The Book of Involutions

Classification of involutions on graded-division simple real algebras

“…It is well known (see [12,Proposition 4.1]) that, given one such h, there exists a unique map σ h : End D (V ) → End D (V ) that satisfies the equation…”

Explicit Construction of Ramanujan Bigraphs

“…By a theorem of Wedderburn [KMRT98,Theorem 19.2], any central simple algebra of degree three is cyclic. From now on we will assume D is in the form given above.…”

“…Let D be a central simple algebra of degree three over E and α be an involution of the second kind, i.e., an anti-automorphism of D that acts on E by conjugation with respect to E/F . By Wedderburn's theorem [KMRT98,Theorem 19.2], D is a cyclic algebra over E. Let N D denote the reduced norm of D. Then, (D, α) defines a special unitary group G by…”

Normal enveloping algebras

“…Over the decades, normal algebras with involutions have been extensively investigated on their own; see, for example, [Beidar et al 1981;Bovdi et al 1985;Bovdi 1990;1997;Bovdi and Siciliano 2007;Brešar and Vukman 1989;Herstein 1976;Knus et al 1998;Lim 1977;1979;Maxwell 1972]. Moreover, they have several applications in linear algebra and functional analysis; see, for example, [Berberian 1959;Fuglede 1950;Maxwell 1972;Mosić and Djordjević 2009;Putnam 1951;Yood 1974].…”