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In algebra many algebraic groups $G$ of finite type over a field $k$ may be realized as closed subgroups of $\operatorname{GL}_k(V)$, where $V$ is a finite dimensional vector space. Hence there is a set of polynomials $I:=\{f_1,..,f_l\}$ with the property that the zero set $Z(I)\subseteq \operatorname{GL}_k(V)$ defines $G$ as a closed subgroup of $\operatorname{GL}_k(V)$ - the general linear group on $V$. Hence we may view $G$ as a "group of matrices" with coefficients in the field $k$. In fact any affine algebraic group $G$ over a field $k$ may be realized as a closed subgroup of $\operatorname{GL}_k(V)$ for some finite dimensional $k$-vector space $V$. There are non-affine algebraic groups: Abelian varieties. If $E\subseteq \mathbb{P}^2_k$ is an elliptic curve over $k$, it follows $E$ has a group structure $m:E \times E \rightarrow E$, making $(E,m)$ into an abelian algebraic group. The mulitiplication map $m$ is a map of algebraic varieties. Since any affine algebraic group is an affine algebraic variety and an elliptic curve $E$ is a projective variety, we cannot embed $E$ as a closed subgroup of $\operatorname{GL}_k(V)$.

Example: If $G=\operatorname{SL}_k(V)$ where $V$ is a finite dimensional $k$-vector space and $H\subseteq G$ is a closed sub group we may construct the "quotient" $G/H$ and $G/H$ is a smooth quasi projective algebraic variety of finite type over $k$. If $H$ is the subgroup of $G$ fixing a $d$-dimensional sub space ($d< dim(V)$) it follows $G/H\cong \mathbb{G}(d,V)$ is the grassmannian variety parametrizing $d$-dimensional vector subspaces of $V$. The group $H$ is a matrix group - it is a closed sub-group of $G$. Any linear representation $\rho:H \rightarrow \operatorname{GL}_k(W)$ gives rise to a finite rank vector bundle $\pi:E(\rho)\rightarrow \mathbb{G}(d,V)$. Hence in geometry such $k$-linear representations of matrix groups arise in the study of vector bundles on the grassmannian (and other flag varieties). In fact the vector bundle $E(\rho)$ has a canonical left $G$-action, the map $\pi$ is invariant with respect to this action and there is an "equivalence of categories" between the category of "$G$-linearized" finite rank algebraic vector bundles on $\mathbb{G}(d,V)$ and finite dimensional $k$-linear representations of $H$. This type of correspondence between geometry and linear representations of algebraic groups is much studied.

Example: Let $V:=k\{e_0,e_1\}$ and $V^*:=k\{x_0,x_1\}$ with $x_0,x_1$ coordinate functions on $V$. It follows $\operatorname{Sym}_k(V^*)=k[x_0,x_1]$ is the polynomial ring in two variables. It follows $\operatorname{Proj}(\operatorname{Sym}_k(V^*)):=\mathbb{P}^1_k$ is the projective line. Let $\mathcal{O}(l)$ be the tautological bundle with $l\geq 1$. It follows

F1. $\Gamma(\mathbb{P}^1_k, \mathcal{O}(l))=\operatorname{Sym}_k^l(V^*)$

is the $l$'th symmetric product of $V^*$. In this case if we choose a line $L$ in $V$ and let $H$ be the subgroup of $\operatorname{SL}(V)$ fixing $L$ it follows $\operatorname{SL}(V)/H \cong \mathbb{P}^1_k$. Formula F1 gives a geometric construction of all irreducible finite dimensional $\operatorname{SL}(V)$-modules. They are all on the form $Sym_k^l(V^*)$ for some $l\geq 1$. This generalize to higher dimension: For any $V$ we may realize all irreducible finite dimensional $\operatorname{SL}(V)$-modules as global sections of invertibel sheaves on flag varieties $\operatorname{SL}(V)/P$. (The Borel-Weil-Bott formula). Hence global sections of invertible sheaves and higher rank vector bundles on grassmannians and flag varieties carries the structure of a linear representation of a matrix group.

There are classification results on connected algebraic groups over a field $k$:

Thm. Every connected algebraic group $G$ may be realized as an extension of an abelian variety $A$ by an affine algebraic group $H$.

And $H$ may be realized as a matrix group. Hence any connected algebraic group is the extension of an abelian variety by a "matrix group". Hence in algebra they appear "everywhere".

In algebra many algebraic groups $G$ of finite type over a field $k$ may be realized as closed subgroups of $\operatorname{GL}_k(V)$, where $V$ is a finite dimensional vector space. Hence there is a set of polynomials $I:=\{f_1,..,f_l\}$ with the property that the zero set $Z(I)\subseteq \operatorname{GL}_k(V)$ defines $G$ as a closed subgroup of $\operatorname{GL}_k(V)$ - the general linear group on $V$. Hence we may view $G$ as a "group of matrices" with coefficients in the field $k$. In fact any affine algebraic group $G$ over a field $k$ may be realized as a closed subgroup of $\operatorname{GL}_k(V)$ for some finite dimensional $k$-vector space $V$. There are non-affine algebraic groups: Abelian varieties. If $E\subseteq \mathbb{P}^2_k$ is an elliptic curve over $k$, it follows $E$ has a group structure $m:E \times E \rightarrow E$, making $(E,m)$ into an abelian algebraic group. The mulitiplication map $m$ is a map of algebraic varieties. Since any affine algebraic group is an affine algebraic variety and an elliptic curve $E$ is a projective variety, we cannot embed $E$ as a closed subgroup of $\operatorname{GL}_k(V)$.

Example: If $G=\operatorname{SL}_k(V)$ where $V$ is a finite dimensional $k$-vector space and $H\subseteq G$ is a closed sub group we may construct the "quotient" $G/H$ and $G/H$ is a smooth quasi projective algebraic variety of finite type over $k$. If $H$ is the subgroup of $G$ fixing a $d$-dimensional sub space ($d< dim(V)$) it follows $G/H\cong \mathbb{G}(d,V)$ is the grassmannian variety parametrizing $d$-dimensional vector subspaces of $V$. The group $H$ is a matrix group - it is a closed sub-group of $G$. Any linear representation $\rho:H \rightarrow \operatorname{GL}_k(W)$ gives rise to a finite rank vector bundle $\pi:E(\rho)\rightarrow \mathbb{G}(d,V)$. Hence in geometry such $k$-linear representations of matrix groups arise in the study of vector bundles on the grassmannian (and other flag varieties). In fact the vector bundle $E(\rho)$ has a canonical left $G$-action, the map $\pi$ is invariant with respect to this action and there is an "equivalence of categories" between the category of "$G$-linearized" finite rank algebraic vector bundles on $\mathbb{G}(d,V)$ and finite dimensional $k$-linear representations of $H$. This type of correspondence between geometry and linear representations of algebraic groups is much studied.

Example: Let $V:=k\{e_0,e_1\}$ and $V^*:=k\{x_0,x_1\}$ with $x_0,x_1$ coordinate functions on $V$. It follows $\operatorname{Sym}_k(V^*)=k[x_0,x_1]$ is the polynomial ring in two variables. It follows $\operatorname{Proj}(\operatorname{Sym}_k(V^*)):=\mathbb{P}^1_k$ is the projective line. Let $\mathcal{O}(l)$ be the tautological bundle with $l\geq 1$. It follows

F1. $\Gamma(\mathbb{P}^1_k, \mathcal{O}(l))=\operatorname{Sym}_k^l(V^*)$

is the $l$'th symmetric product of $V^*$. In this case if we choose a line $L$ in $V$ and let $H$ be the subgroup of $\operatorname{SL}(V)$ fixing $L$ it follows $\operatorname{SL}(V)/H \cong \mathbb{P}^1_k$. Formula F1 gives a geometric construction of all irreducible finite dimensional $\operatorname{SL}(V)$-modules. They are all on the form $Sym_k^l(V^*)$ for some $l\geq 1$. This generalize to higher dimension: For any $V$ we may realize all irreducible finite dimensional $\operatorname{SL}(V)$-modules as global sections of invertibel sheaves on flag varieties $\operatorname{SL}(V)/P$. (The Borel-Weil-Bott formula). Hence global sections of invertible sheaves and higher rank vector bundles on grassmannians and flag varieties carries the structure of a linear representation of a matrix group.

There are classification results on connected algebraic groups over a field $k$:

Thm. Every connected algebraic group $G$ may be realized as an extension of an abelian variety $A$ by an affine algebraic group $H$.

And $H$ may be realized as a matrix group. Hence any connected algebraic group is the extension of an abelian variety by a "matrix group". Hence in algebra they appear "everywhere".

The message has been deleted and the author has left the forum.

In algebra many algebraic groups $G$ of finite type over a field $k$ may be realized as closed subgroups of $\operatorname{GL}_k(V)$, where $V$ is a finite dimensional vector space. Hence there is a set of polynomials $I:=\{f_1,..,f_l\}$ with the property that the zero set $Z(I)\subseteq \operatorname{GL}_k(V)$ defines $G$ as a closed subgroup of $\operatorname{GL}_k(V)$ - the general linear group on $V$. Hence we may view $G$ as a "group of matrices" with coefficients in the field $k$. In fact any affine algebraic group $G$ over a field $k$ may be realized as a closed subgroup of $\operatorname{GL}_k(V)$ for some finite dimensional $k$-vector space $V$. There are non-affine algebraic groups: Abelian varieties. If $E\subseteq \mathbb{P}^2_k$ is an elliptic curve over $k$, it follows $E$ has a group structure $m:E \times E \rightarrow E$, making $(E,m)$ into an abelian algebraic group. The mulitiplication map $m$ is a map of algebraic varieties. Since any affine algebraic group is an affine algebraic variety and an elliptic curve $E$ is a projective variety, we cannot embed $E$ as a closed subgroup of $\operatorname{GL}_k(V)$.

Example: If $G=\operatorname{SL}_k(V)$ where $V$ is a finite dimensional $k$-vector space and $H\subseteq G$ is a closed sub group we may construct the "quotient" $G/H$ and $G/H$ is a smooth quasi projective algebraic variety of finite type over $k$. If $H$ is the subgroup of $G$ fixing a $d$-dimensional sub space ($d< dim(V)$) it follows $G/H\cong \mathbb{G}(d,V)$ is the grassmannian variety parametrizing $d$-dimensional vector subspaces of $V$. The group $H$ is a matrix group - it is a closed sub-group of $G$. Any linear representation $\rho:H \rightarrow \operatorname{GL}_k(W)$ gives rise to a finite rank vector bundle $\pi:E(\rho)\rightarrow \mathbb{G}(d,V)$. Hence in geometry such $k$-linear representations of matrix groups arise in the study of vector bundles on the grassmannian (and other flag varieties). In fact the vector bundle $E(\rho)$ has a canonical left $G$-action, the map $\pi$ is invariant with respect to this action and there is an "equivalence of categories" between the category of "$G$-linearized" finite rank algebraic vector bundles on $\mathbb{G}(d,V)$ and finite dimensional $k$-linear representations of $H$. This type of correspondence between geometry and linear representations of algebraic groups is much studied.

Example: Let $V:=k\{e_0,e_1\}$ and $V^*:=k\{x_0,x_1\}$ with $x_0,x_1$ coordinate functions on $V$. It follows $\operatorname{Sym}_k(V^*)=k[x_0,x_1]$ is the polynomial ring in two variables. It follows $\operatorname{Proj}(\operatorname{Sym}_k(V^*)):=\mathbb{P}^1_k$ is the projective line. Let $\mathcal{O}(l)$ be the tautological bundle with $l\geq 1$. It follows

F1. $\Gamma(\mathbb{P}^1_k, \mathcal{O}(l))=\operatorname{Sym}_k^l(V^*)$

is the $l$'th symmetric product of $V^*$. In this case if we choose a line $L$ in $V$ and let $H$ be the subgroup of $\operatorname{SL}(V)$ fixing $L$ it follows $\operatorname{SL}(V)/H \cong \mathbb{P}^1_k$. Formula F1 gives a geometric construction of all irreducible finite dimensional $\operatorname{SL}(V)$-modules. They are all on the form $Sym_k^l(V^*)$ for some $l\geq 1$. This generalize to higher dimension: For any $V$ we may realize all irreducible finite dimensional $\operatorname{SL}(V)$-modules as global sections of invertibel sheaves on flag varieties $\operatorname{SL}(V)/P$. (The Borel-Weil-Bott formula). Hence global sections of invertible sheaves and higher rank vector bundles on grassmannians and flag varieties carries the structure of a linear representation of a matrix group.

There are classification results on connected algebraic groups over a field $k$:

Thm. Every connected algebraic group $G$ may be realized as an extension of an abelian variety $A$ by an affine algebraic group $H$.

And $H$ may be realized as a matrix group. Hence any connected algebraic group is the extension of an abelian variety by a "matrix group"

In algebra many algebraic groups $G$ of finite type over a field $k$ may be realized as closed subgroups of $\operatorname{GL}_k(V)$, where $V$ is a finite dimensional vector space. Hence there is a set of polynomials $I:=\{f_1,..,f_l\}$ with the property that the zero set $Z(I)\subseteq \operatorname{GL}_k(V)$ defines $G$ as a closed subgroup of $\operatorname{GL}_k(V)$ - the general linear group on $V$. Hence we may view $G$ as a "group of matrices" with coefficients in the field $k$. In fact any affine algebraic group $G$ over a field $k$ may be realized as a closed subgroup of $\operatorname{GL}_k(V)$ for some finite dimensional $k$-vector space $V$. There are non-affine algebraic groups: Abelian varieties. If $E\subseteq \mathbb{P}^2_k$ is an elliptic curve over $k$, it follows $E$ has a group structure $m:E \times E \rightarrow E$, making $(E,m)$ into an abelian algebraic group. The mulitiplication map $m$ is a map of algebraic varieties. Since any affine algebraic group is an affine algebraic variety and an elliptic curve $E$ is a projective variety, we cannot embed $E$ as a closed subgroup of $\operatorname{GL}_k(V)$.

Example: If $G=\operatorname{SL}_k(V)$ where $V$ is a finite dimensional $k$-vector space and $H\subseteq G$ is a closed sub group we may construct the "quotient" $G/H$ and $G/H$ is a smooth quasi projective algebraic variety of finite type over $k$. If $H$ is the subgroup of $G$ fixing a $d$-dimensional sub space ($d< dim(V)$) it follows $G/H\cong \mathbb{G}(d,V)$ is the grassmannian variety parametrizing $d$-dimensional vector subspaces of $V$. The group $H$ is a matrix group - it is a closed sub-group of $G$. Any linear representation $\rho:H \rightarrow \operatorname{GL}_k(W)$ gives rise to a finite rank vector bundle $\pi:E(\rho)\rightarrow \mathbb{G}(d,V)$. Hence in geometry such $k$-linear representations of matrix groups arise in the study of vector bundles on the grassmannian (and other flag varieties). In fact the vector bundle $E(\rho)$ has a canonical left $G$-action, the map $\pi$ is invariant with respect to this action and there is an "equivalence of categories" between the category of "$G$-linearized" finite rank algebraic vector bundles on $\mathbb{G}(d,V)$ and finite dimensional $k$-linear representations of $H$. This type of correspondence between geometry and linear representations of algebraic groups is much studied.

Example: Let $V:=k\{e_0,e_1\}$ and $V^*:=k\{x_0,x_1\}$ with $x_0,x_1$ coordinate functions on $V$. It follows $\operatorname{Sym}_k(V^*)=k[x_0,x_1]$ is the polynomial ring in two variables. It follows $\operatorname{Proj}(\operatorname{Sym}_k(V^*)):=\mathbb{P}^1_k$ is the projective line. Let $\mathcal{O}(l)$ be the tautological bundle with $l\geq 1$. It follows

F1. $\Gamma(\mathbb{P}^1_k, \mathcal{O}(l))=\operatorname{Sym}_k^l(V^*)$

is the $l$'th symmetric product of $V^*$. In this case if we choose a line $L$ in $V$ and let $H$ be the subgroup of $\operatorname{SL}(V)$ fixing $L$ it follows $\operatorname{SL}(V)/H \cong \mathbb{P}^1_k$. Formula F1 gives a geometric construction of all irreducible finite dimensional $\operatorname{SL}(V)$-modules. They are all on the form $Sym_k^l(V^*)$ for some $l\geq 1$. This generalize to higher dimension: For any $V$ we may realize all irreducible finite dimensional $\operatorname{SL}(V)$-modules as global sections of invertibel sheaves on flag varieties $\operatorname{SL}(V)/P$. (The Borel-Weil-Bott formula). Hence global sections of invertible sheaves and higher rank vector bundles on grassmannians and flag varieties carries the structure of a linear representation of a matrix group.

There are classification results on connected algebraic groups over a field $k$:

Thm. Every connected algebraic group $G$ may be realized as an extension of an abelian variety $A$ by an affine algebraic group $H$.

And $H$ may be realized as a matrix group. Hence any connected algebraic group is the extension of an abelian variety by a "matrix group". Hence in algebra they appear "everywhere".