Английская Википедия:Biquaternion algebra
In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field.
The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions 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 of two quaternion algebras.Шаблон:SfnШаблон:Sfn
A biquaternion algebra is a central simple algebra of dimension 16 and degree 4 over the base field: it has exponent (order of its Brauer class in the Brauer group of F)Шаблон:Sfn equal to 1 or 2.
Albert's theorem
Let A = (a1,a2) and B = (b1,b2) be quaternion algebras over F.
The Albert form for A, B is
- <math>q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ . </math>
It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B.Шаблон:Sfn The quaternion algebras are linked if and only if the Albert form is isotropic, otherwise unlinked.Шаблон:Sfn
Albert's theorem states that the following are equivalent:
- A ⊗ B is a division algebra;
- The Albert form is anisotropic;
- A, B are division algebras and they do not have a common quadratic splitting field.Шаблон:SfnШаблон:Sfn
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, then the biquaternion algebra is isomorphic to the algebra M4(F) of 4×4 matrices over F: otherwise, it is isomorphic to the product Шаблон:Nowrap where D is a quaternion division algebra over F.Шаблон:Sfn The Schur index of a biquaternion algebra is 4, 2 or 1 according as the Witt index of the Albert form is 0, 1 or 3.Шаблон:SfnШаблон:Sfn
Characterisation
A theorem of Albert states that every central simple algebra of degree 4 and exponent 2 is a biquaternion algebra.Шаблон:SfnШаблон:Sfn
Citations
References
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