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Volumn 79, Issue 5, 2009, Pages

Uncontrollable quantum systems: A classification scheme based on Lie subalgebras

Author keywords

[No Author keywords available]

Indexed keywords

CLASSIFICATION SCHEME; FINITE-LEVEL QUANTUM SYSTEMS; LIE ALGEBRA; PHYSICAL STRUCTURES; QUANTUM SYSTEM; SUBALGEBRA; SUBALGEBRAS;

EID: 65649137014     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.79.053403     Document Type: Article
Times cited : (40)

References (27)
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    • Every subalgebra of su (N) is reductive. The same holds true for other compact Lie groups as well. The easiest way to prove this is to use the following result: if a Lie algebra has a representation V in which its trace form (defined by (a,b) = trV (ab) is nondegenerate, then the Lie algebra is reductive. This is proved, e.g., in Sec. 5.7 of. But since the trace form in the fundamental representation of su (N) is negative definite, its restriction to any subalgebra is also negative definite.
    • Every subalgebra of su (N) is reductive. The same holds true for other compact Lie groups as well. The easiest way to prove this is to use the following result: if a Lie algebra has a representation V in which its trace form (defined by (a,b) = trV (ab) is nondegenerate, then the Lie algebra is reductive. This is proved, e.g., in Sec. 5.7 of. But since the trace form in the fundamental representation of su (N) is negative definite, its restriction to any subalgebra is also negative definite.
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    • This is known as a Levi decomposition (see, e.g.,).
    • This is known as a Levi decomposition (see, e.g.,).
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    • Because the Hilbert space carries a representation of g= /k gk it carries a representation of G= /k Hk through exponentiation and therefore H= /k Hk, where Hk is a representation for Gk and gk (see, e.g.,).
    • Because the Hilbert space carries a representation of g= /k gk it carries a representation of G= /k Hk through exponentiation and therefore H= /k Hk, where Hk is a representation for Gk and gk (see, e.g.,).


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.