by two elements of order two.

Some "familiar" constructions by generators and relations

1. $𝔽[x_1,...,x_n]$ is the free commutative algebra on the set $\{x_1,...,x_n\}.$
2. The free group on the set $X=\{x_1,...,x_n\}$ is the monoid given by generators $\{x_1,...,x_n,y_1,...,y_n\}$ and relations $$x_1{y}_1=1, x_2{y}_2=1, ..., x_n{y}_n=1.$$
3. Let $V$ and $W$ be $R-$modules, $V$ a right $R-$module and $W$ a left $R-$module. The tensor product is the abelian group generated by $\{v\otimes w | v\in V, w\in W\}$ with relations $$\begin{array}{rcl}(z_1{v}_1+z_2{v}_2)\otimes w& =& z_1(v_1\otimes w)+z_2(v_2\otimes w)\\ v\otimes (z_1{w}_1+z_2{w}_2)& =& z_1(v\otimes w_1)+z_2(v\otimes w_2)\\ vr\otimes w& =& v\otimes rw\end{array}$$ for $v,v_1,v_2\in V, w,w_1,w_2\in W, z_1,z_2\in \mathbb{Z}, r\in R.$
4. $𝔽[x_1^{\pm 1},...,x_n^{\pm 1}]$ is the commutative algebra given by generators $x_1,...,x_n,y_1,...,y_n$ and relations $$x_1{y}_1=1, x_2{y}_2=1, ..., x_n{y}_n=1.$$
5. The free abelian group ${\mathbb{Z}}^n$ is the group given by generators $x_1,...,x_n$ with relations $$x_i+x_j=x_j+x_i,for\; i,j\in \{1,2,...,n\}.$$

The goal of classical invariant theory

Let $V$ be a vector space with basis $\{x_1,...,x_n\}.$ Then $$S\left(V\right)=𝔽[x_1,...,x_n].$$ Let $W_0$ be a subgroup of $GL\left(V\right).$ Since $S\left(V\right)$ is a functor, $W_0$ acts on $S\left(V\right):$ $$w(x_{i_1}\cdots x_{i_k})=\left(w{x}_{i_1}\right)\left(w{x}_{i_2}\right)\cdots \left(w{x}_{i_k}\right)$$ for $w\in W_0,$ $1\le i_1\le \cdots \le i_k\le n.$

The invariant ring of $W_0$ is $$S{\left(V\right)}^{W_0}=\{f\in S\left(V\right) | wf=f \; for\; w\in W_0\}.$$

Find generators and relations for $S{\left(V\right)}^{W_0}.$

Let $V$ be a free $\mathbb{Z}-$module with basis $\{x_1,...,x_n\}.$ Then $$𝔽\left[V\right]=𝔽[x_1^{\pm 1},...,x_n^{\pm 1}].$$ Let $W_0$ be a subgroup of $GL\left(V\right)$ (note that $GL\left(V\right)=G{L}_n\left(\mathbb{Z}\right)$), and $$w{x}^{\lambda }=x^{w\lambda }$$ where $x^{\lambda }=x_1^{{\lambda }_1}\cdots x_n^{{\lambda }_n}$ if $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n.$

The invariant ring of $W_0$ is $$𝔽{\left[V\right]}^{W_0}=\{f\in 𝔽\left[V\right] | wf=f \; for\; w\in W_0\}.$$

1. The cyclic groups are $$\mathbb{Z}=G_{\infty ,1,1}, \; and\; \frac{\mathbb{Z}}{r\mathbb{Z}}=G_{r,1,1}.$$
2. The dihedral groups are $$I_2\left(r\right)=G_{r,r,2} \; and\; I_2\left(\infty \right)=G_{\infty ,\infty ,2}.$$

Notes and References

The definition of presentation is found in [Bou, Alg. Ch.10, §1 no.4 (13)]. The definition of cyclic group is in [Bou, Alg.] and the definition of dihedral group is from [Bou, Lie, Ch.IV, §1 Def.2].

References

References?

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