1. CJM 2009 (vol 62 pp. 52)
|An Algebraic Approach to Weakly Symmetric Finsler Spaces |
In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions $2$ and $3$. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing S-curvature may exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature
Categories:53C60, 58B20, 22E46, 22E60
2. CJM 2006 (vol 58 pp. 1291)
|The General Structure of $G$-Graded Contractions of Lie Algebras I. The Classification |
We give the general structure of complex (resp., real) $G$-graded contractions of Lie algebras where $G$ is an arbitrary finite Abelian group. For this purpose, we introduce a number of concepts, such as pseudobasis, higher-order identities, and sign invariants. We characterize the equivalence classes of $G$-graded contractions by showing that our set of invariants (support, higher-order identities, and sign invariants) is complete, which yields a classification.
Keywords:Lie algebras, graded contractions
3. CJM 1998 (vol 50 pp. 210)
|Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$ |
In this paper, we determine when two simple generalized Cartan type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type $W$ Lie algebras.
Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjecture
Categories:17B40, 17B65, 17B56, 17B68
4. CJM 1997 (vol 49 pp. 133)
|Exterior powers of the adjoint representation |
Exterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.
Keywords:Lie algebras, adjoint representation, exterior algebra
Categories:20G05, 20C30, 22E10, 22E60