Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces
Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces
复制标题
DOI：
10.1016/j.jalgebra.2004.03.016
复制
发表时间：
2001-12
影响因子：
0.9
通讯作者：
S. E. Rao
中科院分区：
文献类型：
--
作者：
S. E. Rao
我们继续研究由Benkart和Osborn在20世纪80年代初开始的8维实数除法代数，其推导代数很大（类型为G2， A2, 2A1或A1）。对于他们所构造的一些实代数族，我们找到了它们是除法代数的充分必要条件，确定了该类代数族中的两个除法代数同构的条件，或确定了该类代数的自同构群。我们用这些族中的一个证明了每一个二维实数除法代数都嵌入到一个四维和一个八维实数除法代数中。构造了由欧几里得空间R6参数化的非同构实8维代数族F6，并对其进行了详细研究。f6中的除法代数对应于参数空间的非空开子集。我们还引入了一个有趣的2参数子族F2∧F6，它包含广义伪八元代数。得到了(A,μ)∈f2是除法代数的充分必要条件。在一般情况下，f6中的代数是1生成的，并且有SO(3)作为自同构群。我们还确定了F6中除法代数的所有（1-、2-和4维）子代数。我们证明存在8维（和4维）实数除法代数，其1维子代数不包含在任何2维子代数中。我们还构造了具有不包含在任何四维子代数中的二维子代数的八维实数除法代数。
We continue the study of the 8-dimensional real division algebras whose derivation algebra is large (of type G2, A2, 2A1, or A1) begun by Benkart and Osborn in the early 1980s. For some of the families of real algebras that they constructed, we find necessary and sufficient conditions for them to be division algebras, determine when two division algebras in such a family are isomorphic or determine the automorphism group of such an algebra. We use one of these families to prove that every 2-dimensional real division algebra embeds in a 4-dimensional and in an 8-dimensional real division algebra. A new family, F6, of non-isomorphic real 8-dimensional algebras, parametrized by the Euclidean space R6, is constructed and studied in detail. The division algebras in F6correspond to a non-empty open subset of the parameter space. We also introduce an interesting 2-parameter subfamily F2⊂ F6, which contains the generalized pseudo-octonion algebras. We obtain necessary and sufficient conditions for (A,μ)∈ F2to be a division algebra. In the generic case, the algebras in F6are 1-generated and have SO(3) as the automorphism group. We also determine all (1-, 2-, and 4-dimensional) subalgebras of the division algebras in F6. We show that there exist 8-dimensional (and 4-dimensional) real division algebras having a 1-dimensional subalgebra not contained in any 2-dimensional subalgebra. We also construct 8-dimensional real division algebras having a 2-dimensional subalgebra not contained in any 4-dimensional subalgebra.