Hyperbolicity of algebras with involution over a given extension.
Problem: given a finite extension \(K\supseteq F\), determine which anisotropic central simple algebras with involution over \(F\) become hyperbolic over \(K\). (Recall that \((A,\sigma)\) is isotropic if \(\sigma(a)a=0\) for some \(a\neq 0\), and hyperbolic if for some right ideal \(I\), \(I^\perp=\{x\mid\sigma(x)I=0\}=I\).) This problem was studied by \textit{J.-P. Tignol} [J. Algebra 181, No. 3, 857-875 (1996; Zbl 0847.16019)], who constructed for a field \(F\) (characteristic not \(2\)) and a separable polynomial \(\pi\in F[t]\) of degree \(2n\), an algebra with involution \(H_\pi\), whose simple quotients are the only anisotropic central simple algebras over \(F\) that become hyperbolic over \(F(\pi)=F[t]/F[t]\pi\). In fact, \(H_\pi\) is the free algebra \(F\langle a_1,\dots,a_n,b_1,\dots,b_n\rangle\) with the relations following from asserting the decomposition \(\pi(t)=(t^n+a_1t^{n-1}+\cdots+a_n)(t^n+b_1t^{n-1}+\cdots+b_n)\). \textit{D. E. Haile} and \textit{J.-P. Tignol} [J. Algebra 199, No. 1, 94-115 (1998; Zbl 0897.16018)] showed that when \(n>2\), \(H_\pi\) contains a noncommutative free subalgebra, and thus introduce a more manageable homomorphic image \(C_\pi\), which covers many examples of algebras that become hyperbolic over \(F(\pi)\). This quotient is described in the current paper as follows. Let \(q=\sum a_{ij}x_ix_j\) be the generic quadratic form of dimension \(n\) over \(F\), and \(f=\sum r_it^{n-i}\) the generic polynomial of degree \(n\). Let \(Z_\pi\) be quotient of \(F[a_{ij},r_i]\) defined by the relation \(f(t)-q(t^{n-1},\dots,1)=\pi(t)\). Then \(C_\pi\) is the even Clifford algebra of the form \(x_0^2-q(x_1,\dots,x_n)\) over \(Z_\pi\). It is shown that for every anisotropic quadratic form \(p\) over a finite extension \(K\) of \(F\), if \(p\) represents \(\pi(t)\) over \(K(t)\), then the even Clifford algebra \(C_0(p)\) is a quotient of \(C_\pi\) (as a ring with involution). Partial converse is given, but a complete description of the simple quotients of \(C_\pi\) remains unknown. Another object studied here is \(A_\pi\), the common Abelianization of \(H_\pi\) and \(C_\pi\). This algebra encodes the solution to the commutative analog of the problem stated above: \textit{D. E. Haile} and \textit{P. J. Morandi} [Commun. Algebra 29, No. 12, 5733-5753 (2001; Zbl 0996.16023)] showed that an anisotropic étale \(F\)-algebra \(E\) with involution \(\sigma\) becomes hyperbolic over \(F(\pi)\) if and only if \(\pi(t)=\sigma(g)g\) for some \(g\in K[t]\). Indeed, this is the case if and only if one of the factors of \(E\) is a quotient of \(A_\pi\). Let \(B_\pi\) be the polynomial ring \(F[\alpha_1,\dots,\alpha_{2n}]\) modulo the relations determined by \(\pi(t)=(t-\alpha_1)\cdots(t-\alpha_{2n})\), with the involution exchanging \(\alpha_i\) and \(\alpha_{n+i}\). The authors prove when \(\pi\) is irreducible, that \(A_\pi\) is the invariant subring of \(B_\pi\) under the natural action of \(S_n\times S_n\) on the generators. It follows that the dimension of \(A_\pi\) over \(F\) is the binomial coefficient \(\binom{2n}{n}\). Conditions for \(A_\pi\) to be isotropic or hyperbolic, in terms of the action of the Galois group of \(\pi\) on the roots, are also given.
scientific article; zbMATH DE number 5882427