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 ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...

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 (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian group ...

exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

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 and x_0 is an element of X called the base point, also spelled basepoint. Maps between pointed sets (X, x_0) and (Y, y_0) – called based ma ...

with distinguished element , and is a symmetric

surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

satisfying axioms :

\begin\text \quad &q(a,(-1)a) = 1,\\
\text \quad &q(a,b) = q(a,c) \Leftrightarrow q(a,bc) = 1,\\
\text \quad &q(a,b) = q(c,d) \Rightarrow \exists x\mid q(a,b) = q(a,x), q(c,d) = q(c,x)\end.

Every field gives rise to a -structure by taking to be , the set of

Brauer class Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Br ...

es of quaternion algebras in the

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

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 To ...

, 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