## 数学代写|霍普夫代数代写Hopf algebra代考|CO739

statistics-lab™ 为您的留学生涯保驾护航 在代写霍普夫代数Hopf algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写霍普夫代数Hopf algebra代写方面经验极为丰富，各种代写霍普夫代数Hopf algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|霍普夫代数代写Hopf algebra代考|Constructions in the category of coalgebras

Definition 1.4.1 Let $(C, \Delta, \varepsilon)$ be a coalgebra. A k-subspace $D$ of $C$ is called a subcoalgebra if $\Delta(D) \subseteq D \otimes D$.

It is clear that if $D$ is a subcoalgebra, then $D$ together with the map $\Delta_D: D \rightarrow D \otimes D$ induced by $\Delta$ and with the restriction $\varepsilon_D$ of $\varepsilon$ to $D$ is a coalgebra.

Proposition 1.4.2 If $\left(C_i\right){i \in I}$ a family of subcoalgebras of $C$, then $\sum{i \in I} C_i$ is a subcoalgebra.

Proof: $\Delta\left(\sum_{i \in I} C_i\right)=\sum_{i \in I} \Delta\left(C_i\right) \subseteq \sum_{i \in I} C_i \otimes C_i \subseteq\left(\sum_{i \in I} C_i\right) \otimes\left(\sum_{i \in I} C_i\right)$.
In the category $k-\operatorname{Cog}$ the notion of subcoalgebra coincides with the notion of subobject. We describe now the factor objects in this category.

Definition 1.4.3 Let $(C, \Delta, \varepsilon)$ be a coalgebra and I a k-subspace of $C$. Then $I$ is called:
i) a left (right) coideal if $\Delta(I) \subseteq C \otimes I$ (respectively $\Delta(I) \subseteq I \otimes C$ ).
ii) a coideal if $\Delta(I) \subseteq I \otimes C+C \otimes I$ and $\varepsilon(I)=0$.
Exercise 1.4.4 Show that if $I$ is a coideal it does not follow that $I$ is a left or right coideal.

Lemma 1.4.5 Let $V$ and $W$ two k-vector spaces, and $X \subseteq V, Y \subseteq W$ vector subspaces. Then $(V \otimes Y) \cap(X \otimes W)=X \otimes Y$.

Proof: Let $\left(x_j\right){j \in J}$ be a basis in $X$ which we complete with $\left(x_j\right){j \in J^{\prime}}$ up to a basis of $V$. Also consider $\left(y_p\right){p \in P}$ a basis of $Y$, which we complete with $\left(y_p\right){p \in P}$ to get a basis of $W$. Consider an element
\begin{aligned} q &=\sum_{j \in J, p \in P} a_{j p} x_j \otimes y_p+\sum_{j \in J, p \in P^{\prime}} b_{j p} x_j \otimes y_p+\ &+\sum_{j \in J^{\prime}, p \in P} c_{j p} x_j \otimes y_p+\sum_{j \in J^{\prime}, p \in P^{\prime}} d_{j p} x_j \otimes y_p \end{aligned}
in $(V \otimes Y) \cap(X \otimes W)$, where $a_{j p}, b_{j p}, c_{j p}, d_{j p}$ are scalars. Fix $j_0 \in J, p_0 \in P$ and choose $f \in V^, g \in W^$ such that $f\left(x_{j_0}\right)=1, f\left(x_j\right)=0$ for any $j \in J \cup J^{\prime}, j \neq j_0$, and $g\left(y_{p_0}\right)=1, g\left(y_p\right)=0$ for any $p \in P \cup P^{\prime}, p \neq p_0$. Since $q \in V \otimes Y$, it follows that $(f \otimes g)(q)=0$. But then denoting by $\phi: k \otimes k \rightarrow k$ the canonical isomorphism, we have $\phi(f \otimes g)(q)=b_{j_0 p_0}$, hence $b_{j_0 p_0}=0$.
Similarly, we obtain that all of the $b_{j p}, c_{j p}, d_{j p}$ are zero, and thus $q=0$. It follows that $(V \otimes Y) \cap(X \otimes W) \subseteq X \otimes Y$. The reverse inclusion is clear.

## 数学代写|霍普夫代数代写Hopf algebra代考|THE FINITE DUAL

Remark 1.4.23 The coalgebra defined in the previous proposition is called the co-opposite coalgebra of $C$ and it is denoted by $C^{c o p}$. This concept is dual to the one of opposite algebra of an algebra. We recall that if $(A, M, u)$ is a $k$-algebra, then the multiplication $M T: A \otimes A \rightarrow A$ and the unit $u$ define an algebra structure on the space $A$, called the opposite algebra of $A$. This is denoted by $A^{o p}$.

Proposition 1.4.24 Let $C$ be a coalgebra. Then the algebras $\left(C^{c o p}\right)^$ and $\left(C^\right)^{o p}$ are equal.

Proof: Denote by $M_1$ and $M_2$ the multiplications in $\left(C^{c o p}\right)^$ and $\left(C^\right)^{o p}$. Then for any $c^, d^ \in C^$ and $c \in C$ we have $$\begin{gathered} M_1\left(c^ \otimes d^\right)(c)=\left(c^ \otimes d^\right)(T \Delta(c))=\sum c^\left(c_2\right) d^\left(c_1\right) \ M_2\left(c^ \otimes d^\right)(c)=\sum d^\left(c_1\right) c^*\left(c_2\right) \end{gathered}$$
which ends the proof.
We close this section by giving the dual version for coalgebras of the extension of scalars for algebras. Let $(C, \Delta, \varepsilon)$ be a $k$-coalgebra and $\phi: k \rightarrow$ $K$ a morphism of fields. We define $\Delta^{\prime}: K \otimes_k C \rightarrow\left(K \otimes_k C\right) \otimes_K\left(K \otimes_k C\right)$ and $\varepsilon^{\prime}: K \otimes_k C \rightarrow K$ by $\Delta^{\prime}(\alpha \otimes c)=\sum\left(\alpha \otimes c_1\right) \otimes\left(1 \otimes c_2\right)$ and $\varepsilon^{\prime}(\alpha \otimes c)=\alpha \phi(\varepsilon(c))$ for any $\alpha \in K, c \in C$. The following result is again easily checked
Proposition 1.4.25 $\left(K \otimes_k C, \Delta^{\prime}, \varepsilon^{\prime}\right)$ is a $K$-coalgebra.

# 霍普夫代数代考

## 数学代写|霍普夫代数代写Hopf algebra代考|Constructions in the category of coalgebras

i) 一个左 (右) 共理想如果 $\Delta(I) \subseteq C \otimes I$ (分别 $\Delta(I) \subseteq I \otimes C$ ).
ii) 一个理想的如果 $\Delta(I) \subseteq I \otimes C+C \otimes I$ 和 $\varepsilon(I)=0$.

$$q=\sum_{j \in J, p \in P} a_{j p} x_j \otimes y_p+\sum_{j \in J, p \in P^{\prime}} b_{j p} x_j \otimes y_p+\quad+\sum_{j \in J^{\prime}, p \in P} c_{j p} x_j \otimes y_p+\sum_{j \in J^{\prime}, p \in P^{\prime}} d_{j p} x_j \otimes y_p$$

$p \in P \cup P^{\prime}, p \neq p_0$. 自从 $q \in V \otimes Y$ ，它遵循 $(f \otimes g)(q)=0$. 但随后表示 $\phi: k \otimes k \rightarrow k$ 规范同构，我 们有 $\phi(f \otimes g)(q)=b_{j_0 p_0}$ ，因此 $b_{j_0 p_0}=0$.

## 数学代写|霍普夫代数代写Hopf algebra代考|THE FINITE DUAL

$\Delta^{\prime}(\alpha \otimes c)=\sum\left(\alpha \otimes c_1\right) \otimes\left(1 \otimes c_2\right)$ 和 $\varepsilon^{\prime}(\alpha \otimes c)=\alpha \phi(\varepsilon(c))$ 对于任何 $\alpha \in K, c \in C$. 下面的结果再 次很容易验证

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|霍普夫代数代写Hopf algebra代考|MathG6250

statistics-lab™ 为您的留学生涯保驾护航 在代写霍普夫代数Hopf algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写霍普夫代数Hopf algebra代写方面经验极为丰富，各种代写霍普夫代数Hopf algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|霍普夫代数代写Hopf algebra代考|The dual (co)algebra

We will often use the following simple fact: if $X$ and $Y$ are $k$-vector spaces, and $t$ is an element of $X \otimes Y$, then $t$ can be represented as $t=\sum_{i=1}^n x_i \otimes y_i$ for some positive integer $n$, some linearly independent $\left(x_i\right){i=1, n}$ in $X$, and some $\left(y_i\right){i=1, n} \subset Y$. Similarly, $t$ can be written as a sum of tensor monomials with the elements appearing on the second tensor position being linearly independent.

Exercise 1.3.1 Let $t$ be a non-zero element of $X \otimes Y$. Show that there exist a positive integer $n$, some linearly independent $\left(x_i\right){i=1, n} \subset X$, and some linearly independent $\left(y_i\right){i=1, n} \subset Y$ such that $t=\sum_{i=1}^n x_i \otimes y_i$
The following lemma is well known from linear algebra.
Lemma 1.3.2 Let $k$ be a field, $M, N, V$ three $k$-vector spaces, and the linear maps $\phi: M^* \otimes V \rightarrow \operatorname{Hom}(M, V), \phi^{\prime}: \operatorname{Hom}\left(M, N^\right) \rightarrow(M \otimes N)^, \rho:$ $M^* \otimes N^* \rightarrow(M \otimes N)^$ defined by $$\begin{gathered} \phi(f \otimes v)(m)=f(m) v \text { for } f \in M^, v \in V, m \in M, \ \phi^{\prime}(g)(m \otimes n)=g(m)(n) \text { for } g \in H o m\left(M, N^\right), m \in M, n \in N, \ \rho(f \otimes g)(m \otimes n)=f(m) g(n) \text { for } f \in M^, g \in N^*, m \in M, n \in N . \end{gathered}$$
Then:
i) $\phi$ is injective. If moreover $V$ is finite dimensional, then $\phi$ is an isomorphism.
ii) $\phi^{\prime}$ is an isomorphism.
iii) $\rho$ is injective. If moreover $N$ is finite dimensional, then $\rho$ is an isomorphism.

## 数学代写|霍普夫代数代写Hopf algebra代考|THE DUAL (CO)ALGEBRA

$H^$ is isomorphic to the algebra of formal power series $k[[X]]$, a canonical isomorphism being given by $$\phi: H^ \rightarrow k[[X]], \phi(f)=\sum_{n \geq 0} f\left(c_n\right) X^n .$$
The dual problem is the following: having an algebra $(A, M, u)$ can one introduce a canonical structure of a coalgebra on $A^$ ? We remark that is is not possible to perform a construction similar to the one of the dual algebra, due to the inexistence of a canonical morphism $(A \otimes A)^ \rightarrow A^* \otimes A^$. However, if $A$ is finite dimensional, the canonical morphism $\rho: A^ \otimes A^* \rightarrow$ $(A \otimes A)^$ is bijective and we can use $\rho^{-1}$. Thus, if the algebra $(A, M, u)$ is finite dimensional, we define the maps $\Delta: A^ \rightarrow A^* \otimes A^$ and $\varepsilon: A^ \rightarrow k$ by $\Delta=\rho^{-1} M^$ and $\varepsilon=\psi u^$, where $\psi: k^* \rightarrow k$ is the canonical isomorphism, $\psi(f)=f(1)$ for $f \in k^$. We remark that if $\Delta(f)=\sum_i g_i \otimes h_i$, where $g_i, h_i \in A^$, then $f(a b)=$ $\sum_i g_i(a) h_i(b)$ for any $a, b \in A$. Also if $\left(g_j^{\prime}, h_j^{\prime}\right)_j$ is a finite family of elements in $A^$ such that $f(a b)=\sum_j g_j^{\prime}(a) h_j^{\prime}(b)$ for any $a, b \in A$, then $\sum_i g_i \otimes h_i=$ $\sum_j g_j^{\prime} \otimes h_j^{\prime}$, following from the injectivity of $\rho$. In conclusion, we can define $\Delta(f)=\sum g_i \otimes h_i$ for any $\left(g_i, h_i\right) \in A^$ with the property that $f(a b)=\sum_i g_i(a) h_i(b)$ for any $a, b \in A$.

Proposition 1.3.9 If $(A, M, u)$ is a finite dimensional algebra, then we have that $\left(A^*, \Delta, \varepsilon\right)$ is a coalgebra.

# 霍普夫代数代考

## 数学代写|霍普夫代数代写Hopf algebra代考|THE DUAL (CO)ALGEBRA

^同构于形式幂级数的代数 $k[[X]]$, 由下式给出的规范同构
$$\phi: H^{\rightarrow} k[[X]], \phi(f)=\sum_{n \geq 0} f\left(c_n\right) X^n .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|霍普夫代数代写Hopf algebra代考|Math6329

statistics-lab™ 为您的留学生涯保驾护航 在代写霍普夫代数Hopf algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写霍普夫代数Hopf algebra代写方面经验极为丰富，各种代写霍普夫代数Hopf algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

Let $k$ be a field. All unadorned tensor products are over $k$. The following alternative definition for the classical notion of a $k$-algebra sheds a new light on this concept, the ingredients of the new definition being objects (vector spaces), morphisms (linear maps), tensor products and commutative diagrams.

Definition 1.1.1 A $k$-algebra is a triple $(A, M, u)$, where $A$ is a $k$-vector space, $M: A \otimes A \longrightarrow A$ and $u: k \longrightarrow A$ are morphisms of $k$-vector spaces such that the following diagrams are commutative:

Remark 1.1.2 The definition is equivalent to the classical one, requiring $A$ to be a unitary ring, and the existence of a unitary ring morphism $\phi$ : $k \longrightarrow A$, with $\operatorname{Im} \phi \subseteq Z(A)$. Indeed, the multiplication $a \cdot b=M(a \otimes b)$ defines on A a structure of unitary ring, with identity element $u(1)$; the role of $\phi$ is played by $u$ itself. For the converse, we put $M(a \otimes b)=a \cdot b$ and $u=\phi$.
Due to the above, the map $M$ is called the multiplication of the algebra $A$, and $u$ is called its unit. The commutativity of the first diagram in the definition is just the associativity of the multiplication of the algebra.
The importance of the above definition resides in the fact that, due to its categorical nature, it can be dualized. We obtain in this way the notion of a coalgebra.

Example 1.1.4 1) Let $S$ be a nonempty set; $k S$ is the $k$-vector space with basis $S$. Then $k S$ is a coalgebra with comultiplication $\Delta$ and counit $\varepsilon$ defined by $\Delta(s)=s \otimes s, \varepsilon(s)=1$ for any $s \in S$. This shows that any vector space can be endowed with a $k$-coalgebra structure.
2) Let $H$ be a $k$-vector space with basis $\left{c_m \mid m \in \mathbf{N}\right}$. Then $H$ is a coalgebra with comultiplication $\Delta$ and counit $\varepsilon$ defined by
$$\Delta\left(c_m\right)=\sum_{i=0, m} c_i \otimes c_{m-i}, \quad \varepsilon\left(c_m\right)=\delta_{0, m}$$
for any $m \in \mathbf{N}$ ( $\delta_{i j}$ will denote throughout the Kronecker symbol). This coalgebra is called the divided power coalgebra, and we will come back to it later.
3) Let $(S, \leq)$ be a partially ordered locally finite set (i.e. for any $x, y \in S$, with $x \leq y$, the set ${z \in S \mid x \leq z \leq y}$ is finite). Let $T={(x, y) \in S \times S \mid$ $x \leq y}$ and $V k$-vector space with basis $T$. Then $V$ is a coalgebra with
$$\begin{gathered} \Delta(x, y)=\sum_{x \leq z \leq y}(x, z) \otimes(z, y) \ \varepsilon(x, y)=\delta_{x ; y} \end{gathered}$$
for any $(x, y) \in T$.

## 数学代写|霍普夫代数代写Hopf algebra代考|The finite topology

Let $X$ and $Y$ be non-empty sets and $Y^X$ the set of all mappings from $X$ to $Y$. It is clear that we can regard $Y^X$ as the product of the sets $Y_x=Y$, where $x$ ranges over the index set $X$. The finite topology of $Y^X$ is obtained by taking the product space in the category of topological spaces, where each $Y_x$ is regarded as a discrete space. A basis for the open sets in this topology is given by the sets of the form
$$\left{g \in Y^X \mid g\left(x_i\right)=f\left(x_i\right), 1 \leq i \leq n\right}$$
where $\left{x_i \mid 1 \leq i \leq n\right}$ is a finite set of elements of $X$, and $f$ is a fixed element of $Y^X$, so that every open set is a union of open sets of this form.
Assume now that $k$ is a field, and $X$ and $Y$ are two $k$-vector spaces. The set $\operatorname{Hom}k(X, Y)$ of all $k$-homomorphisms from $X$ to $Y$, which is also a $k$-vector space, is a subset of $Y^X$. Thus we can consider on $\operatorname{Hom}_k(X, Y)$ the topology induced by the finite topology on $Y^X$. This topology on $\operatorname{Hom}_k(X, Y)$ is also called the finite topology. If $f \in \operatorname{Hom}_k(X, Y)$, the the sets $$\mathcal{O}\left(f, x_1, \ldots, x_n\right)=\left{g \in \operatorname{Hom}_k(X, Y) \mid g\left(x_i\right)=f\left(x_i\right), 1 \leq i \leq n\right}$$ form a basis for the filter of neighbourhoods of $f$, where $\left{x_i \mid 1 \leq i \leq\right.$ $n}$ ranges over the finite subsets of $X$. Note that $\mathcal{O}\left(f, x_1, \ldots, x_n\right)=$ $\bigcap{i=1}^n \mathcal{O}\left(f, x_i\right)$, and $\mathcal{O}\left(f, x_1, \ldots, x_n\right)=f+\mathcal{O}\left(0, x_1, \ldots, x_n\right)$

Proposition 1.2.1 With the above notation we have the following results. a) $\operatorname{Hom}_k(X, Y)$ is a closed subspace of $Y^X$ (in the finite topology).
b) $\operatorname{Hom}_k(X, Y)$, with the finite topology, is a topological $k$-vector space (the topology of $k$ is the discrete topology).
c) If $\operatorname{dim}_k(X)<\infty$, then the finite topology on Hom ${ }_k(X, Y)$ is discrete.
Proof: a) Pick $f$ in the closure of $\operatorname{Hom}_k(X, Y)$, and let $x_1, x_2 \in X$, and $\lambda, \mu \in k$. The open set $U=\left{g \in Y^X \mid g\left(x_1\right)=f\left(x_1\right), g\left(x_2\right)=\right.$ $\left.f\left(x_2\right), g\left(\lambda x_1+\mu x_2\right)=f\left(\lambda x_1+\mu x_2\right)\right}$ is a neighbourhood of $f$, and therefore $U \cap \operatorname{Hom}_k(X, Y) \neq \emptyset$. If $h \in U \cap \operatorname{Hom}_k(X, Y)$, then $h\left(x_1\right)=f\left(x_1\right)$, $h\left(x_2\right)=f\left(x_2\right)$, and $h\left(\lambda x_1+\mu x_2\right)=f\left(\lambda x_1+\mu x_2\right)$. Since $h\left(\lambda x_1+\mu x_2\right)=$ $\lambda h\left(x_1\right)+\mu h\left(x_2\right)=\lambda f\left(x_1\right)+\mu f\left(x_2\right)$, we obtain that $f\left(\lambda x_1+\mu x_2\right)=\lambda f\left(x_1\right)+$ $\mu f\left(x_2\right)$, so $f \in \operatorname{Hom}_k(X, Y)$.

# 霍普夫代数代考

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

$$\Delta\left(c_m\right)=\sum_{i=0, m} c_i \otimes c_{m-i}, \quad \varepsilon\left(c_m\right)=\delta_{0, m}$$

3) 让 $(S, \leq)$ 是部分有序的居部有限集（即对于任何 $x, y \in S$ ，和 $x \leq y$ ，集合 $z \in S \mid x \leq z \leq y$ 是有限

$$\Delta(x, y)=\sum_{x \leq z \leq y}(x, z) \otimes(z, y) \varepsilon(x, y)=\delta_{x ; y}$$

## 数学代写|霍普夫代数代写Hopf algebra代考|The finite topology

c) 如果 $\operatorname{dim}_k(X)<\infty$ ，那么 Hom 上的有限拓扑 $k(X, Y)$ 是离散的。

$h\left(x_2\right)=f\left(x_2\right)$ ，和 $h\left(\lambda x_1+\mu x_2\right)=f\left(\lambda x_1+\mu x_2\right)$. 自从 $h\left(\lambda x_1+\mu x_2\right)=$
$\lambda h\left(x_1\right)+\mu h\left(x_2\right)=\lambda f\left(x_1\right)+\mu f\left(x_2\right)$ ， 我们得到 $f\left(\lambda x_1+\mu x_2\right)=\lambda f\left(x_1\right)+\mu f\left(x_2\right)$ ，所以 $f \in \operatorname{Hom}_k(X, Y)$

## 有限元方法代写

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## MATLAB代写

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