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Else from the quotient algebra partition considered in the preceding episodes, two kinds of partitions on unitary Lie algebras are created by nonabelian bi-subalgebras. It is of interest that there exists a partition duality between the two kinds of partitions. With an application of an appropriate coset rule, the two partitions return to a quotient algebra partition when the generating bi-subalgebra is abelian. Procedures are proposed to merge or detach a co-quotient algebra, which help deliver type-AIII Cartan decompositions of more varieties. In addition, every Cartan decomposition is obtainable from the quotient algebra partition of the highest rank. Of significance is the universality of the quotient algebra partition to classical and exceptional Lie algebras.
An algebraic structure, Quotient Algebra Partition or QAP, is introduced in a serial of articles. The structure QAP is universal to Lie Algebras and enables algorithmic and exhaustive Cartan decompositions. The first episode draws the simplest form o
In the 3rd episode of the serial exposition, quotient algebra partitions of rank zero earlier introduced undergo further partitions generated by bi-subalgebras of higher ranks. The refin
This is the sequel exposition following [1]. The framework quotient algebra partition is rephrased in the language of the s-representation. Thanks to this language, a quotient algebra partition of the simplest form is established under a minimum numb
A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cart
In a previous paper [{it J. Phys. A: Math. Theor.} {bf 40} (2007) 11105], we constructed a class of coherent states for a polynomially deformed $su(2)$ algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed