## 数学代写|李群和李代数代写lie group and lie algebra代考|Math735

statistics-lab™ 为您的留学生涯保驾护航 在代写李群和李代数lie group and lie 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。我们的专家在代写李群和李代数lie group and lie algebra代写方面经验极为丰富，各种代写李群和李代数lie group and lie 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|李群和李代数代写lie group and lie algebra代考|THE EXPONENTIAL MAPPING

The classical matrix groups which have been discussed so far are all defined by imposing various algebraic conditions on the matrices belonging to the group. When these conditions are written out in terms of the matrix elements, we obtain complicated nonlinear equations to solve. It is remarkable that in each case these nonlinear conditions can be replaced by an equivalent set of linear conditions. This is in fact the essential idea behind the passage from Lie groups to Lie algebras. The actual relation between Lie groups and Lie algebras can be described in several different ways. To expedite matters, we shall consider here the fastest way to get to the Lie algebra by using the concept of matrix exponentiation. Although this method does not work for all Lie groups, it is sufficient for the classical matrix groups.
For any square matrix $A$, we define $\exp (A)$ by the power series
$$I+A+A^2 / 2 !+\cdots+A^n / n !+\cdots,$$
where $I$ is the identity matrix, and each matrix element converges absolutely. Thus, for example, we have $\exp (0)=I$. From the inverse function theorem, it is easily seen that every matrix in some neighborhood of the identity in $G L(n, C)$ can be expressed as $\exp (A)$ for some $n \times n$ complex matrix $A$. For $G L(n, \mathbb{R})$, we can restrict the matrix $A$ to be real. In the case of $G L(n, \mathbb{R})$, the exponential map does not yield the whole group, as can be seen by considering the case where $n=1$. The group $G L(1, \mathbb{R})$ is the real line with the origin removed, while the image of the exponential map is only the positive real axis. This is the price that must be paid for using the exponential map; that is, we may sometimes lose the global structure of the group. The local structure however is completely preserved.

To assure that the matrix $\exp (A)$ will belong to a given one of the classical matrix Lie groups we have discussed, we require the matrix $A$ to satisfy a suitable linear condition. For the special linear groups, we can replace the nonlinear condition on $\exp (A)$ involving the determinant by a linear condition on $A$ involving the trace. If $\lambda$ is an eigenvalue of $A$, then $\exp (\lambda)$ is an eigenvalue of $\exp (A)$ with the same eigenspace. The determinant of $\exp (A)$, being the product of its eigenvalues, is equal to $\exp (\operatorname{Tr}(A))$ since the trace $\operatorname{Tr}(A)$ is the sum of the eigenvalues of $A$. Hence the determinant of $\exp (A)$ is one if the trace of $A$ is zero.

## 数学代写|李群和李代数代写lie group and lie algebra代考|LIE GROUPS

The general theory of Lie groups seeks to unify and to extend the discussion of the various classical groups. The extension of the theory comes about by treating not only matrix groups, but other types of groups as well.

Basically, a Lie group is the structure which naturally results when analytic machinery is coupled with abstract group theory. Curvilinear coordinates, derivatives, and power series become available as tools for the study of the resulting structure. Technically, a Lie group is first of all required to be a topological group, that is, it must be possible to assign a topology under which the group operation and taking inverses are both continuous functions $[129],[164],[192]$. The general concept of a topological group already leads to a very rich theory in which some of the methods of calculus, especially integration theory, can be applied. To make use of the full power of all the methods of calculus, including differentiation, we must introduce an analytic structure as well as a topology. A proper discussion of the analytic structure of a Lie group requires the concept of an analytic manifold [38], [42], [54], $[64]$

Historically, the concept of an analytic manifold grew out of the theory of Riemann surfaces in complex variable theory [238]. The theory of differentiable and analytic manifolds has since become the basis for modern differential geometry $[127],[230]$. The idea of defining a Lie group in terms of analytic manifolds did not gain widespread acceptance until almost half a century after Lie’s original work [163]. Moreover, it turned out that any connected real analytic manifold may always be analytically embedded in a Euclidean space, so that it appears after all that manifold theory is somewhat of a luxury [171]. Therefore, instead of using the intrinsic definition, we shall regard a connected analytic manifold more intuitively as a smooth surface in a Euclidean space of suitable dimensions. In general, an analytic manifold will consist of several pieces if it is not connected. About each point of the manifold there is required to be an open set with all points within this open set being located by a curvilinear coordinate system. It is also required that the assignment of coordinates to points be a continuous mapping with a continuous inverse. Such an open subset of a manifold is called a coordinate neighborhood or coordinate patch, and the coordinate system is said to give a chart of this neighborhood.

# 李群和李代数代写

## 数学代写|李群和李代数代写lie group and lie algebra代考|THE EXPONENTIAL MAPPING

$$I+A+A^2 / 2 !+\cdots+A^n / n !+\cdots,$$

## 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|李群和李代数代写lie group and lie algebra代考|MATH6390

statistics-lab™ 为您的留学生涯保驾护航 在代写李群和李代数lie group and lie 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。我们的专家在代写李群和李代数lie group and lie algebra代写方面经验极为丰富，各种代写李群和李代数lie group and lie 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|李群和李代数代写lie group and lie algebra代考|GROUPS IN GEOMETRY

Matrix Lie groups also arise in various types of non-Euclidean geometry, and are sometimes encountered in applications. An important example is the Lorentz group, which is an indefinite-metric version of the orthogonal group for the geometry of space-time in the special theory of relativity. Another group, the symplectic group, plays an important role in the geometry of phase space in classical mechanics. To obtain a deeper understanding of the geometrical significance of the orthogonal and symplectic groups, we turn to some elementary notions from the theory of bilinear forms [134].

A bilinear form on a vector space $V$ over a field $\mathbb{F}$ is a function which assigns to each ordered pair of vectors $x$ and $y$ in $V$ a scalar $(x, y) \in \mathbb{F}$, and which satisfies the following properties for any scalars $\alpha$ and $\beta$ and any vectors $x, y$ and $z$ in $V$ :
\begin{aligned} &(\alpha x+\beta y, z)=\alpha(x, z)+\beta(y, z), \ &(x, \alpha y+\beta z)=\alpha(x, y)+\beta(x, z) . \end{aligned}

A bilinear form $(x, y)$ can be completely described by its matrix $\beta_{i j}$ $=\left(e_i, e_j\right)$ with respect to any basis $e_1, \cdots, e_n$. Indeed, if we are given this matrix, then for any vectors
$$x=\sum_i \xi^i e_i \text { and } y=\sum_j \eta^j e_j,$$
we can compute
$$(x, y)=\sum_{i, j} \beta_{i j} \xi^{\xi i} \eta^j .$$
A bilinear form $(x, y)$ is symmetric if $(x, y)=(y, x)$ for all $x$ and $y$ in $V$. If $(x, y)$ is a symmetric bilinear form, then its matrix $\beta$ is equal to its own transpose. Similarly, we say that a bilinear form $(x, y)$ is antisymmetric if $(x, y)$ $=-(y, x)$ for all $x$ and $y$ in $V$, and in this case $\beta$ is equal to the negative of its transpose. A familiar example of a symmetric bilinear form is, of course, the inner product in Euclidean vector geometry. In Riemannian geometry, the metric tensor and the curvature tensor are both examples of symmetric bilinear forms.

Symmetric bilinear forms also occur naturally in the study of linear operators in vector spaces of finite dimension. We may define such a bilinear form by using the trace operation $\operatorname{Tr}_V()$ for linear operators on a vector space $V$. If we introduce a basis, the trace of a linear operator is the sum of its diagonal matrix elements
$$\operatorname{Tr}_V(\alpha)=\sum_i \alpha_i^i$$

## 数学代写|李群和李代数代写lie group and lie algebra代考|Lie Groups and Lie Algebras

In the real case, if det $\beta \neq 0$, we can find a matrix $\sigma$ so that $\beta^{\prime}$ is diagonal and its diagonal elements are $\pm 1$. The signature $(p, q)$, where $p$ and $q$ are the numbers of $+1$ ‘s and $-1$ ‘s respectively on the diagonal, is an invariant of the real symmetric bilinear form. In the special case where $p=n$ and $q=0$, the real symmetric bilinear form is said to be positive definite because $(x, x)>0$ for all $x \neq 0$, and we obtain the ordinary real orthogonal group $O(n, \mathbb{R})$. In the general case, the group $O(p, q ; \mathbb{R})$ which we obtain may be described as an orthogonal group with an indefinite metric of signature $(p, q)$ in a space of dimension $n=p+q$. An example of this is the Lorentz group $O(3,1 ; \mathbb{R})$, which is just the orthogonal group of the Minkowski space-time $\mathbb{R}^4$ $(+++)$

The definition of the symplectic groups is somewhat similar to that of the orthogonal groups. Just as the orthogonal groups consist of linear operators which leave invariant a symmetric nonsingular bilinear form, the symplectic groups consist of linear operators which leave invariant an antisymmetric nonsingular bilinear form. A symplectic linear operator $\alpha$ is an operator satisfying
$$(\alpha x, \alpha y)=(x, y)$$
for an antisymmetric nonsingular bilinear form. Nonsingular antisymmetric bilinear forms can only occur in vector spaces $V$ which ha ve even dimension, $\operatorname{dim} V=2 n$. In any such space we can find a hyperbolic basis $e_{\pm 1}, \cdots, e_{\pm n}$ such that
\begin{aligned} &\left(e_i, e_j\right)=\left(e_{-i}, e_{-j}\right)=0, \ &\left(e_i, e_{-j}\right)=\delta_{i j} \end{aligned}
for all $i, j=1, \cdots, n$.
To describe the matrices of symplectic linear operators with respect to such a basis, we introduce the $2 n \times 2 n$ matrix
$$J=\left[\begin{array}{rr} 0 & 1 \ -1 & 0 \end{array}\right]$$
where 1 denotes the $n \times n$ identity matrix. The matrix group $S p(n, \mathbb{C})$ is then the set of all complex $2 n \times 2 n$ matrices $A$ which satisfy
$$A^T J A=J,$$
where $A^T$ denotes the transpose of the matrix $A$. Closely related to this complex symplectic group $S p(n, \mathbb{C})$ are some other matrix groups obtained by taking intersections. Since we can find hyperbolic bases in both real and complex vector spaces, there is also only one real symplectic group. This real symplectic group is the intersection
$$S p(n, \mathbb{R})=S p(n, \mathbb{C}) \cap G L(2 n, \mathbb{R})$$

# 李群和李代数代写

## 数学代写|李群和李代数代写lie group and lie algebra代考|GROUPS IN GEOMETRY

$$(\alpha x+\beta y, z)=\alpha(x, z)+\beta(y, z), \quad(x, \alpha y+\beta z)=\alpha(x, y)+\beta(x, z) .$$

$$x=\sum_i \xi^i e_i \text { and } y=\sum_j \eta^j e_j,$$

$$(x, y)=\sum_{i, j} \beta_{i j} \xi^{\xi i} \eta^j .$$

$$\operatorname{Tr}_V(\alpha)=\sum_i \alpha_i^i$$

## 数学代写|李群和李代数代写lie group and lie algebra代考|Lie Groups and Lie Algebras

$$(\alpha x, \alpha y)=(x, y)$$

$$\left(e_i, e_j\right)=\left(e_{-i}, e_{-j}\right)=0, \quad\left(e_i, e_{-j}\right)=\delta_{i j}$$

$$J=\left[\begin{array}{lll} 0 & 1-1 & 0 \end{array}\right]$$

$$A^T J A=J,$$

$$S p(n, \mathbb{R})=S p(n, \mathbb{C}) \cap G L(2 n, \mathbb{R})$$

## 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|李群和李代数代写lie group and lie algebra代考|MAT552

statistics-lab™ 为您的留学生涯保驾护航 在代写李群和李代数lie group and lie 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。我们的专家在代写李群和李代数lie group and lie algebra代写方面经验极为丰富，各种代写李群和李代数lie group and lie 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|李群和李代数代写lie group and lie algebra代考|THE GENERAL LINEAR GROUP

Since many of the central ideas of Lie theory arose in the study of geometry and linear algebra, it is fitting to begin with a review of some topics in vector space theory so that we can begin to talk about some of the essential tools, such as the Lie algebra and tensor algebra constructions. It is also instructive to examine the classical linear Lie groups before getting involved with the rather technical general definition of a Lie group.

We recall that a linear mapping $\alpha: V_1 \rightarrow V_2$ from a vector space $V_1$ into a vector space $V_2$ is a mapping which preserves vector addition and scalar multiplication: and
$$\alpha(x+y)=\alpha(x)+\alpha(y), \quad \text { where } x, y \in V_1,$$
$$\alpha(\lambda x)=\lambda \alpha(x),$$
where $x \in V_1$ and $\lambda$ is a scalar.
In particular, a linear operator on a vector space $V$ is a linear mapping from $V$ into itself. A linear operator has an inverse if and only if it is both oneto-one and onto as a mapping. The set of all invertible linear operators on a vector space $V$ is a group called the general linear group on $V$.

In the general linear group we take the group operation to be the ordinary composition of linear operators. A linear operator on a finite-dimensional vector space is invertible if and only if its determinant is nonzero. The general linear group on a finite-dimensional vector space consists therefore of all linear operators with nonzero determinants. The set of all linear operators with determinant one forms a subgroup called the special linear group on $V$.
If we introduce a basis into the vector space, each linear operator can be represented by a matrix and we can talk in terms of matrix groups. Thus, the matrix of a linear operator $\alpha$ with respect to a basis $e_1, \cdots, e_n$ for $V$ is the array of scalars $\left[\alpha_j^i\right]$ determined by
$$\alpha e_j=\sum_{i=1}^n \alpha_j^i e_i .$$
The general and special linear groups may then be regarded as groups of matrices, the group operation being matrix multiplication.

## 数学代写|李群和李代数代写lie group and lie algebra代考|ORTHOGONAL AND UNITARY GROUPS

Other important subgroups of the general linear group $G L(n, \mathbb{C})$ are familiar from elementary matrix theory. These include the group of complex orthogonal matrices $O(n, \mathbb{C})$ and the group of unitary matrices $U(n)$. We recall that a matrix is orthogonal if its transpose is its inverse, while a matrix is unitary if its transposed complex conjugate is its inverse. We obtain “special” versions of these groups by taking their intersections with the complex special linear group. The special complex orthogonal group is
$$S O(n, \mathbb{C})=O(n, \mathbb{C}) \cap S L(n, \mathbb{C}) .$$
The special unitary group is
$$S U(n)=U(n) \cap S L(n, \mathbb{C}) .$$
The determinant of an orthogonal matrix can only be $+1$ or $-1$. This is because the determinant of the transpose of a matrix is the same as that of the matrix, while the determinant of the inverse matrix is the reciprocal of its determinant. Orthogonal matrices are classified as proper or improper, depending upon whether their determinants are $+1$ or $-1$. Thus we may also describe the special orthogonal group as the group of all proper orthogonal matrices.

By considering further intersections of the above groups with the real general linear group, we obtain some groups which are important in Euclidean geometry. The real orthogonal group is defined as
\begin{aligned} O(n, \mathbb{R}) &=O(n, \mathbb{C}) \cap G L(n, \mathbb{R}) \ &=U(n) \cap G L(n, \mathbb{R}) \ &=U(n) \cap O(n, \mathbb{C}) \end{aligned}

# 李群和李代数代写

## 数学代写|李群和李代数代写lie group and lie algebra代考|THE GENERAL LINEAR GROUP

$$\alpha(\lambda x)=\lambda \alpha(x),$$

$$\alpha e_j=\sum_{i=1}^n \alpha_j^i e_i .$$

## 数学代写|李群和李代数代写lie group and lie algebra代考|ORTHOGONAL AND UNITARY GROUPS

$$S O(n, \mathbb{C})=O(n, \mathbb{C}) \cap S L(n, \mathbb{C}) .$$

$$S U(n)=U(n) \cap S L(n, \mathbb{C}) .$$

$$O(n, \mathbb{R})=O(n, \mathbb{C}) \cap G L(n, \mathbb{R}) \quad=U(n) \cap G L(n, \mathbb{R})=U(n) \cap O(n, \mathbb{C})$$

## 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。