{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T11:50:52Z","timestamp":1775821852150,"version":"3.50.1"},"reference-count":42,"publisher":"American Mathematical Society (AMS)","issue":"5","license":[{"start":{"date-parts":[[2009,12,16]],"date-time":"2009-12-16T00:00:00Z","timestamp":1260921600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"

\n The\n \n \n \n \n \n S<\/mml:mi>\n L<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n \n C<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {SL}(3,\\mathbb {C})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -representation variety\n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak {R}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of a free group\n \n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n r<\/mml:mi>\n <\/mml:msub>\n \\mathtt {F}_r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n arises naturally by considering surface group representations for a surface with boundary. There is an\n \n \n \n \n \n S<\/mml:mi>\n L<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n \n C<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {SL}(3,\\mathbb {C})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -action on the coordinate ring of\n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak {R}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by conjugation. The geometric points of the subring of invariants of this action is an affine variety\n \n \n \n \n X<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak {X}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The points of\n \n \n \n \n X<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak {X}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n parametrize isomorphism classes of completely reducible representations. We show the coordinate ring\n \n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n \n X<\/mml:mi>\n <\/mml:mrow>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\mathbb {C}[\\mathfrak {X}]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a complex Poisson algebra with respect to a presentation of\n \n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n r<\/mml:mi>\n <\/mml:msub>\n \\mathtt {F}_r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n imposed by the surface. Lastly, we work out the bracket on all generators when the surface is a three-holed sphere or a one-holed torus.\n <\/p>","DOI":"10.1090\/s0002-9947-08-04777-6","type":"journal-article","created":{"date-parts":[[2009,1,23]],"date-time":"2009-01-23T11:05:40Z","timestamp":1232708740000},"page":"2397-2429","source":"Crossref","is-referenced-by-count":18,"special_numbering":"888","title":["Poisson geometry of SL(3,\u2102)-character varieties relative to a surface with boundary"],"prefix":"10.1090","volume":"361","author":[{"given":"Sean","family":"Lawton","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,12,16]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1080\/00927878908823740","article-title":"On a minimal set of generators for the invariants of 3\u00d73 matrices","volume":"17","author":"Abeasis, Silvana","year":"1989","journal-title":"Comm. 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