The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the p...The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the present paper, we obtain a matrix representation for each of these Lie algebras. We are able to find such representations by exploiting properties of the radical, principally, when it has a trivial center, in which case we can obtain such a representation by restricting the adjoint representation. Another important subclass of algebras is where the radical has a codimension one abelian nilradical and for which a representation can readily be found. In general, finding matrix representations for abstract Lie algebras is difficult and there is no algorithmic process, nor is it at all easy to program by computer, even for algebras of low dimension. The present paper represents another step in our efforts to find linear representations for all the low dimensional abstract Lie algebras.展开更多
The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be ...The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be easily incorporated into a first course in abstract algebra and without appealing to the Sylow Theorems. The paper concludes with some exercises for students.展开更多
文摘The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the present paper, we obtain a matrix representation for each of these Lie algebras. We are able to find such representations by exploiting properties of the radical, principally, when it has a trivial center, in which case we can obtain such a representation by restricting the adjoint representation. Another important subclass of algebras is where the radical has a codimension one abelian nilradical and for which a representation can readily be found. In general, finding matrix representations for abstract Lie algebras is difficult and there is no algorithmic process, nor is it at all easy to program by computer, even for algebras of low dimension. The present paper represents another step in our efforts to find linear representations for all the low dimensional abstract Lie algebras.
文摘The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be easily incorporated into a first course in abstract algebra and without appealing to the Sylow Theorems. The paper concludes with some exercises for students.
