My name is Hanzhang Yin, and I have developed this website as a resource to facilitate the review of key concepts in abstract algebra and it is not fully completed. Feel free to send me an email at hanyin@ku.edu if there are any errors or suggestions for improvement.
\(\textbf{Definition. }\) A vector space \(L\) over a field \(\mathbb{F}\), with an operation \(L \times L \rightarrow L\), denoted \((x, y) \mapsto [xy]\) and called the \(\textbf{bracket}\) or \(\textbf{commutator}\) of \(x\) and \(y\), is called a \(\textbf{Lie algebra}\) over \(\mathbb{F}\) if the following axioms are satisfied:
\(\textit{James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Third Printing, P.1}\)
Now fix a \(\textbf{maximal toral subalgebra}\) \(H\) of \(L\), i.e., a toral subalgebra not properly included in any other. (The notation \(H\) is less natural than \(T\), but more traditional.) For example, if \(L = \mathfrak{sl}(n, \mathbb{F})\), it is easy to verify (Exercise 1) that \(H\) can be taken to be the set of diagonal matrices (of trace 0). Since \(H\) is abelian (by the above lemma), \(\operatorname{ad}_L H\) is a commuting family of semisimple endomorphisms of \(L\). According to a standard result in linear algebra, \(\operatorname{ad}_L H\) is simultaneously diagonalizable. In other words, \(L\) is the direct sum of the subspaces \(L_\alpha = \{x \in L \mid [hx] = \alpha(h)x \text{ for all } h \in H\}\), where \(\alpha\) ranges over \(H^*\). Notice that \(L_0\) is simply \(C_L(H)\), the centralizer of \(H\); it includes \(H\), thanks to the lemma. The set of all nonzero \(\alpha \in H^*\) for which \(L_\alpha \neq 0\) is denoted by \(\Phi\); the elements of \(\Phi\) are called the roots of \(L\) relative to \(H\) (and are finite in number). With this notation we have a root space decomposition (or Cartan decomposition): \[ L = C_L(H) \oplus \bigoplus_{\alpha \in \Phi} L_\alpha. \] When \(L = \mathfrak{sl}(n, \mathbb{F})\), for example, the reader will observe that \((*)\) corresponds to the decomposition of \(L\) given by the standard basis (1.2). Our aim in what follows is first to prove that \(H = C_L(H)\), then to describe the set of roots in more detail, and ultimately to show that \(\Phi\) characterizes \(L\) completely.
\(\textit{James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Third Printing, P.35}\)
\(\textbf{Definition 1.}\) A hyperplane in a vector space \( V \) is a subspace of codimension 1. In other words, a hyperplane is a linear variety of the form \( \{ v \in V : f(v) = 0 \} \) where \( f \) is a linear functional on \( V \).
\(\textbf{Definition 2.}\) The ambient space is \(n\)-dimensional Euclidean space, in which case the hyperplanes are the \((n-1)\)-dimensional flats, each of which separates the space into two half spaces.
\(\textbf{Example. }\)More generally, a hyperplane in \(\mathbb{P}_n\) is given by an equation \[a_0x_0 + a_1x_1 + \dots + a_nx_n =0\] with \(a_i\in K \) not all zero.
Fix a vector space \(V\) over \(\mathbb{R}\) with a nondegenerate positive definite inner product \((\cdot, \cdot)\). Define the \(\textit{reflection}\) relative to a nonzero vector \(\alpha \in V\) to be the linear transformation on \(V\) given by \[ s_\alpha(v) = v - \frac{2(v, \alpha)}{(\alpha, \alpha)}\alpha. \] Note, \(s_\alpha\) maps \(\alpha\) to \(-\alpha\) and fixes the hyperplane perpendicular to \(\alpha\). Following the literature, we will use the notation \[ \langle v, \alpha \rangle = \frac{2(v, \alpha)}{(\alpha, \alpha)}. \] so \[ \alpha^* = \frac{2\alpha}{(\alpha, \alpha)} \] such that \[ (v, \alpha^*) = (v, \alpha). \]
\(\textbf{Definition. }\)Let \( \mathbf{x} = (x_1, x_2, \ldots, x_n) \), \( \mathbf{y} = (y_1, y_2, \ldots, y_n) \in \mathbb{R}^n \). Then the \(\textbf{Euclidean Inner Product}\) between \( \mathbf{x} \) and \( \mathbf{y} \), denoted \( \mathbf{x} \cdot \mathbf{y} \), is defined to be \[ \mathbf{x} \cdot \mathbf{y} = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n = \sum_{i=1}^n x_i y_i. \]
\(\textbf{Remark. }\)In short, the Euclidean inner product is the dot product of two vectors.
\(\textbf{Definition 1. }\)A subset \(\Phi\) of the Euclidean space \(E\) is called a root system in \(E\) if the following axioms are satisfied:
\(\textbf{Definition 2. }\)A set \(\Phi\) of non-zero vectors (roots) in a Euclidean space \(E\) equipped with an inner product \(\langle \cdot, \cdot \rangle\) forms a root system if it satisfies the following conditions:
\(\textbf{Definition 3. }\) Let \( E \) be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by \( (\cdot, \cdot) \). A root system \( \Phi \) in \( E \) is a finite set of non-zero vectors (called roots) that satisfy the following conditions:
Now, let up go through an example of a root system: \(A_2\).
\(\textbf{Example. }\)Let \( V \) be the hyperplane in \(\mathbb{R}^3\) consisting of \(\{ (x_1, x_2, x_3) \in \mathbb{R}^3 \mid \sum x_i = 0 \}\) equipped with the usual inner product. Let \(\{ e_1, e_2, e_3 \}\) denote the standard basis for \(\mathbb{R}^3\). Let \( R = \{ \pm(e_1 - e_2), \pm(e_2 - e_3), \pm(e_1 - e_3) \} \). Then \( R \) is the root system of type \( A_2 \) (of rank 2) in \( V \). The set \( S := \{ \alpha_1, \alpha_2 \} = \{ e_1 - e_2, e_2 - e_3 \} \) is a basis of the root system \( R \). The Cartan integer \( \langle \alpha_i, \alpha_j \rangle \) is 2 or -1 according as to whether \( i = j \) or \( i \neq j \).
\(\textbf{Definition. }\)Let \(\Phi\) be a root system in a Euclidean space \(V\) with an inner product \(\langle \cdot, \cdot \rangle\). For each root \(\alpha \in \Phi\), the reflecting hyperplane \(P_\alpha\) associated with the root \(\alpha\) is defined as: \[ P_\alpha = \{ x \in V \mid \langle x, \alpha \rangle = 0 \} \]
Introduction to Lie Algebra and Representation Theory P.42A root \(\alpha \in \Phi\) is called simple if it cannot be written as a sum of two distinct roots in \(\Phi\). The set of simple roots is denoted by \(\Pi\).
A root \(\alpha \in \Phi\) is called positive if it can be written as a sum of simple roots with nonnegative coefficients. The set of positive roots is denoted by \(\Phi^+\).
The Cartan matrix \(A = (a_{ij})\) of a root system \(\Phi\) with respect to the basis \(\Pi\) is defined by \[ a_{ij} = \langle \alpha_i, \alpha_j \rangle. \] And each entry \(a_{ij}\) is an integer and we call it the Cartan integer.
The Weyl group \(W\) of a root system \(\Phi\) is the group generated by the reflections \(\sigma_{\alpha}\) for \(\alpha \in \Phi\). The Weyl group is a finite reflection group acting on the Euclidean space \(E\).
The Dynkin diagram is a graph with vertices indexed by the simple roots and the number of edges between \(a_i\) and \(a_j\) for \(i\neq j\) is \(\langle a_i, a_j\rangle\langle a_j, a_i\rangle\) with an arrow pointing to the smaller of the two roots if they have different lengths.
Let \( V \) be the hyperplane in \(\mathbb{R}^3\) consisting of \(\{ (x_1, x_2, x_3) \in \mathbb{R}^3 \mid \sum x_i = 0 \}\) equipped with the usual inner product. Let \(\{ e_1, e_2, e_3 \}\) denote the standard basis for \(\mathbb{R}^3\). Let \( R = \{ \pm(e_1 - e_2), \pm(e_2 - e_3), \pm(e_1 - e_3) \} \). Then \( R \) is the root system of type \( A_2 \) (of rank 2) in \( V \). The set \( S := \{ \alpha_1, \alpha_2 \} = \{ e_1 - e_2, e_2 - e_3 \} \) is a basis of the root system \( R \). The Cartan integer \( \langle \alpha_i, \alpha_j \rangle \) is 2 or -1 according as to whether \( i = j \) or \( i \neq j \).
\(\textbf{Definition.}\) \(\textit{Affine}\) \( n \)-space (over \( K \) where \(K\) is a perfect field) is the set of \( n \)-tuples \[ \mathbb{A}^n = \mathbb{A}^n(\overline{K}) = \{ P = (x_1, \ldots, x_n) : x_i \in \overline{K} \}. \] Similarly, the set of \( K \)-rational points of \( \mathbb{A}^n \) is the set \[ \mathbb{A}^n(K) = \{ P = (x_1, \ldots, x_n) \in \mathbb{A}^n : x_i \in K \}. \]
An affine subspace of a vector space \( V \) is a subset of \( V \) of the form \( v + W \) where \( v \in V \) and \( W \) is a subspace of \( V \).
We shall denote by \(\mathbb{A}_K^n\) or just \(\mathbb{A}^n\), the affine \(n\)-space, consisting of \((a_1, \ldots, a_n), a_i \in K\).
For \(P=(a_1, \ldots, a_n) \in \mathbb{A}^n\), the \(a_i\)'s are called the affine coordinates of \(P\). Affine varieties Given an ideal \(I\) in the polynomial algebra \(K[x_1, \ldots, x_n]\), let \[ V(I)=\{(a_1, \ldots, a_n) \in \mathbb{A}^n \mid f(a_1, \ldots, a_n)=0 \text{ for all } f \in I\}. \] The set \(V(I)\) is called an affine variety. Clearly \(V(I)=V(\sqrt{I})\). Fixing a (finite) set of generators \(\{f_1, \ldots, f_r\}\) for \(I, V(I)\) can be thought of as the set of common zeros of \(f_1, \ldots, f_r\). Conversely, given a subset \(X \subset \mathbb{A}^n\), let \[ \mathcal{I}(X)=\{f \in K[x_1, \ldots, x_n] \mid f(x)=0 \text{ for all } x \in X\}. \]
A linear variety in a vector space \( V \) is a subset of \( V \) of the form \( W \) where \( W \) is a subspace of \( V \).
A linear variety corresponding to a subspace of codimension 1 is called a hyperplane.\(\textbf{Definition.}\) An affine algebraic set \( V \) is called an (affine) variety if \( I(V) \) is a prime ideal in \( \overline{K}[X] \). Note that if \( V \) is defined over \( K \), it is not enough to check that \( I(V/K) \) is prime in \( K[X] \). For example, consider the ideal \( (X_1^2 - 2X_2^2) \) in \( \mathbb{Q}[X_1, X_2] \).
Define a topology on \(\mathbb{A}^n\) by declaring \(\{V(I) \mid I\) an ideal in \(K[x_1, \ldots, x_n]\}\) as the set of closed sets. We now check that this defines a topology on \(\mathbb{A}^n\). We have:
Statements (1) and (3) are clear. For (2): The inclusion \(V(I) \cup V(J) \subset V(I \cap J)\) is clear. To see the reverse inclusion, let \(a \in V(I \cap J)\). If possible, let us assume that \(a \notin V(I)\) and \(a \notin V(J)\). This assumption implies that there exist \(f \in I\) and \(g \in J\) such that \(f(a) \neq 0\) and \(g(a) \neq 0\). Since \(fg \in I \cap J\), we have \(f(a)g(a) = 0\), which is a contradiction.
Hence, our assumption is wrong, and the reverse inclusion follows.
\(\textbf{Definition. }\) A polynomial \( f \in \overline{K}[X] = \overline{K}[X_0, \ldots, X_n] \) is homogeneous of degree \( d \) if \[ f(\lambda X_0, \ldots, \lambda X_n) = \lambda^d f(X_0, \ldots, X_n) \text{ for all } \lambda \in \overline{K}. \] An ideal \( I \subseteq \overline{K}[X] \) is homogeneous if it is generated by homogeneous polynomials.
Let \( f \) be a homogeneous polynomial and let \( P \in \mathbb{P}^n \). It makes sense to ask whether \( f(P) = 0 \), since the answer is independent of the choice of homogeneous coordinates for \( P \). To each homogeneous ideal \( I \) we associate a subset of \( \mathbb{P}^n \) by the rule \[ V_I = \{ P \in \mathbb{P}^n : f(P) = 0 \text{ for all homogeneous } f \in I \}. \]
The arithemtic of Elliptic Curves P.7.\(\textbf{Definition. }\) A (projective) algebraic set is any set of the form \( V_I \) for a homogeneous ideal \( I \). If \( V \) is a projective algebraic set, the (homogeneous) ideal of \( V \), denoted by \( I(V) \), is the ideal of \( \overline{K}[X] \) generated by \[ \{ f \in \overline{K}[X] : f \text{ is homogeneous and } f(P) = 0 \text{ for all } P \in V \}. \] Such a \( V \) is defined over \( K \), denoted by \( V/K \), if its ideal \( I(V) \) can be generated by homogeneous polynomials in \( K[X] \). If \( V \) is defined over \( K \), then the set of \( K \)-rational points of \( V \) is the set \[ V(K) = V \cap \mathbb{P}^n(K). \] As usual, \( V(K) \) may also be described as \[ V(K) = \{ P \in V : P^\sigma = P \text{ for all } \sigma \in G_{\overline{K}/K} \}. \]
The arithemtic of Elliptic Curves P.7.\(\textbf{Definition.}\) A projective algebraic set is called a (projective) variety if its homogeneous ideal \( I(V) \) is a prime ideal in \( \overline{K}[X] \).
The arithemtic of Elliptic Curves P.9.In most cases, it is reasonable to just think of an algebraic variety as a set of solutions to a system of polynomial equations (either over R, C, or some algebraically closed field).
\(\textbf{Definition. }\) An algebraic group is an algebraic variety \( G \) which is at the same time a group, in such a way that the following conditions are satisfied: the maps \( \varphi : G \to G \) given by \( \varphi(g) = g^{-1} \) and \( \psi : G \times G \to G \) given by \( \psi(g_1, g_2) = g_1 g_2 \) are regular maps (here \( g^{-1} \) and \( g_1 g_2 \) are the inverse and product in the group \( G \)).
Let \( T, B, Q, W \), etc. be as above. Given any \( w \in W \) there exists a well-defined coset \( wQ \) in \( G/Q \) which we will denote by \( e_{w,Q} \). Then the set of \( T \)-fixed points in \( G/Q \) for the action given by left multiplication is precisely \( \{ e_{w,Q} \mid w \in W_Q^{\text{min}} \} \). Let \( w \in W_Q^{\text{min}} \), and let \( X_Q(w) \) be the Zariski closure of \( Be_{w,Q} \) in \( G/Q \). Then \( X_Q(w) \) with the canonical reduced scheme structure is called the Schubert variety in \( G/Q \) associated to \( wW_Q \).