Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Sean Lawton
Journal: Trans. Amer. Math. Soc. 361 (2009), 2397-2429
MSC (2000): Primary 58D29; Secondary 14D20
DOI: https://doi.org/10.1090/S0002-9947-08-04777-6
Published electronically: December 16, 2008
MathSciNet review: 2471924
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AP] Abeasis, A., and Pittaluga, M., On a minimal set of generators for the invariants of $ 3\times 3$ matrices, Comm. Alg. $ \mathbf{17(2)}$ (1989), 487-499. MR 978487 (90d:15021)
  • [ADS] Aslaksen, H., Drensky, V., and Sadikova, L., Defining relations of invariants of two $ 3\times 3$ matrices, arXiv:math.RA/0405389 v1, 2004. MR 2155550 (2006b:16034)
  • [A] Artin, M., On Azumaya algebras and finite dimensional representations of rings, J. of Alg. $ \mathbf{11}$ (1969), 532-563. MR 0242890 (39:4217)
  • [B] Brown, K., ``Cohomology of groups,'' Graduate Texts in Mathematics No. 87, Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
  • [Br] Bredon, G., ``Topology and Geometry'' Graduate Texts in Mathematics No. 139, Springer-Verlag, New York, 1993. MR 1224675 (94d:55001)
  • [CLO] Cox, D., Little, J., and O'Shea, D., ``Using Algebraic Geometry,'' Graduate Texts in Mathematics No. 185, Springer-Verlag, New York, 1998. MR 1639811 (99h:13033)
  • [CS] Chas, M., and Sullivan, D., String Topology, arXiv.org/math.GT/9911159, 1999.
  • [CSM] Carter, R., Segal, G., and Maconald, I., ``Lectures on Lie Groups and Lie Algebras,'' London Mathematical Society Student Texts No. 32, Cambridge University Press, Cambridge, 1995. MR 1356712 (97f:17002)
  • [D] Dolgachev, I.,``Lectures on Invariant Theory,'' London Mathematical Lecture Notes Series 296, Cambridge University Press, 2003. MR 2004511 (2004g:14051)
  • [DF] Drensky, V., and Formanek, E., ``Polynomial Identity Rings,'' Advanced Courses in Mathematics. CRM Barcelona, Birkh $ \ddot{\mathrm{a}}$user Verlag. Basel, 2004. MR 2064082 (2005c:16040)
  • [Du] 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.
  • [DZ] Dufour, J., and Zung, N.T., ``Poisson Structures and their Normal Forms,'' Progress in Mathematics, Vol. 242, Birkhäuser Verlag, Berlin, 2005. MR 2178041 (2007b:53170)
  • [E] Eisenbud, D., ``Commutative Algebra with a View Toward Algebraic Geometry,'' Graduate Texts in Mathematics No. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [Fo] Foth, Philip A., Geometry of moduli spaces of flat bundles on punctured surfaces, Internat. J. Math. $ \mathbf{9}$ (1998), no. 1, 63-73. MR 1612318 (99i:14014)
  • [F] Fox, R., Free differential calculus. I, Ann. of Math. $ \mathbf{57}$ (1953), 547-560. MR 0053938 (14:843d)
  • [G1] Goldman, W., Geometric structures on manifolds and varieties of representations, Contemp. Math. $ \mathbf{74}$ (1988), 169-197. MR 957518 (90i:57024)
  • [G2] Goldman, W., Convex real projective structures on compact surfaces, J. Diff. Geo. $ \mathbf{31}$ (1990), 791-845. MR 1053346 (91b:57001)
  • [G3] Goldman, W., Introduction to Character Varieties, unpublished notes, 2003.
  • [G4] Goldman, W., The Symplectic Nature of Fundamental Groups of Surfaces, Advances in Math. $ \mathbf{54}$ (1984), 200-225. MR 762512 (86i:32042)
  • [G5] Goldman, W., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. $ \mathbf{85}$ (1986), 263-302. MR 846929 (87j:32069)
  • [G6] Goldman, W., Mapping class group dynamics on surface group representations in ``Problems on Mapping Class Groups and Related Topics,'' ed. by B. Farb, Proc. Symp. Pure and Applied Math., Vol. $ \mathbf{74}$ (2006), 198-225. MR 2264541 (2007h:57020)
  • [GHJW] Guruprasad, K., Huebschmann, J., Jeffrey, L., and Weinstein, A., Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. $ \mathbf{89}$ (1997), no. 2, 377-412. MR 1460627 (98e:58034)
  • [GM] Goldman, W., and Millson, J., The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. No. $ \mathbf{67}$ (1988), 43-96. MR 972343 (90b:32041)
  • [Ha] Hartshorne, Robin, ``Algebraic geometry,'' Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. MR 0463157 (57:3116)
  • [H] Hungerford, T., ``Algebra,'' Graduate Texts in Mathematics No. 73, Springer-Verlag, New York, 1974. MR 600654 (82a:00006)
  • [Ka] Karshon, Y., An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc. $ \mathbf{116}$ (1992), 591-605. MR 1112494 (93a:14010)
  • [Ki] Kim, H., The symplectic global coordinates on the moduli space of real projective structures, J. Diff. Geo. $ \mathbf{53}$ (1999), 359-401. MR 1802726 (2002f:53144)
  • [Ko] Kostov, Vladimir Petrov, The connectedness of some varieties and the Deligne-Simpson problem, J. Dyn. Control Syst. $ \mathbf{11}$ (2005), no. 1, 125-155. MR 2122469 (2006c:15029)
  • [L] Lawton, S., Generators, Relations and symmetries in pairs of 3$ \times$3 unimodular matrices, J. Algebra $ \mathbf{313}$ (2007), 782-801. MR 2329569
  • [LP] 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).
  • [MS] Marincuk, A., and Sibirskii, K., Minimal polynomial bases of affine invariants of square matrices of order three, Mat. Issled. $ \mathbf{6}$ (1971), 100-113. MR 0289548 (44:6736)
  • [MKS] Magnus, W., Karrass, A., and Solitar, D., ``Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations,'' Pure and Applied Mathematics Vol. XIII, Interscience Publishers, 1966. MR 0207802 (34:7617)
  • [N] Nakamoto, K., The structure of the invariant ring of two matrices of degree $ 3$, J. Pure and Applied Alg. $ \mathbf{166}$ (2002), 125-148. MR 1868542 (2002h:16038)
  • [P1] Procesi, C., Invariant theory of $ n\times n$ matrices, Advances in Mathematics 19 (1976), 306-381. MR 0419491 (54:7512)
  • [P2] Procesi, C., Finite dimensional representations of algebras, Israel J. Math. 19 (1974), 169-182. MR 0357507 (50:9975)
  • [R] Razmyslov, Y., Trace identities of full matrix algebras over a field of characteristic zero. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. $ \mathbf{38}$ (1974), 723-756. MR 0506414 (58:22158)
  • [S] Shafarevich, I., ``Basic Algebraic Geometry $ 1$,'' $ 2^{\text{nd}}$ edition, Springer-Verlag, Berlin, 1994. MR 1328833 (95m:14001)
  • [Si] Sikora, A., $ SL_n$-character varieties as space of graphs, Trans. Amer. Math. Soc. $ \mathbf{353}$ (2001), no. 7, 2773-2804. MR 1828473 (2003b:57004)
  • [SR] Spencer, A., and Rivlin, R., Further results in the theory of matrix polynomials, Arch. Rational Mech. Anal. $ \mathbf{4}$ (1960), 214-230. MR 0109830 (22:715)
  • [T] Teranishi, Y., The ring of invariants of matrices, Nagoya Math. J. $ \mathbf{104}$ (1986), 149-161. MR 868442 (88a:16038)
  • [Wa] Warner, Frank W., ``Foundations of differentiable manifolds and Lie groups,'' Graduate Texts in Mathematics No. 94. Springer-Verlag, New York-Berlin, 1983. ix+272. MR 722297 (84k:58001)
  • [W] Wolpert, S., On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) $ \mathbf{117}$ (1983), 207-234. MR 690844 (85e:32028)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 58D29, 14D20

Retrieve articles in all journals with MSC (2000): 58D29, 14D20


Additional Information

Sean Lawton
Affiliation: Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: slawton@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9947-08-04777-6
Received by editor(s): March 23, 2007
Published electronically: December 16, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society