In mathematics, a biquaternion algebra is a compound of

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 ...

s over a field. The

biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...

s of

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

(1844) and the related

split-biquaternion In mathematics, a split-biquaternion is a hypercomplex number of the form :q = w + xi + yj + zk where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient ''w'', ''x' ...

s and

dual quaternion In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of ...

s do not form biquaternion algebras in this sense.

Definition

Let ''F'' be a field of characteristic not equal to 2. A ''biquaternion algebra'' over ''F'' is a

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

of two

s.Lam (2005) p.60Szymiczek (1997) p.452 A biquaternion algebra is a

central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...

of dimension 16 and

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

4 over the base field: it has exponent (order of its

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 ...

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 ''F'') equal to 1 or 2.

Albert's theorem

Let ''A'' = (''a''₁,''a''₂) and ''B'' = (''b''₁,''b''₂) be quaternion algebras over ''F''. The Albert form for ''A'', ''B'' is :

q = \left\langle\right\rangle \ .

It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''.Knus et al (1991) p.192 The quaternion algebras are linked if and only if the Albert form is

isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

, otherwise unlinked.Lam (2005) p.70

Albert Albert may refer to: Companies * Albert (supermarket), a supermarket chain in the Czech Republic * Albert Heijn, a supermarket chain in the Netherlands * Albert Market, a street market in The Gambia * Albert Productions, a record label * Albert ...

's theorem states that the following are equivalent: * ''A''⊗''B'' is a

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

; * The Albert form is

anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...

; * ''A'', ''B'' are division algebras and they do not have a common quadratic splitting field.Jacobson (1996) p.77 In the case of linked algebras we can further classify the other possible structures for the tensor product in terms of the Albert form. If the form is

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

, then the biquaternion algebra is isomorphic to the algebra M₄(''F'') of 4×4 matrices over ''F'': otherwise, it is isomorphic to the product M₂(''F'')⊗''D'' where ''D'' is a quaternion division algebra over ''F''.Szymiczek (1997) p.452 The

Schur index In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...

of a biquaternion algebra is 4, 2 or 1 according as the

Witt index :''"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.'' In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isom ...

of the Albert form is 0, 1 or 3.Lam (2005) p.437Knus et al (1991) p.236

Characterisation

A theorem of Albert states that every central simple algebra of degree 4 and exponent 2 is a biquaternion algebra.Knus et al (1991) p.233

References

* * * * * {{cite book , last=Szymiczek , first=Kazimierz , title=Bilinear algebra. An introduction to the algebraic theory of quadratic forms , zbl=0890.11011 , series=Algebra, Logic and Applications , volume=7 , location=Langhorne, PA , publisher=Gordon and Breach Science Publishers , year=1997 , isbn=9056990764 Quaternions