an involution or anti-involution is a self-inverse linear mapping. in this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. moreover, properties and geometrical meanings of these matrices will be given as reflections in r^3.

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

We investigate various aspects of involutions of groups, i.e, anti-automorphisms of order at most two. The emphasis is on finite abelian groups. We count the number of involutions for the cyclic groups, and consider the problem for direct products of groups. We also give a characterization for the set of skewed squares of finitely generated abelian groups with identity as the involution. The pr...

We investigate various aspects of involutions of groups, i.e, anti-automorphisms of order at most two. The emphasis is on finite abelian groups. We count the number of involutions for the cyclic groups, and consider the problem for direct products of groups. We also give a characterization for the set of skewed squares of finitely generated abelian groups with identity as the involution. The pr...

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with invo...

We classify anti-involutions of Lie superalgebra ŜD preserving the principal gradation, where ŜD is the central extension of the Lie superalgebra of differential operators on the super circle S. We clarify the relations between the corresponding subalgebras fixed by these anti-involutions and subalgebras of ĝl∞|∞ of types OSP and P . We obtain a criterion for an irreducible highest weight modul...

We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group. Also we present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of...

It is shown that the quotients of a complex surface under free holomorphic and anti-holomorphic involutions are homeomorphic but not diffeomorphic. This gives a way to construct exotic smooth structures on some complex surfaces.