    in mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted  and sp(n). the latter is sometimes called the compact symplectic group to distinguish it from the former. 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. in cartan's classification of the simple lie algebras, the lie algebra of the complex group  is denoted cn, and sp(n) is the compact real form of . the name "symplectic group" is due to hermann weyl (details) as a replacement for the previous confusing names of (line) complex group and abelian group, and is the greek analog of "complex". ==sp(2n, f)== the symplectic group of degree 2n over a field f, denoted , 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 determinant 1, the symplectic group is a subgroup of the special linear group . 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). when , the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that . for , there are additional conditions, i.e.  is then a proper subgroup of . typically, the field f is the field of real numbers, r, or complex numbers, c. in this case  is a real/complex lie group of real/complex dimension . these groups are connected but noncompact.  is simply connected while  has a fundamental group isomorphic to z. the lie algebra of {{nowrap|sp(2n, f)}
Lie_groups Symplectic_geometry