# 数学代写|李群和李代数代写lie group and lie algebra代考|Math3550

## 数学代写|李群和李代数代写lie group and lie algebra代考|DIRECT SUMS OF VECTOR SPACES

The direct sum, like the tensor product, is a fundamental vector space operation which finds many applications in the theory of Lie algebras and their representations. The direct sum operation in vector space theory is useful both as an analytical tool and as a constructive procedure. Corresponding to these two modes of usage, there are actually two slightly different definitions of the direct sum, known as the internal and the external direct sum. In practice there is little danger in being a bit careless on this point since these two variants are to a large extent equivalent, and the distinction between them can usually be understood from context. If $V_1$ and $V_2$ are vector spaces over a field $\mathbb{F}$, their external direct sum consists of all the ordered pairs $\left(v_1, v_2\right)$, where $v_1$ is in $V_1$ and $v_2$ is in $V_2$. The external direct sum, denoted by $V_1+V_2$, may be regarded as a vector space if vector addition and multiplication of vectors by scalars are defined component wise. A familiar example of the use of the external direct sum is in the construction of the $n$-dimensional vector space
$$\mathbb{F}^n=\mathbb{F}+\cdots+\mathbb{F} \quad \text { ( } n \text { copies) }$$
over a field $\mathbb{F}$. Another example of the use of the external direct sum is in the construction of the tensor algebra, to be discussed later.

The concept of internal direct sum is used when we are talking about the lattice of all subspaces of a given vector space. The set of all subspaces of a vector space is closed under the operation of intersection, but not under the operation of union. The sum $S_1+S_2$ of subspaces $S_1$ and $S_2$ of a vector space is the set of all elements of the form $x+y$, where $x$ is in $S_1$, and $y$ is in $S_2$. Equivalently, the sum of two subspaces may be described as the subspace spanned by their union. The set of all subspaces of a vector space is said to be a lattice under the two operations $\cap$ and $+$. This means that these operations satisfy certain axioms somewhat reminiscent of Boolean algebra, but not quite as strong [112], [134]. In particular, the distributive laws between $\cap$ and + do not hold, and there is no analogue of the de Morgan laws of complementation. More formally, a lattice is a partially ordered set in which every pair of elements has a least upper bound and a greatest lower bound. In the case of the lattice of subspaces of a vector space, the partial ordering is just the inclusion relation, while $S_1 \cap S_2$ is the greatest lower bound of $S_1$ and $S_2$ and $S_1+S_2$ is the least upper bound of $S_1$ and $S_2$.

## 数学代写|李群和李代数代写lie group and lie algebra代考|THE LATTICE OF IDEALS OF A LIE ALGEBRA

For the structure and classification of Lie algebras, the concepts of subalgebra and ideal play the same fundamental roles that subgroups and normal subgroups play in Lie group theory. If $A$ and $B$ are subspaces of a Lie algebra, we denote by $[A, B]$ the subspace spanned by all vectors $[a, b]$, where $a \in A$ and $b \in B$. A subalgebra $S$ of a Lie algebra $L$ is a subspace which is closed under the Lie multiplication, that is, which satisfies $[S, S] \subset S$. An ideal $I$ of a Lie algebra $L$ is a subalgebra such that the Lie product of an element of $L$ with any element of $I$ is in the subalgebra $I$, that is, for all $x \in L$ and all $y \in I$, we have $[x, y] \in I$. Thus, an ideal $I$ of a Lie algebra $L$ may be defined as a subspace which satisfies $[L, I] \subset I$. As may be expected, subalgebras and ideals figure in the fundamental homomorphism theorems for Lie algebras. In particular, the kernel of any Lie algebra homomorphism is an ideal, while the image is a subalgebra.

The sum and intersection of ideals of a Lie algebra are again ideals, and the ideals of a $\mathrm{Lie}$ algebra form a lattice under these two operations. In addition, it follows from the Jacobi identity that the Lie product $\left[I_1, I_2\right]$ of two ideals is again an ideal. The situation regarding subalgebras is a little bit different since the sum of two subalgebras need not be a subalgebra, although the intersection of any set of subalgebras is still a subalgebra. Nevertheless, the subalgebras of a Lie algebra still form a lattice. For both the lattice of ideals and the lattice of subalgebras, the partial ordering is inclusion and the greatest lower bound is intersection. In the lattice of ideals, the least upper bound for a pair of ideals is the sum of those ideals. In the lattice of subalgebras, the least upper bound for two subalgebras is not their sum, but rather the intersection of all subalgebras containing their union. Just as in the case of finite groups, the structure of a Lie algebra can be studied by investigating the properties of its lattice of subalgebras [22], [37], [101], [105], [141], [142].

## 数学代写|李群和李代数代写lie group and lie algebra代考|DIRECT SUMS OF VECTOR SPACES

$$\mathbb{F}^n=\mathbb{F}+\cdots+\mathbb{F} \quad(n \text { copies })$$

## 数学代写|李群和李代数代写lie group and lie algebra代考|THE LATTICE OF IDEALS OF A LIE ALGEBRA

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