Algebras with involution that become hyperbolic under a given extension
Let \(K\) be a field of characteristic not 2. For a central simple algebra with involution, \((A,\sigma)\), the authors allow besides the usual sort also algebras of the form \(B\times B^o\), where \(B\) is a central simple algebra over \(K\) and \(\sigma\) is the switch involution: \(\sigma(a,b^o)=(b,a^o)\). The subfield of the centre of \(A\) fixed by \(\sigma\) is written \(F\). If \(A\) contains \(a\) such that \(\sigma(a)a=0\), \(A\) is called isotropic and \(A\) is hyperbolic if \(A\) contains a right ideal \(I\) such that \(I=I^\perp\), where \(I^\perp=\{y\in A\mid\sigma(y)I=0\}\). E.g. a product algebra with switch involution is hyperbolic. Let \(\pi(t)\) be a monic separable polynomial over \(F\) of even degree \(2n\). Then the ring \(F(\pi)=F[t]/(\pi(t))\) is a direct product of field extensions of \(F\) and \((A\otimes F(\pi),\sigma\otimes 1)\) is a direct product of algebras. If all these algebras are hyperbolic, \((A,\sigma)\) is said to become hyperbolic over \(F(\pi)\). \textit{J.-P. Tignol} [in J. Algebra 181, No. 3, 857-875 (1996; Zbl 0847.16019)]\ gave a hyperbolicity criterion by proving that an anisotropic algebra \((A,\sigma)\) becomes hyperbolic over \(F(\pi)\) if and only if there is a homomorphism \((H_\pi,\sigma_\pi)\to(A,\sigma)\) of \(F\)-algebras with involution, where \(H_\pi\) is the universal algebra over which \(\pi\) splits into two factors of degree \(n\) and \(\sigma_n\) interchanges these factors. The authors' aim is to describe the structure of \(H_\pi\) and reformulate the hyperbolicity criterion. Consider an algebra \((A,\sigma)\) where \(A\) is the endomorphism ring of an anisotropic quadratic space \((V,q)\) over \(F\) and \(\sigma\) is the adjoint map associated to \(q\). The algebra \(A\) becomes hyperbolic over \(F(\pi)\) precisely if \((V,q)\) becomes hyperbolic over \(F(\pi)\); the criterion stated earlier (in terms of \(H_\pi\)) can now be formulated as follows. Theorem. If an anisotropic quadratic space \((V,q)\) becomes hyperbolic over \(F(\pi)\), then there exist finite-dimensional anisotropic simple homomorphic images \((B_i,\sigma_i)\) of \((H_\pi,\sigma_\pi)\), Hermitian spaces \((V_i,h_i)\) over \((B_i,\sigma_i)\) (\(i=1,\dots,r\)) and \(F\)-linear maps \(s_i\colon B_i\to F\) such that \(s_i(bb')=s_i(b'b)\), \(s_i(\alpha_i(b))=s_i(b)\) for all \(b,b'\in B\) and \(q=s_{1^*}(h_1)\perp\cdots\perp s_{r^*}(b_r)\). Conversely every quadratic form of this type becomes hyperbolic over \(F(\pi)\). This result can be used to describe the Witt kernel as the subgroup of \(WF\) generated by the images of the transfer maps \(s_{i^*}\). The structure of \((H_\pi,\sigma_\pi)\) is also described more closely as quotient of a skew polynomial ring over a free algebra. In the particular case \(n=2\), \(B_i\) is identified as a Clifford algebra and this leads to the known splitting criteria for quaternion algebras.
central simple algebras with involutions
splitting criteria for quaternion algebras
scientific article; zbMATH DE number 1115984