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In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. In this article, we shall denote these two groups Sp(2n, F) and Sp(n). The latter is sometimes called the compact symplectic group to distinguish it from the former. Note that many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the matrices used to represent the groups. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
For the square matrix section, see square matrix. ...
Sp(2n, F)
The symplectic group of degree 2n over a field F, denoted Sp(2n, F), is the group of 2n by 2n symplectic matrices with entries in F, and with the group operation that of matrix multiplication. Since all symplectic matrices have unit determinant, the symplectic group is a subgroup of the special linear group SL(2n, F). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, a symplectic matrix is a 2n×2n matrix M (whose entries are typically either real or complex) satisfying the condition where MT denotes the transpose of M and Ω is the 2n×2n skew-symmetric matrix Here In is the n×n identity matrix. ...
This article gives an overview of the various ways to multiply matrices. ...
In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
More abstractly, the symplectic group can be defined as the set of linear transformations of a 2n-dimensional vector space over F that preserve a nondegenerate, skew-symmetric, bilinear form. Such a vector space is called a symplectic vector space. The symplectic group of an abstract symplectic vector space V is also denoted Sp(V). In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...
In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...
In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. ...
When n = 1, the symplectic condition on a matrix is satisfied iff the determinant is one so that Sp(2, F) = SL(2, F). For n > 1, there are additional conditions. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...
Typically, the field F is the field of real numbers, R, or complex numbers, C. In this case Sp(2n, F) is a real/complex Lie group of real/complex dimension n(2n + 1). These groups are connected but noncompact. Sp(2n, C) is simply connected while Sp(2n, R) has a fundamental group isomorphic to Z. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ...
This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ...
In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
The Lie algebra of Sp(2n, F) is given by the set of 2n×2n matrices A (with entries in F) that satisfy In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
- ΩA + ATΩ = 0
where AT is the transpose of A and Ω is the skew-symmetric matrix In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
Sp(n) The symplectic group, Sp(n), is the subgroup of GL(n, H) (invertible quaternionic matrices) which preserves the standard hermitian form on Hn: In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...
 That is, Sp(n) is just the quaternionic unitary group, U(n, H). Indeed, it is sometimes called the hyperunitary group. Note that Sp(n) is not a symplectic group in the sense of the previous section—it does not preserve a skew-symmetric form on Hn (in fact, there is no such form). The justification for calling this group symplectic is explained in the next section. In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ...
Sp(n) is a real Lie group of dimension n(2n + 1). It is compact, connected, and simply connected. The Lie algebra of Sp(n) is given by the set of n by n quaternionic matrices that satisfy Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
 where  is the conjugate transpose of A (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator. In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...
Relationships between the symplectic groups The relationship between the groups Sp(2n, R), Sp(2n, C), and Sp(n) is most evident at the level of their Lie algebras. It turns out the Lie algebras of these three groups, when considered as real Lie groups, all share the same complexification. In Cartan's classification of the simple Lie algebras, this algebra is denoted Cn. In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ...
Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...
In mathematics, a simple Lie group is a Lie group which is also a simple group. ...
Stated slightly differently, the complex Lie algebra Cn is just the algebra sp(2n, C) of the complex Lie group Sp(2n, C). This algebra has two different real forms: In mathematics, the Killing form, named for Wilhelm Killing (1847-1923), is a bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. ...
- the compact form, sp(n), which is the Lie algebra of Sp(n), and
- the normal form, sp(2n, R), which is the Lie algebra of Sp(2n, R).
Comparison of the symplectic groups | | matrices | Lie group | dim/R | dim/C | compact | π1 | | Sp(2n, R) | R | real | n(2n + 1) | – | no | Z | | Sp(2n, C) | C | complex | 2n(2n + 1) | n(2n + 1) | no | 1 | | Sp(n) | H | real | n(2n + 1) | – | yes | 1 | The term normal form is used in a variety of contexts. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
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