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Register a new userQuotient Algebra Partition and Cartan Decomposition for su(N) II
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Zheng-Yao Su
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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 number of conditions governed by a bi-subalgebra of rank zero, i.e., a Cartan subalgebra. Within the framework, all Cartan subalgebras of su(N) are classified and generated recursively through the process of the subalgebra extension.
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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 of the structure in terms of the spinor representation.
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.
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
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 Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.
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 $su(1,1)$ algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1,1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU(1,1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.
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