Poisson geometry of 𝑆𝐿(3,ℂ)-character varieties relative to a surface with boundary

Lawton, Sean

doi:10.1090/S0002-9947-08-04777-6

Poisson geometry of $\mathrm {SL}(3,\mathbb {C})$-character varieties relative to a surface with boundary
HTML articles powered by AMS MathViewer

by Sean Lawton PDF

Trans. Amer. Math. Soc. 361 (2009), 2397-2429 Request permission

Abstract:

The $\mathrm {SL}(3,\mathbb {C})$-representation variety $\mathfrak {R}$ of a free group $\mathtt {F}_r$ arises naturally by considering surface group representations for a surface with boundary. There is an $\mathrm {SL}(3,\mathbb {C})$-action on the coordinate ring of $\mathfrak {R}$ by conjugation. The geometric points of the subring of invariants of this action is an affine variety $\mathfrak {X}$. The points of $\mathfrak {X}$ parametrize isomorphism classes of completely reducible representations. We show the coordinate ring $\mathbb {C}[\mathfrak {X}]$ is a complex Poisson algebra with respect to a presentation of $\mathtt {F}_r$ 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.

References

Silvana Abeasis and Marilena Pittaluga, On a minimal set of generators for the invariants of $3\times 3$ matrices, Comm. Algebra 17 (1989), no. 2, 487–499. MR 978487, DOI 10.1080/00927878908823740
H. Aslaksen, V. Drensky, and L. Sadikova, Defining relations of invariants of two $3\times 3$ matrices, C. R. Acad. Bulgare Sci. 58 (2005), no. 6, 617–622. MR 2155550
M. Artin, On Azumaya algebras and finite dimensional representations of rings, J. Algebra 11 (1969), 532–563. MR 242890, DOI 10.1016/0021-8693(69)90091-X
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. MR 1224675, DOI 10.1007/978-1-4757-6848-0
David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
Chas, M., and Sullivan, D., String Topology, arXiv.org/math.GT/9911159, 1999.
Roger Carter, Graeme Segal, and Ian Macdonald, Lectures on Lie groups and Lie algebras, London Mathematical Society Student Texts, vol. 32, Cambridge University Press, Cambridge, 1995. With a foreword by Martin Taylor. MR 1356712, DOI 10.1017/CBO9781139172882
Igor Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. MR 2004511, DOI 10.1017/CBO9780511615436
Vesselin Drensky and Edward Formanek, Polynomial identity rings, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2004. MR 2064082, DOI 10.1007/978-3-0348-7934-7
Dubnov, J., Sur une généralisation de l’équation de Hamilton-Cayley et sur les invariants simultanés de plusieurs affineurs, Proc. Seminar on Vector and Tensor Analysis, Mechanics Research Inst., Moscow State Univ. $\mathbf {2/3}$ (1935), 351-367.
Jean-Paul Dufour and Nguyen Tien Zung, Poisson structures and their normal forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, Basel, 2005. MR 2178041, DOI 10.1007/3-7643-7335-0
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Philip A. Foth, Geometry of moduli spaces of flat bundles on punctured surfaces, Internat. J. Math. 9 (1998), no. 1, 63–73. MR 1612318, DOI 10.1142/S0129167X9800004X
Ralph H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560. MR 53938, DOI 10.2307/1969736
William M. Goldman, Geometric structures on manifolds and varieties of representations, Geometry of group representations (Boulder, CO, 1987) Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 169–198. MR 957518, DOI 10.1090/conm/074/957518
William M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), no. 3, 791–845. MR 1053346
Goldman, W., Introduction to Character Varieties, unpublished notes, 2003.
William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225. MR 762512, DOI 10.1016/0001-8708(84)90040-9
William M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), no. 2, 263–302. MR 846929, DOI 10.1007/BF01389091
William M. Goldman, Mapping class group dynamics on surface group representations, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 189–214. MR 2264541, DOI 10.1090/pspum/074/2264541
K. Guruprasad, J. Huebschmann, L. Jeffrey, and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89 (1997), no. 2, 377–412. MR 1460627, DOI 10.1215/S0012-7094-97-08917-1
William M. Goldman and John J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96. MR 972343, DOI 10.1007/BF02699127
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR 600654, DOI 10.1007/978-1-4612-6101-8
Yael Karshon, An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc. 116 (1992), no. 3, 591–605. MR 1112494, DOI 10.1090/S0002-9939-1992-1112494-2
Hong Chan Kim, The symplectic global coordinates on the moduli space of real projective structures, J. Differential Geom. 53 (1999), no. 2, 359–401. MR 1802726
Vladimir Petrov Kostov, The connectedness of some varieties and the Deligne-Simpson problem, J. Dyn. Control Syst. 11 (2005), no. 1, 125–155. MR 2122469, DOI 10.1007/s10883-005-0004-4
Sean Lawton, Generators, relations and symmetries in pairs of $3\times 3$ unimodular matrices, J. Algebra 313 (2007), no. 2, 782–801. MR 2329569, DOI 10.1016/j.jalgebra.2007.01.003
Lawton, S., and Peterson, E., Spin Networks and $\mathrm {SL}(2,\mathbb {C})$-Character Varieties, to appear in the “Handbook of Teichmüller Theory” (A. Papadopoulos, editor), Volume II, EMS Publishing House, Zürich, 2008 (in press).
A. V. Marinčuk and K. S. Sibirskiĭ, Minimal polynomial bases of affine invariants of square matrices of order three, Mat. Issled. 6 (1971), no. vyp. 1 (19), 100–113 (Russian). MR 0289548
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Kazunori Nakamoto, The structure of the invariant ring of two matrices of degree 3, J. Pure Appl. Algebra 166 (2002), no. 1-2, 125–148. MR 1868542, DOI 10.1016/S0022-4049(01)00004-4
C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
Claudio Procesi, Finite dimensional representations of algebras, Israel J. Math. 19 (1974), 169–182. MR 357507, DOI 10.1007/BF02756630
Ju. P. Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756 (Russian). MR 0506414
Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
Adam S. Sikora, $\textrm {SL}_n$-character varieties as spaces of graphs, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2773–2804. MR 1828473, DOI 10.1090/S0002-9947-01-02700-3
A. J. M. Spencer and R. S. Rivlin, Further results in the theory of matrix polynomials, Arch. Rational Mech. Anal. 4 (1960), 214–230 (1960). MR 109830, DOI 10.1007/BF00281388
Yasuo Teranishi, The ring of invariants of matrices, Nagoya Math. J. 104 (1986), 149–161. MR 868442, DOI 10.1017/S0027763000022728
Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297, DOI 10.1007/978-1-4757-1799-0
Scott Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), no. 2, 207–234. MR 690844, DOI 10.2307/2007075