## 数学代写|霍普夫代数代写Hopf algebra代考|Sweedler’s 4-dimensional Hopf algebra

Assume that $\operatorname{char}(k) \neq 2$. Let $H$ be the algebra given by generators and relations as follows: $H$ is generated as a $k$-algebra by $c$ and $x$ satisfying the relations
$$c^2=1, x^2=0, x c=-c x$$
Then $H$ has dimension 4 as a $k$-vector space, with basis ${1, c, x, c x}$. The coalgebra structure is induced by
$$\Delta(c)=c \otimes c, \Delta(x)=c \otimes x+x \otimes 1$$ $$\varepsilon(c)=1, \varepsilon(x)=0$$
In this way, $H$ becomes a bialgebra, which also has an antipode $S$ given by $S(c)=c^{-1}, S(x)=-c x$.
This was the first example of a non-commutative and non-cocommutative Hopf algebra.

Let $n \geq 2$ be an integer, and $\lambda$ a primitive $n$-th root of unity. Consider the algebra $H_{n^2}(\lambda)$ defined by the generators $c$ and $x$ with the relations
$$c^n=1, x^n=0, x c=\lambda c x$$
On this algebra we can introduce a coalgebra structure induced by
$$\begin{gathered} \Delta(c)=c \otimes c, \Delta(x)=c \otimes x+x \otimes 1 \ \varepsilon(c)=1, \varepsilon(x)=0 . \end{gathered}$$
In this way, $H_{n^2}(\lambda)$ becomes a bialgebra of dimension $n^2$, having the basis $\left{c^i x^j \mid 0 \leq i, j \leq n-1\right}$. The antipode is defined by $S(c)=c^{-1}$ and $S(x)=-c^{-1} x$. We note that for $n=2$ and $\lambda=-1$ we obtain Sweedler’s 4-dimensional Hopf algebra.

## 数学代写|霍普夫代数代写Hopf algebra代考|Hopf modules

Throughout this section $H$ will be a Hopf algebra.
Definition 4.4.1 A $k$-vector space $M$ is called a right $H$-Hopf module if $H$ has a right $H$-module structure (the action of an element $h \in H$ on an element $m \in M$ will be denoted by $m h$ ), and a right $H$-comodule structure, given by the map $\rho: M \rightarrow M \otimes H, \rho(m)=\sum m_{(0)} \otimes m_{(1)}$, such that for any $m \in M, h \in H$
$$\rho(m h)=\sum m_{(0)} h_1 \otimes m_{(1)} h_2 .$$

Remark 4.4.2 It is easy to check that $M \otimes H$ has a right module structure over $H \otimes H$ (with the tensor product of algebras structure) defined by $(m \otimes$ $h)(g \otimes p)=m g \otimes h p$ for any $m \otimes h \in M \otimes H, g \otimes p \in H \otimes H$. Considering then the morphism of algebras $\Delta: H \rightarrow H \otimes H$, we obtain that $M \otimes H$ becomes a right $H$-module by restriction of scalars via $\Delta$. This structure is given by $(m \otimes h) g=\sum m g_1 \otimes h g_2$ for any $m \otimes h \in M \otimes H, g \in H$. With this structure in hand, we remark that the compatibility relation from the preceding definition means that $\rho$ is a morphism of right $H$-modules.
There is a dual interpretation of this relation. Consider $H \otimes H$ with the tensor product of coalgebras structure. Then $M \otimes H$ has a natural structure of a right comodule over $H \otimes H$, defined by $m \otimes h \mapsto \sum m_{(0)} \otimes h_1 \otimes m_{(1)} \otimes h_2$. The multipliction $\mu: H \otimes H \rightarrow H$ of the algebra $H$ is a morphism of coalgebras, and then by corestriction of scalars $M \otimes H$ becomes a right $H$ comodule, with $m \otimes h \mapsto \sum m_{(0)} \otimes h_1 \otimes m_{(1)} h_2$. Then the compatibility relation from the preceding definition may be expressed by the fact that the map $\phi: M \otimes H \rightarrow H$, giving the right $H$-module structure of $M$, is a morphism of $H$-comodules.

We can define a category having as objects the right $H$-Hopf modules, and as morphisms between two such objects all linear maps which are also morphisms of right $H$-modules and morphisms of right $H$-comodules. This category is denoted by $\mathcal{M}_H^H$, and will be called the category of right $H$-Hopf modules. It is clear that in this category a morphism is an isomorphism if and only if it is bijective.

# 霍普夫代数代考

## 数学代写|霍普夫代数代写Hopf algebra代考|Sweedler’s 4-dimensional Hopf algebra

$$c^2=1, x^2=0, x c=-c x$$

$$\begin{gathered} \Delta(c)=c \otimes c, \Delta(x)=c \otimes x+x \otimes 1 \ \varepsilon(c)=1, \varepsilon(x)=0 \end{gathered}$$

$$c^n=1, x^n=0, x c=\lambda c x$$

$$\Delta(c)=c \otimes c, \Delta(x)=c \otimes x+x \otimes 1 \varepsilon(c)=1, \varepsilon(x)=0 .$$

## 数学代写|霍普夫代数代写Hopf algebra代考|Hopf modules

$$\rho(m h)=\sum m_{(0)} h_1 \otimes m_{(1)} h_2 .$$

$h)(g \otimes p)=m g \otimes h p$ 对于任何 $m \otimes h \in M \otimes H, g \otimes p \in H \otimes H$. 然后考虑代数的态射
$\Delta: H \rightarrow H \otimes H$ ，我们得到 $M \otimes H$ 成为一种权利 $H$ – 通过标量限制的模块 $\Delta$. 这个结构由
$(m \otimes h) g=\sum m g_1 \otimes h g_2$ 对于任何 $m \otimes h \in M \otimes H, g \in H$. 有了这个结构，我们注意到前面定义

$H \otimes H_{｝ \quad \text { 被定义为 } m \otimes h \mapsto \sum m_{(0)} \otimes h_1 \otimes m_{(1)} \otimes h_2 \text {. 乘法 } \mu: H \otimes H \rightarrow H \text { 代数的 } H \text { 是余代数 }$ 的态射，然后通过标量的共限 $M \otimes H$ 成为一种权利 $H$ 余模块，与 $m \otimes h \mapsto \sum m_{(0)} \otimes h_1 \otimes m_{(1)} h_2$. 那么上述定义中的相容关系可以表示为映射 $\phi: M \otimes H \rightarrow H$ ，赋予权利 $H$-模块结构 $M$, 是一个态射 $H$ commodules

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