On the existence of orthogonal decompositions of the simple Lie algebra of type \(C_3\)
The paper deals with a simple complex Lie algebra \(C_{3}.\) The possibility to decompose \(C_{3}\) into a direct sum of Cartan subalgebras which are orthogonal to each other with respect to Killing form is explored. In the case of \(C_{3}\) it is necessary to find 7 orthogonal to each other sets of matrices each containing 3 commuting, linearly independent, diagonalizable matrices. The author concentrates on the existence of monomial orthogonal decompositions. An orthogonal decomposition is called monomial if there exists a basis consisting of monomial matrices for each Cartan subalgebra that occurs. A nonsingular matrix is called monomial if it can be written as a product of a diagonal matrix and a permutation matrix. The author proves, using computational methods (Maple), that \(C_{3}\) has no monomial orthogonal decomposition. At the end of the paper Maple scripts are presented.
scientific article; zbMATH DE number 1558145