Quaternionic vector space
Encyclopedia
In mathematics
, a left (or right) quaternionic vector space is a left (or right) H-module
where H denotes the noncommutative ring
of the quaternion
s.
The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:
for quaternions q and q1, q2, ... qn.
Since H is a division algebra
, every finitely generated
(left or right) H-module has a basis
, and hence is isomorphic to Hn for some n.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a left (or right) quaternionic vector space is a left (or right) H-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
where H denotes the noncommutative ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s.
The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:
for quaternions q and q1, q2, ... qn.
Since H is a division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
, every finitely generated
Finitely generated
In mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...
(left or right) H-module has a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
, and hence is isomorphic to Hn for some n.
See also
- Vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
- General linear groupGeneral linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
- Special linear groupSpecial linear groupIn mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
- SL(n,H)
- Symplectic groupSymplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...