Maps On Matrix Algebras Preserving K-Potent Relation Over Fields

Matrix algebra is a branch of algebra. It has wide applications in other areas, for example, computer, graph theory, economics, control, etc. Preserver Problem is an interesting research area of matrix algebra, and problem of preserving idempotence is one of important research subject. The results of problem of preserving idempotence are used to resolve some physics problems, characterize general preserver problems on matrix semi-group and Lie algebra. Therefore,in recent years, the study on this subject is active.In this thesis, we first introduce the background and developing state of Preserver Problems, then study the problem of preserving k -potence on matrix algebras ( k > 1is an integer). The main works in this thesis are as follows:1. We give the definition of the map which preserving k -potence, then characterize the maps preserving k -potence on full matrix algebras by method of induction. As k is any integer greater than 1, the arbitrarity of k makes the characterizing more complicative than characterizing the maps preserving idempotence before. Of course, our results make the relevant results of preserving idempotence before as a special case of this thesis. The consequence on the maps which preservies k -potence is that the maps have the standard forms.2. Let n≥3, we solve the problem of characterizing the maps preserving k -potence from upper-triangular matrix algebras to full matrix algebras. The characteristic elements which will be used to characterize the maps preserving k -potence on upper-triangular matrix algebras are less than full matrix algebras, this makes the work more difficult. The method we used is different from the full matrix algebras. We first characterize the image of the matrix which is a direct sum of 3×3 upper-triangular matrix and O_n-3, then characterize the image of the matrix which is direct sum of 2×2 upper-triangular matrix and O_n-2, futhermore we describe the images of all n×n upper-triangular matrices. It is proved that the maps have the standard forms.3. We characterize maps preserving tripotence from symmetric matrix space to full matrix space. As the special nature of symmetric matrix, the coefficients in the (i , j )-position are same to that in the ( j , i )-position, there exist some relations on the images of the elements located in the symmetric position. Hence, investigating the relations between the images of these elements is necessary for describing the forms of this kind of maps. Our works show that the maps preserving tripotence from symmetric matrix space to full matrix space are standard.