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.^[1][2]

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)^[3] 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

$q=\left\langle {-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle \ .$

It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B.^[4] The quaternion algebras are linked if and only if the Albert form is isotropic, otherwise unlinked.^[5]

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.^[6][7]

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 M₂(F)⊗D where D is a quaternion division algebra over F.^[2] 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.^[8][9]

Characterisation

A theorem of Albert states that every central simple algebra of degree 4 and exponent 2 is a biquaternion algebra.^[8][10]

References

1. Lam (2005) p.60
2. Szymiczek (1997) p.452
3. Cohn, Paul M. (2003). Further Algebra and Applications. Springer-Verlag. p. 208. ISBN 1852336676.
4. Knus et al (1991) p.192
5. Lam (2005) p.70
6. Albert, A.A. (1972). "Tensor products of quaternion algebras". Proc. Am. Math. Soc. 35: 65–66. doi:10.1090/s0002-9939-1972-0297803-6. Zbl 0263.16012.
7. Jacobson (1996) p.77
8. Lam (2005) p.437
9. Knus et al (1991) p.236
10. Knus et al (1991) p.233

• Albert, A.Adrian (1932). "Normal division algebras of degree four over an algebraic field". Trans. Am. Math. Soc. 34: 363–372. doi:10.2307/1989546. Zbl 0004.10002.
• Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874.16002.
• Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The book of involutions. Colloquium Publications. 44. With a preface by J. Tits. Providence, RI: American Mathematical Society. ISBN 0-8218-0904-0. Zbl 0955.16001.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Szymiczek, Kazimierz (1997). Bilinear algebra. An introduction to the algebraic theory of quadratic forms. Algebra, Logic and Applications. 7. Langhorne, PA: Gordon and Breach Science Publishers. ISBN 9056990764. Zbl 0890.11011.