×

(Co)isotropic triples and poset representations. (English) Zbl 1482.37061

Summary: We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we assume only to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is “2 + 2 + 2” consisting of three independent ordered pairs, with the involution exchanging the members of each pair.
A key feature of the classification is that any indecomposable (co)isotropic triple is either “split” or “non-split.” The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the “split” case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a “symplectification.”
In the course of the paper we develop the framework of “symplectic poset representations,” which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples.

MSC:

37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
37J06 General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants
15A21 Canonical forms, reductions, classification
15A03 Vector spaces, linear dependence, rank, lineability

References:

[1] M. B , Introduction to the representation theory of algebras, Springer, Cham, 2015. · Zbl 1330.16001
[2] K. J. B , On the number of square classes of a field of finite level, in J. W. Hoff-man, J. Hurrelbrink, J. Morales, R. Perlis, and P. van Wamelen (eds.), Proceedings of the Conference on Quadratic Forms and Related Topics, held at Louisiana State Uni-versity, Baton Rouge, LA, March 26-30, 2001, Doc. Math. 2001, Extra Vol. Deutsche Mathematiker Vereinigung, Berlin, 2001, pp. 65-84. · Zbl 0982.11018
[3] D. J. B , Representations and cohomology, Volume 1: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics, 30, Cambridge University Press, Cambridge, 1991. · Zbl 0718.20001
[4] I. N. B ȋ -I. M. G ’ -V. A. P , Coxeter functors, and
[5] Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), pp. 19-33, in Russian; · Zbl 0269.08001
[6] English translation, Russian Math. Surveys 28 (1973), no. 2, pp. 17-32. · Zbl 0279.08001
[7] S. H. D , Generalized Jordan normal forms of linear operators, J. Math. Sci. (N.Y.) 198 (2014), no. 5, pp. 498-504.
[8] H. D -J. W . An introduction to quiver representations, Graduate Stud-ies in Mathematics, 184, American Mathematical Society, Providence, R.I., 2017. · Zbl 1426.16001
[9] N. C. D -M. A. D G -J. N. P , Metaplectic formulation of the Wigner transform and applications, Rev. Math. Phys. 25 (2013), no. 10, 1343010, 19 pp. · Zbl 1282.35429
[10] V. D -C.-M. R , Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57 pp. · Zbl 0332.16015
[11] P. D -M. R. F , The representation theory of finite graphs and associated algebras, Carleton Math. Lecture Notes 5 (1973). · Zbl 0304.08006
[12] J. J. D , On the Morse index in variational calculus, Advances in Math. 21 (1976), no. 2, pp. 173-195. · Zbl 0361.49026
[13] P. E -O. G -S. H -T. L -A. S -D. V -E. Y , Introduction to representation theory, with historical interludes by S. Gerovitch, Student Mathematical Library, 59, American Mathematical Society, Providence, R.I., 2011. · Zbl 1242.20001
[14] P. G , Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), pp. 71-103. · Zbl 0232.08001
[15] P. G -A. V. R , Representations of finite dimensional algebras, with a chapter by B. Keller, in A. I. Kostrikin, I. R. Shafarevich, R. V. Gamkrelidze, P. Gabriel, and A. V. Roiter (eds.), Algebra VIII, translated from the Russian, transla-tion edited by A. I. Kostrikin and I. R. Shafarevich, Encyclopaedia of Mathematical Sciences, 73, Springer-Verlag, Berlin, 1992. (Co)isotropic triples and poset representations 161 · Zbl 0839.16001
[16] I. M. G ’ -V. A. P , Problems of linear algebra and classifica-tion of quadruples of subspaces in a finite-dimensional vector space, in B. Sz.-Nagy (ed.), Hilbert space operators and operator algebras, Proceedings of an International Conference held at Tihany, 14-18 September 1970. Colloquia Mathematica Soci-etatis János Bolyai, 5, North-Holland Publishing Co., Amsterdam and London, 1972, pp. 163-237. · Zbl 0294.15002
[17] V. K , Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, pp. 57-92. · Zbl 0427.17001
[18] H. K , Normal forms and versal deformations of linear Hamiltonian systems, J. Differential Equations 51 (1984), no. 3, pp. 359-407. · Zbl 0488.34034
[19] T. Y. L , Lectures on modules and rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999. · Zbl 0911.16001
[20] J. L , Lectures on rings and modules, with an appendix by I. G. Connell, Blaisdell Publishing Co. Ginn and Co., Waltham, MA, etc., 1966. · Zbl 0143.26403
[21] A. L -K. M , Canonical forms for symplectic and Hamiltonian matrices, Celestial Mech. 9 (1974), pp. 213-238. · Zbl 0316.15005
[22] G. L -M. V , The Weil representation, Maslov index and theta series, Progress in Mathematics, 6, Birkhäuser, Boston, MA, 1980. · Zbl 0444.22005
[23] J. L -A. W , Co isotropic pairs in Poisson and presymplectic vector spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 072, 10 pp. · Zbl 1327.15023
[24] J. L -A. W , Decomposition of co isotropic relations, Lett. Math. Phys. 106 (2016), no. 12, pp. 1837-1847. · Zbl 1362.18005
[25] A. I. M ’ , Foundations of linear algebra, translated from the Russian by Th. Brown, edited by J. B. Roberts, W. H. Freeman & Co., San Francisco, CA, and London, 1963.
[26] J. N , Continuous geometry, with a foreword by I. Halperin, Reprint of the 1960 original, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, N.J., 1998. · Zbl 0919.51002
[27] H.-G. Q -W. S -M. S , Quadratic and Hermitian forms in additive and abelian categories, J. Algebra 59 (1979), no. 2, pp. 264-289. · Zbl 0412.18016
[28] A. R. R , Squares, London Mathematical Society Lecture Note Series, 171, Cambridge University Press, Cambridge, 1993. · Zbl 0785.11022
[29] D. W. R , Classroom notes the generalized Jordan canonical form, Amer. Math. Monthly 77 (1970), no. 4, pp. 392-395. · Zbl 0196.05702
[30] W. S , On subspaces of inner product spaces, in Proceedings of the Interna-tional Congress of Mathematicians, Vol. 1, Held in Vancouver, August 21-29, 1974, Canadian Mathematical Congress, Montreal, 1975, pp. 331-335. · Zbl 0364.15019
[31] Ch. Herrmann -J. Lorand -A. Weinstein
[32] R. S , Quiver representations, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2014.
[33] V. V. S , Classification problems for systems of forms and linear mappings, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 6, pp. 1170-1190, 1358, in Russian; · Zbl 0653.15006
[34] English translation, Math. USSR-Izv. 31 (1988), no. 3, pp. 481-501. · Zbl 0678.15011
[35] R. S , On admissible and perfect elements in modular lattices, preprint, 2004. arXiv:math/0404171
[36] R. S , Gel’fand-Ponomarev and Herrmann constructions for quadruples and sextuples, J. Pure Appl. Algebra 211 (2007), no. 1, pp. 95-202. · Zbl 1169.06006
[37] A. W , The invariance of Poincaré’s generating function for canonical trans-formations, Invent. Math. 16 (1972), pp. 202-213. · Zbl 0235.70008
[38] J. W , On the normal forms of linear canonical transformations in dynam-ics, Amer. J. Math. 59 (1937), no. 3, pp. 599-617. · JFM 63.0845.02
[39] Manoscritto pervenuto in redazione il 17 novembre 2019.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.