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We show how to count their number and study their dispersion relations. Their number is less than the number of broken generators when some of them form canonically conjugate pairs. The pairing occurs when the generators have a nonzero expectation value of their commutator. For non-semi-simple algebras, central extensions are possible. The underlying geometry of the coset space in general is partially symplectic."}]},{"notation":[{"@value":"5 pages, REVTeX 4.1; v3: published version"}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1380284921030026113","@type":"Researcher","foaf:name":[{"@value":"Haruki Watanabe"}]},{"@id":"https://cir.nii.ac.jp/crid/1420564276160319872","@type":"Researcher","personIdentifier":[{"@type":"KAKEN_RESEARCHERS","@value":"20222341"},{"@type":"NRID","@value":"1000020222341"},{"@type":"CINII_AUTHOR_ID","@value":"DA17002634"},{"@type":"URI","@value":"https://ci.nii.ac.jp/author/DA17002634#entity"},{"@type":"URI","@value":"https://viaf.org/viaf/NII%7CDA17002634"}],"foaf:name":[{"@value":"Hitoshi Murayama"}]}],"publication":{"publicationIdentifier":[{"@type":"PISSN","@value":"00319007"},{"@type":"EISSN","@value":"10797114"},{"@type":"PISSN","@value":"http://id.crossref.org/issn/10797114"}],"prism:publicationName":[{"@value":"Physical Review Letters"}],"dc:publisher":[{"@value":"American 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