Cyclic unitary involutions on central simple algebras

S. Tikhonov,   V. Yanchevskiĭ

Let \(k\) be a field and \(K/k\) be a quadratic separable field extension. An involution on a central simple \(K\)-algebra \(A\) is called a unitary involution if its restriction to \(K\) is nontrivial and a \(K/k\)-involution if this restriction is a nontrivial \(k\)-automorphism. The set of \(K/k\)-involutions of \(A\) will be denoted by \(Inv_{K/k}(A)\).

Definition. Let \(G\) be a finite cyclic group and \(Inv_{K/k}(A)\neq\emptyset\). A central simple \(K\)-algebra \(A\) is called a cyclic involutorial crossed product with the group \(G\) if there exists a strictly maximal subfield \(N\) of \(A\) such that \(N\) is a composite of \(\chi\) and \(K\), where \(\chi/k\) is a cyclic extension with the group \(G\), \(\chi\) and \(K\) are linear disjoint over \(k\), and there exists \(\tau \in Inv_{K/k}(A)\) such that \(\tau |_{\chi}\) is trivial. In this case, the corresponding involution \(\tau\) is called cyclic.

Cyclic involutions play an important role in investigating the structure of linear algebraic groups (especially, outer forms of groups of type \(A_n\)).

The problem of the existence of cyclic involutions on central simple algebras over global fields was solved affirmatively by Albert in [1]. He has also proved that, for quaternion algebras, all \(K/k\)-involutions are cyclic. Unfortunately, it is known that in general there are cyclic algebras such that all \(K/k\)-involutions are not cyclic. In [5,4], it is proved that all \(K/k\)-involutions on central simple \(K\)-algebras \(A\) are cyclic in the following cases:

(i) \(k\) is finite and the degree of \(A\) is odd;
(ii) \(k\) has the property that any quadratic \(k\)-form in 8 variables is isotropic, the degree of \(A\) is 3, and \(k\) contains a primitive 3th root of unity if \(char k \ne 3\);
(iii) \(k\) is global and \(A\) is a split \(K\)-algebra of odd degree;
(iv) \(k\) is local non-dyadic and \(A\) is of odd degree.

The aim of the talk is to present the following two results.

Theorem 1. Let \(k\) be a global field. Then all \(K/k\)-involutions on central simple \(K\)-algebras of odd degree are cyclic.

By using results and methods from [2,3], we prove the following

Theorem 2. Let \(k\) be the field of fractions of a two-dimensional excellent Henselian local domain with an algebraically closed residue field of characteristic zero. Then all \(K/k\)-involutions on central simple \(K\)-algebras of odd degree are cyclic.

References

1. A. A. Albert, Involutorial simple algebras and real Riemann matrices, Ann. of Math., 46, N4 (1935), 886–964.
2. J.-L. Colliot-Thélène, P. Gille, R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J., 121, N2 (2004), 285–341.
3. J.-L. Colliot-Thélène, M. Ojanguren, R. Parimala, Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes, Proceedings of the International Colloquium on Algebra, arithmetic, and geometry (Mumbai, January 4–12, 2000), Parts I and II, Bombay. Stud. Math., Tata Inst. Fundam. Res., 16 (2002), 185–217.
4. A. V. Prokopchuk, S. V. Tikhonov , V. I. Yanchevskiĭ, On cyclic unitary involutions of split simple central algebras of odd degree, Vesti NAN Belarusi, N1 (2013), 36–40. (Russian)
5. V. I. Yanchevskiĭ, Cyclic unitary involutions of simple central algebras, Dokl. NAN Belarusi, 2, N2 (2013), 32–37. (Russian)

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