Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n). The latter is called the compact symplectic group. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n. S p ( 2 n )   =   { M ∈ M 2 n × 2 n ( F )    with    M T Ω M = Ω } , {displaystyle mathrm {Sp} (2n) = {,Min M_{2n imes 2n}(F) { ext{ with }} M^{T}!Omega ,M=Omega ,},} In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n). The latter is called the compact symplectic group. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n. The name 'symplectic group' is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of 'complex'. The symplectic group of degree 2n over a field F, denoted Sp(2n, F), is the group of 2n × 2n symplectic matrices with entries in F with the operation of matrix multiplication: where MT is the transpose of M, and Ω is a nonsingular skew-symmetric matrix, often taken to be: where In is the identity matrix. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, F). Notational warning: What is here called Sp(2n, F) is often denoted Sp(n, F). Defined more abstractly as a classical group, the symplectic group is the set of linear transformations of a 2n-dimensional vector space over F which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the symplectic group of an abstract symplectic vector space V is denoted Sp(V). Typically, the field F is the field of real numbers R or complex numbers C. In these cases Sp(2n, F) is a real/complex Lie group of real/complex dimension n(2n + 1). These groups are connected but non-compact. The center of Sp(2n, F) consists of the matrices I2n and −I2n as long as the characteristic of the field is not 2. Since the center of Sp(2n, F) is discrete and its quotient modulo the center is a simple group, Sp(2n, F) is considered a simple Lie group.