Quaternionic vector space: Difference between revisions

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{{Short description|Module over the algebra of quaternions.}}

In [[mathematics]], a '''left''' (or '''right''') '''quaternionic vector space''' is a left (or right) <math>\mathbb H</math>-[[module (mathematics)|module]] where <math>\mathbb H</math> is the [[division ring]] of [[quaternion]]s. One must distinguish between left and right quaternionic vector spaces since <math>\mathbb H</math> is non-commutative. Further, <math>\mathbb H</math> is not a field, so quaternionic vector spaces are not [[vector space]]s, but merely modules.

In [[noncommutative algebra]], a branch of [[mathematics]], a '''quaternionic vector space''' is a [[Module (mathematics)|module]] over the [[Quaternion|quaternions]]. Since the quaternion algebra is [[division ring]], these modules are referred to as "vector spaces". However, the quaternion algebra is [[Noncommutative ring|noncommutative]] so we must distinguish left and right vector spaces. In left vector spaces, linear compositions of vectors <math> v</math> and <math> w</math> have the form <math> av+bw</math> where <math> a</math>, <math> b\in H</math>. In right vector spaces, linear compositions of vectors <math> v</math> and <math> w</math> have the form <math> va+wb</math>.

TheSimilar spaceto <math>\mathbb[[Vector H^n</math>space|vector isspaces bothover a leftfield]], andif righta quaternionic vector space usinghas componentwisefinite multiplication.dimension Namely, for<math> n</math>q, \inthen \mathbbit is isomorphic to the direct sum <math> H^n</math> andof <math>(r_1, \ldots,n</math> r_n)copies \inof \mathbbthe quaternion algebra <math> H^n</math>,. In this case we can use a standard basis which has the form

:<math> q e_1=(r_11, \ldots0, r_n) = (q r_1, \ldots, q r_n0),</math>

:<math> (r_1, \ldots, r_n) q = (r_1 q, \ldots, r_n q).</math>

:<math>e_n=(0,\ldots,0,1)</math>

In a left quaternionic vector space <math> H^n</math> we use componentwise sum of vectors and product of vectors over scalars

:<math> (p_1, \ldots, p_n)+(r_1, \ldots, r_n) = (p_1+ r_1, \ldots, p_n+ r_n)</math>

:<math> q (r_1, \ldots, r_n) = (q r_1, \ldots, q r_n)</math>

In a right quaternionic vector space <math> H^n</math> we also use componentwise sum of vectors and product of vectors over scalars

:<math> (p_1, \ldots, p_n)+(r_1, \ldots, r_n) = (p_1+ r_1, \ldots, p_n+ r_n)</math>

:<math> (r_1, \ldots, r_n)q = ( r_1q, \ldots, r_nq)</math>

Since <math>\mathbb H</math> is a [[division algebra]], every [[Finitely generated module|finitely generated]] (left or right) <math>\mathbb H</math>-module has a [[basis (linear algebra)|basis]], and hence is [[isomorphic]] to <math>\mathbb H^n</math> for some <math>n</math>.

==See also==