In mathematics, a quaternionic structure or -structure is an axiomatic system that abstracts the concept of a
quaternion algebra
In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
A ''quaternionic structure'' is a triple where is an
elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in whic ...
of
exponent
In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
with a distinguished element , is a
pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a Set (mathematics), set and x_0 is an element of X called the base point (also spelled basepoint).
Map (mathematics), Maps between pointed sets ...
with distinguished element , and is a symmetric
surjection
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
satisfying axioms
:
Every field gives rise to a -structure by taking to be , the set of
Brauer class
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist Ri ...
es of quaternion algebras in the
Brauer group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
of with the
split quaternion algebra as distinguished element and the quaternion algebra .
References
* {{cite book , title=Introduction to Quadratic Forms over Fields , volume=67 , series=
Graduate Studies in Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats.
List of books
*1 ''The General T ...
, first=Tsit-Yuen , last=Lam , author-link=T. Y. Lam , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, year=2005 , isbn=0-8218-1095-2 , zbl=1068.11023 , mr = 2104929
Field (mathematics)
Quadratic forms
Quaternions