# 数学代写|李群和李代数代写lie group and lie algebra代考|MATH741

## 数学代写|李群和李代数代写lie group and lie algebra代考|THE BAKER-CAMPBELL-HAUSDORFF FORMULA

As an application of the Levi structure theorem, we shall discuss some applications to the theory of differential equations. We have of course already discussed the original work of Lie on differential equations. Recent work on the application of Lie groups and Lie algebras to differential equations has dealt with problems somewhat different from those studied by Sophus Lie. One example is the study of equations of the type
$$d \mathbf{U}(t) / d t=\mathbf{A}(t) \mathbf{U}(t),$$
where $\mathbf{U}$ and $\mathbf{A}$ are operators and $\mathbf{U}(0)=\mathbf{1}$ is the identity. This equation comes up in quantum mechanics as the Schrödinger equation for the evolution operator $\mathbf{U}(t)$ of a system whose Hamiltonian operator is $\mathbf{A}(t)$, except for a constant factor. Magnus gave a formal solution
$$\mathbf{U}(t)=\exp \Omega(t)$$
for this problem, which converges in some interval about $t=0$ if the operators are finite matrices and $\mathbf{A}$ is a continuous function of $t$ (see [159]).
Wichmann and Norman and Wei treated the case
$$\mathbf{A}(t)=\sum_{i=1}^n a_i(t) \mathbf{X}i,$$ where the constant operators $\mathbf{X}_1, \cdots, \mathbf{X}_n$ span a finite-dimensional Lie algebra [184], [185], [236], [246]. This situation arises in certain problems of control theory [150]. Here the functions $a_i(t)$, known as scalar controls, are piecewise constant functions of time rather than continuous functions. In control theory, the operators $\mathbf{U}(t)$ act on the states of the control system, and describe how the states are transformed as a function of time. In this case, the solution can be given locally in the form treated by Magnus as $$\mathbf{U}(t)=\exp \left(\sum{i=1}^n f_i(t) \mathbf{X}_i\right) .$$

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

One of the main tools in applications of the Lie theory is the concept of a representation. A representation of a group is a homomorphism of the group into a group of linear operators on a vector space. In many applications it is sufficient to treat representations in a fairly loose manner. For example, one speaks of vectors, tensors, pseudoscalars, spinors and the like as being various types of geometrical objects, such as directed arrows, ellipsoids, and so on [39], [208]. Representations of the rotation group crop up throughout mathematical physics in the form of spherical harmonics and Legendre functions, multipole expansions and so forth. The actual representation concept is often held in the background. We must of course make this intuitive conception more precise, and moreover we want to make our discussion apply to any Lie group, not just to the rotation group. However, we shall restrict our attention to analytic homomorphisms, and we assume that all vector spaces have finite dimension.

We define a finite-dimensional representation of a Lie group to be an analytic homomorphism of the Lie group into the general linear group of invertible linear operators on a finite-dimensional vector space. By fixing a basis in the vector space, we obtain a corresponding homomorphism onto a matrix group. Since analytic homomorphisms of Lie groups induce homomorphisms of the corresponding Lie algebras, every finite-dimensional representation of a Lie group induces a representation of its Lie algebra.
A representation of a Lie algebra $L$ consists of a vector space $V$ and a homomorphism $f$ from $L$ into the Lie algebra of all linear operators on $V$. The term “representation” strictly refers to the pair $(V, f)$, but colloquially it is often used to refer just to the homomorphism $f$ alone. The requirement that $f$ be a homomorphism means that if $x$ is in $L$, then $f(x)$ is a linear operator on $V$, and $f(x)$ depends on $x$ linearly:
$$f\left(c_1 x_1+c_2 x_2\right)=c_1 f\left(x_1\right)+c_2 f\left(x_2\right)$$
for all $x_1, x_2$ in $L$ and all scalars $c_1$ and $c_2$. Also we must have
$$f\left(\left[x_1, x_2\right]\right)=f\left(x_1\right) f\left(x_2\right)-f\left(x_2\right) f\left(x_1\right)$$

## 数学代写|李群和李代数代写lie group and lie algebra代考|THE BAKER-CAMPBELL-HAUSDORFF FORMULA

$$d \mathbf{U}(t) / d t=\mathbf{A}(t) \mathbf{U}(t),$$

$$\mathbf{U}(t)=\exp \Omega(t)$$

$$\mathbf{A}(t)=\sum_{i=1}^n a_i(t) \mathbf{X} i$$

$$\mathbf{U}(t)=\exp \left(\sum i=1^n f_i(t) \mathbf{X}_i\right) .$$

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

$$f\left(c_1 x_1+c_2 x_2\right)=c_1 f\left(x_1\right)+c_2 f\left(x_2\right)$$

$$f\left(\left[x_1, x_2\right]\right)=f\left(x_1\right) f\left(x_2\right)-f\left(x_2\right) f\left(x_1\right)$$

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