{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:57Z","timestamp":1740134517266,"version":"3.37.3"},"reference-count":11,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["10871056","10971150","HITC200708"],"award-info":[{"award-number":["10871056","10971150","HITC200708"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["10871056","10971150","HITC200708"],"award-info":[{"award-number":["10871056","10971150","HITC200708"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science Research Foundation in Harbin Institute of Technology","award":["10871056","10971150","HITC200708"],"award-info":[{"award-number":["10871056","10971150","HITC200708"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>l<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>be the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mn>2<\/mml:mn><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>l<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u00d7<\/mml:mo><mml:mn>2<\/mml:mn><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>l<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>matrix symplectic Lie algebra over a commutative ring<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>with 2 invertible. Then<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>l<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:msubsup><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>\u2009\u2009=\u2009\u2009{<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mtable class=\"smallmatrix\"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mo>-<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mo>-<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mtd><\/mml:mtr><mml:mtr><mml:mtd><mml:mn>0<\/mml:mn><\/mml:mtd><mml:mtd><mml:mo>-<\/mml:mo><mml:msubsup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mo>-<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mtd><\/mml:mtr><\/mml:mtable><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math>\u2009\u2223\u2009<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mo>\u0305<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>is an<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>l<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>upper triangular matrix,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:msubsup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mo>\u0305<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><mml:mo>\u0305<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mi>\u2009\u2009<\/mml:mi><mml:mtext>over<\/mml:mtext><mml:mi>\u2009\u2009<\/mml:mi><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math>is the solvable subalgebra of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>l<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. 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