Let gl,,（R） be the general linear Lie algebra of all n×n matrices over a unital commutative ring R with 2 invertible, dn（R） be the Cartan subalgebra of gln（R） of all diagonal matrices. The maximal subalgebr...Let gl,,（R） be the general linear Lie algebra of all n×n matrices over a unital commutative ring R with 2 invertible, dn（R） be the Cartan subalgebra of gln（R） of all diagonal matrices. The maximal subalgebras of gln（R） that contain dn（F：） are classified completely.展开更多
For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all...For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.展开更多
The decomposition of matrices corresponding to the 2-qutrit logic gate by succes-sive Cartan decomposition is investigated, and written in an exponential form based on the relationship between Lie group and Lie algebr...The decomposition of matrices corresponding to the 2-qutrit logic gate by succes-sive Cartan decomposition is investigated, and written in an exponential form based on the relationship between Lie group and Lie algebra, thus making them able to relate with the control field and the Hamiltonian of the system to perform the gate. Finally the decomposition of the ternary SWAP gate is presented in detail.展开更多
基金supported by National Natural Science Foundation of China (Grant No.11171343)the Fundamental Research Funds for the Central Universities (Grant No. 2010LKSX05)
文摘Let gl,,（R） be the general linear Lie algebra of all n×n matrices over a unital commutative ring R with 2 invertible, dn（R） be the Cartan subalgebra of gln（R） of all diagonal matrices. The maximal subalgebras of gln（R） that contain dn（F：） are classified completely.
文摘For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.
基金the National Natural Science Foundation of China (Grant No. 60433050)the Key Project of the Science Foundation of Xuzhou Normal University, China (Grant No. 06XLA05)
文摘The decomposition of matrices corresponding to the 2-qutrit logic gate by succes-sive Cartan decomposition is investigated, and written in an exponential form based on the relationship between Lie group and Lie algebra, thus making them able to relate with the control field and the Hamiltonian of the system to perform the gate. Finally the decomposition of the ternary SWAP gate is presented in detail.
