Glossary of Common Terms 1
Submodule generated by a subset of a module
p. 129 Lang (2002).
Let \(M\) be a module over a ring \(A\) and let \(S\) be a subset of \(M\). By a linear combination of elements of \(S\) (with coefficients in \(A\)) one means a sum \[ \sum_{x \in S} a_x x \] where \(\{a_x\}\) is a set of elements of \(A\), almost all of which are equal to \(0\). These elements \(a_x\) are called the coefficients of the linear combination. Let \(N\) be the set of all linear combinations of elements of \(S\). Then \(N\) is a submodule of \(M\), for if \[\sum_{x \in S} a_x x \hspace{0.5 in} \text{and} \hspace{0.5 in} \sum_{x \in S} b_x x\] are two linear combinations, then their sum is equal to \[\sum_{x \in S} (a_x + b_x) x,\] and if \(c \in A\), then \[ c \left( \sum_{x \in S} a_x x \right) = \sum_{x \in S} ca_x x, \] and these elements are again linear combinations of elements of \(S\). We shall call \(N\) the submodule generated by \(S\), and we call \(S\) a set of generators or \(N\). We sometimes write \(N = A \langle S \rangle\).
Orthogonal group of matrices
According to Wikipedia, we have that for a field \(F\) and a dimension \(n\), \[ O(n,F) = \{Q \in \text{GL}(n,F) \mid Q^TQ = QQ^T = I \}. \]
According to Lang (2002) page 535, the group of automorphisms of a symmetric form on a vector space is called the orthogonal group of the form.
Lang continues with the following on page 536. There is a standard form described over the real numbers in terms of coordinates by \[f(x,x) = x_1^2 + \cdots + x_n^2\] and over \(\mathbb C\) by \[f(x,x) = x_1 \overline{x_1} + \cdots + x_n \overline{x_n}\] and over the quaternions \(K\) by the same formula as in the complex case. The group of automorphisms of this form would be called the unitary group, and would be denoted by \(U_n\). The points of this group in the reals (respectively, complex, resp. quaternions) would be denoted by \[U_n(\mathbb R) \hspace{0.5 in} U_n(\mathbb C) \hspace{0.5 in} U_n(K),\] and these three groups would be called the real unitary group (resp. complex unitary group, resp. quaternion unitary group).
On page 533 of Lang (2002), the following is written. Let \(f: E \times E \to R\) be sesquilinear. By an automorphism of \(f\) we shall mean a linear automorphism \(A: E \to E\) such that \[\langle Ax, Ay \rangle = \langle x, y \rangle,\] just as for bilinear forms. The meaning of this equation is explained by page 522: \(\langle x,y \rangle\) or \(\langle x, y \rangle\) is written instead of \(f(x,y)\).
Note also Proposition 7.1 on page 533 of Lang (2002). Let \(f: E \times E \to R\) be a non-singular sesquilinear form. Let \(A: E \to E\) be a linear map. Then \(A\) is an automorphism of \(f\) if and only if \(A^* A = id\), and \(A\) is invertible.
On page 551 of Lang (2002), it is written that \(O(n) = U_n(\mathbb R)\).
Symplectic Group
The group of automorphisms of an alternating form on a vector space is called the symplectic group of the form (Lang (2002) page 535). Recall that a matrix \(M\) is said to be alternating if \(M^t = -M\) and the diagonal elements of \(M\) are 0 (Lang (2002) page 530). Recall also that a bilinear form \(f: E \times E \to R\) is said to be alternating if \(f(x,x) = 0\) for all \(x \in E\) (Lang (2002) page 530).
Prop 6.5. Let \(E\) be a free module of dimension \(n\) over \(R\), and let \(\mathcal{B}\) be a fixed basis. The map \[f \mapsto M_{\mathcal{B}}^{\mathcal{B}}(f)\] induces an isomorphism between the module of alternating forms on \(E \times E\) and the module of alternating \(n \times n\) matrices over \(R\). (Lang (2002) page 530).
If \(f\) is the standard alternating form whose matrix is \[ \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix} \] one might denote its group of automorphisms by \(A_{2n}\) and call it the alternating form group (Lang (2002) p. 536).
Here is exercise 22-3 of Lee (2013). The real symplectic group is the subgroup \(\text{Sp}(2n,\mathbb R) \subset \text{GL}(2n, \mathbb R)\) consisting of all \(2n \times 2n\) matrices that leave the standard symplectic tensor \(\omega = \sum_{i=1}^n dx^i \wedge dy^i\) invariant, that is, the set of invertible linear maps \(Z: \mathbb R^{2n} \to \mathbb R^{2n}\) such that \(\omega(Zx, Zy) = \omega(x,y)\) for all \(x,y \in \mathbb R^{2n}\).
- Show that \(Z \in \text{Sp}(2n, \mathbb R)\) if and only if it takes the standard basis to a symplectic basis.
- Show that \(Z \in \text{Sp}(2n, \mathbb R)\) if and only if \(Z^TJZ = J\), where \(J\) is the \(2n \times 2n\) block diagonal matrix \[J = \begin{bmatrix} j & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0 & \cdots & j \end{bmatrix}\] with copies of the \(2 \times 2\) block \(j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) along the main diagonal, and zeros elsewhere.
- Show that \(\text{Sp}(2n, \mathbb R)\) is an embedded Lie subgroup of \(\text{GL}(2n, \mathbb R)\), and determine its dimension.
- Determine the Lie algebra of \(\text{Sp}(2n, \mathbb R)\) as a subalgebra of \(\mathfrak{gl}(2n, \mathbb R)\).
- Is \(\text{Sp}(2n, \mathbb R)\) compact?
The following is the notion of symplectic group from Willard Miller (1972). Let \(J\) be the \(2m \times 2m\) skew-symmetric matrix \[J = \begin{bmatrix} 0 & E_m \\ -E_m & 0 \end{bmatrix}.\] A symplectic matrix is a \(2m \times 2m\) complex matrix \(A\) such that \(A^t J A = J\). The symplectic group \(Sp(m) = Sp(m, \mathbb C)\) is the set of all \(2m \times 2m\) symplectic matrices. (p. 174) More generally, we can consider \(Sp(m)\) as the set of linear operators \(A\) on a \(2m\)-dimensional complex vector space \(V\) such that \(\langle Au, Av \rangle = \langle u, v \rangle\) for all \(u,v \in V\), where \(\langle -, - \rangle\) is a non-singular skew-symmetric bilinear form on \(V\). (p. 344)
Special Group
Let \(Aut(f)\) denote the group of automorphisms of a (sesquilinear or bilinear) form \(f: E \times E \to R\), where \(E\) is an \(R-\)module, and \(R\) is a commutative ring. (Lang (2002) p. 536).
Suppose \(f\) is the standard symmetric bilinear form described over the real numbers by \(f(x,x) = x_1x_1 + \cdots + x_nx_n\). Then \(Aut(f) = U_n(\mathbb R) = O(n, \mathbb R)\). And the subgroup of \(Aut(f)\) consisting of those elements whose determinant is 1 is denoted by \(SU_n(\mathbb R) = SO_n(\mathbb R) = SO(n)\). This is called the special group. (Lang (2002) p. 536).
Similarly, we obtain the special unitary group \(SU(n) = SU_n(\mathbb C)\) as the subgroup of \(U(n) = U_n(\mathbb C) = Aut(f)\) for \(f(x,x) = x_1\overline{x_1} + \cdots + x_n \overline{x_n}\) the standard Hermitian sesquilinear form described in coordinates. (Lang (2002) p. 536).
Let \(F\) be a field. Let \(n\) be a positive integer. By \(GL_n(F)\) we mean the group of \(n \times n\) invertible matrices over \(F\). By \(SL_n(F)\) we mean the subgroup of those matrices whose determinant is equal to 1. (Lang (2002) p. 536).
\(Aut(f)\) for \(f\) a form on a vector space
Matrices Indexed by Objects
We follow the appendix in Grinberg (2020).
Let \(\mathbf k\) be a commutative ring. (Why is commutativity needed?) For my use, it will usually be \(\mathbb R\) or \(\mathbb C\).
Let \(S\) and \(T\) be two sets of objects, for example, edges of a lattice.
An \(S \times T\) matrix over \(\mathbf k\) is a family \[
(a_{s,t})_{(s,t) \in S \times T}
\] where \(a_{s,t} \in \mathbf k\) for all \((s,t) \in S \times T\). We can also view an \(S \times T\) matrix as a function \(S \times T \to \mathbf k\). That is, \[
(a_{s,t})_{(s,t) \in S \times T} \in \mathbf{k}^{S \times T} =
\text{Func}(S \times T, \mathbf k).
\]
The main idea is that each entry of an \(S \times T\) matrix over \(\mathbf k\) is uniquely identified by a pair \((s,t)\). This makes it easy to define matrix addition and scalar multiplication.