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More specifically, if <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\dim (V)&gt;1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>dim<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then all the transposed Poisson algebra structures on <jats:inline-formula><jats:alternatives><jats:tex-math>$$W(A,V,\\langle \\cdot ,\\cdot \\rangle )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>W<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u27e8<\/mml:mo>\n                    <mml:mo>\u00b7<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u00b7<\/mml:mo>\n                    <mml:mo>\u27e9<\/mml:mo>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are trivial; and if <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\dim (V)=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>dim<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then such structures are, up to isomorphism, mutations of the group algebra structure on <jats:italic>FA<\/jats:italic>. The transposed Poisson algebra structures on <jats:italic>L<\/jats:italic>(<jats:italic>A<\/jats:italic>,\u00a0<jats:italic>g<\/jats:italic>,\u00a0<jats:italic>f<\/jats:italic>) are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of <jats:italic>L<\/jats:italic>(<jats:italic>A<\/jats:italic>,\u00a0<jats:italic>g<\/jats:italic>,\u00a0<jats:italic>f<\/jats:italic>) with values in the center of <jats:italic>L<\/jats:italic>(<jats:italic>A<\/jats:italic>,\u00a0<jats:italic>g<\/jats:italic>,\u00a0<jats:italic>f<\/jats:italic>). In particular, all of them are usual Poisson structures on <jats:italic>L<\/jats:italic>(<jats:italic>A<\/jats:italic>,\u00a0<jats:italic>g<\/jats:italic>,\u00a0<jats:italic>f<\/jats:italic>). This generalizes earlier results about transposed Poisson structures on Block Lie algebras <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {B}(q)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>B<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00025-023-01962-y","type":"journal-article","created":{"date-parts":[[2023,7,15]],"date-time":"2023-07-15T05:01:46Z","timestamp":1689397306000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Transposed Poisson Structures on Generalized Witt Algebras and Block Lie Algebras"],"prefix":"10.1007","volume":"78","author":[{"given":"Ivan","family":"Kaygorodov","sequence":"first","affiliation":[]},{"given":"Mykola","family":"Khrypchenko","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,15]]},"reference":[{"key":"1962_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.geomphys.2020.103939","volume":"160","author":"H Albuquerque","year":"2021","unstructured":"Albuquerque, H., Barreiro, E., Benayadi, S., Boucetta, M., S\u00e1nchez, J.M.: Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras. 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