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Journal articles on the topic 'Coadjoint orbits'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

Kurniadi, Edi. "Ruang Fase Tereduksi Grup Lie Aff (1)." Jambura Journal of Mathematics 3, no. 2 (June 24, 2021): 180–86. http://dx.doi.org/10.34312/jjom.v3i2.10653.

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ABSTRAKDalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari melalui orbit coadjoint buka di titik tertentu pada ruang dual dari aljabar Lie . Aksi dari grup Lie pada ruang dual menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine tepat mempunyai dua buah orbit coadjoint buka. Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine berdimensi dan untuk kasus grup Lie lainnya.ABSTRACTIn this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

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2

GORSKY, A., and A. JOHANSEN. "LIOUVILLE THEORY AND SPECIAL COADJOINT VIRASORO ORBITS." International Journal of Modern Physics A 10, no. 06 (March 10, 1995): 785–99. http://dx.doi.org/10.1142/s0217751x95000371.

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We describe the Hamiltonian reduction of the coadjoint Kac–Moody orbits to the Virasoro coadjoint orbits explicitly in terms of the Lagrangian approach for the Wess–Zumino–Novikov–Witten theory. While a relation of the coadjoint Virasoro orbit Diff S1/ SL (2, R) to the Liouville theory has already been studied, we analyze the role of special coadjoint Virasoro orbits Diff [Formula: see text]corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros xi of the vector field of stabilizer [Formula: see text] and additional parameters qi, i = 1, …, n, in terms of quantum mechanics for n-point particles on the circle. We argue that the special orbits are generated by insertions of "wrong sign" Liouville exponential into the path integral. The additional parmeters qi are naturally interpreted as accessory parameters for the uniformization map. Summing up the contributions of the special Virasoro orbits we get the integrable sinh–Gordon type theory.

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3

LIEDÓ, M. A. "DEFORMATION QUANTIZATION OF COADJOINT ORBITS." International Journal of Modern Physics B 14, no. 22n23 (September 20, 2000): 2397–400. http://dx.doi.org/10.1142/s0217979200001916.

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A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored.

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4

BOŽIČEVIĆ, MLADEN. "A LIMIT FORMULA FOR EVEN NILPOTENT ORBITS." International Journal of Mathematics 19, no. 02 (February 2008): 223–36. http://dx.doi.org/10.1142/s0129167x08004650.

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Let Gℝ be a real form of a complex, semisimple Lie group G. Assume [Formula: see text] is an even nilpotent coadjoint Gℝ-orbit. We prove a limit formula, expressing the canonical measure on [Formula: see text] as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with [Formula: see text].

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5

Arnal, D., M. Cahen, and S. Gutt. "Deformations on coadjoint orbits." Journal of Geometry and Physics 3, no. 3 (January 1986): 327–51. http://dx.doi.org/10.1016/0393-0440(86)90013-6.

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6

Robinson, P. L. "Equivariant prequantization and admissible coadjoint orbits." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 1 (July 1993): 131–42. http://dx.doi.org/10.1017/s0305004100071462.

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The orbit method has as its primary goal the construction and parametrization of the irreducible unitary representations of a (simply-connected) Lie group in terms of its coadjoint orbits. This goal was achieved with complete success for nilpotent groups by Kirillov[8] and for type I solvable groups by Auslander and Kostant[l] but is known to encounter difficulties when faced with more general groups. Geometric quantization can be viewed as an outgrowth of the orbit method aimed at providing a geometric passage from classical mechanics to quantum mechanics. Whereas the original geometric quantization scheme due to Kostant[9] and Souriau[14] enabled such a passage in a variety of situations, it too encounters difficulties in broader contexts.

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7

Božičević, Mladen. "Invariant measures on nilpotent orbits associated with holomorphic discrete series." Representation Theory of the American Mathematical Society 25, no. 24 (August 18, 2021): 732–47. http://dx.doi.org/10.1090/ert/580.

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Let G R G_\mathbb R be a real form of a complex, semisimple Lie group G G . Assume G R G_\mathbb R has holomorphic discrete series. Let W \mathcal W be a nilpotent coadjoint G R G_\mathbb R -orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on W \mathcal W as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the positive chamber defined by the Borel subalgebra associated with holomorphic discrete series.

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8

Esposito, Chiara, Philipp Schmitt, and Stefan Waldmann. "Comparison and continuity of Wick-type star products on certain coadjoint orbits." Forum Mathematicum 31, no. 5 (September 1, 2019): 1203–23. http://dx.doi.org/10.1515/forum-2018-0302.

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AbstractIn this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

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9

Vi�a, A. "Cohomological splitting of coadjoint orbits." Archiv der Mathematik 82, no. 1 (January 1, 2004): 13–15. http://dx.doi.org/10.1007/s00013-003-4819-5.

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10

Le Bruyn, Lieven. "Noncommutative smoothness and coadjoint orbits." Journal of Algebra 258, no. 1 (December 2002): 60–70. http://dx.doi.org/10.1016/s0021-8693(02)00533-1.

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11

Mauro, D. "Coadjoint orbits, spin and dequantization." Physics Letters B 597, no. 1 (September 2004): 94–104. http://dx.doi.org/10.1016/j.physletb.2004.07.016.

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12

Landsman, N. P. "Strict quantization of coadjoint orbits." Journal of Mathematical Physics 39, no. 12 (December 1998): 6372–83. http://dx.doi.org/10.1063/1.532644.

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13

Lledó, M. A. "Star products on coadjoint orbits." Physics of Atomic Nuclei 64, no. 12 (December 2001): 2136–38. http://dx.doi.org/10.1134/1.1432913.

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14

Lang, Honglei, and Zhangju Liu. "Coadjoint orbits of Lie groupoids." Journal of Geometry and Physics 129 (July 2018): 217–32. http://dx.doi.org/10.1016/j.geomphys.2018.03.011.

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15

Aratyn, H., E. Nissimov, S. Pacheva, and A. H. Zimerman. "Symplectic actions on coadjoint orbits." Physics Letters B 240, no. 1-2 (April 1990): 127–32. http://dx.doi.org/10.1016/0370-2693(90)90420-b.

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16

Oh, Phillial, and Chaiho Rim. "Holstein–Primakoff Realizations on Coadjoint Orbits." Modern Physics Letters A 12, no. 03 (January 30, 1997): 163–72. http://dx.doi.org/10.1142/s0217732397000169.

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We derive the Holstein–Primakoff oscillator realization on the coadjoint orbits of the SU (N+1) and SU (1,N) group by treating the coadjoint orbits as a constrained system and performing the symplectic reduction. By using the action-angle variables transformations, we transform the original variables into Darboux variables. The Holstein–Primakoff expressions emerge after quantization in a canonical manner with a suitable normal ordering. The corresponding Dyson realizations are also obtained and some related issues are discussed.

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17

Akylzhanov, Rauan, and Alexis Arnaudon. "Contractions of group representations via geometric quantization." Letters in Mathematical Physics 110, no. 1 (November 21, 2019): 43–59. http://dx.doi.org/10.1007/s11005-019-01212-9.

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AbstractWe propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their coadjoint orbits, using geometric quantization. The sufficient condition for the contractibility of a representation is expressed via cocycles on coadjoint orbits. This condition is verified explicitly for the contraction of SU$$_2$$2 into $$\mathbb {H}$$H. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.

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18

AVAN, JEAN, and ANTAL JEVICKI. "STRING FIELD ACTIONS FROM W∞." Modern Physics Letters A 07, no. 04 (February 10, 1992): 357–70. http://dx.doi.org/10.1142/s0217732392000306.

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Starting from W∞ as a fundamental symmetry and using the coadjoint orbit method, we derive an action for one-dimensional strings. It is shown that on the simplest non-trivial orbit this gives the single scalar collective field theory. On higher orbits one finds generalized KdV type field theories with increasing number of components.

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19

Martínez Torres, David. "Semisimple coadjoint orbits and cotangent bundles." Bulletin of the London Mathematical Society 48, no. 6 (September 16, 2016): 977–84. http://dx.doi.org/10.1112/blms/bdw058.

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20

Cushman, Richard, and Mark Roberts. "Poisson structures transverse to coadjoint orbits." Bulletin des Sciences Mathématiques 126, no. 7 (August 2002): 525–34. http://dx.doi.org/10.1016/s0007-4497(02)01118-1.

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21

Arnaudon, Alexis, Alex L. De Castro, and Darryl D. Holm. "Noise and Dissipation on Coadjoint Orbits." Journal of Nonlinear Science 28, no. 1 (July 17, 2017): 91–145. http://dx.doi.org/10.1007/s00332-017-9404-3.

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22

Bolsinov, Alexey V., and Božidar Jovanović. "Magnetic geodesic flows on coadjoint orbits." Journal of Physics A: Mathematical and General 39, no. 16 (March 31, 2006): L247—L252. http://dx.doi.org/10.1088/0305-4470/39/16/l01.

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23

Eshmatov, Alimjon, and Philip Foth. "On sums of admissible coadjoint orbits." Proceedings of the American Mathematical Society 142, no. 2 (November 14, 2013): 727–35. http://dx.doi.org/10.1090/s0002-9939-2013-11799-5.

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24

Castro, Alexander Caviedes. "Calabi quasimorphisms for monotone coadjoint orbits." Journal of Topology and Analysis 09, no. 04 (August 10, 2017): 689–706. http://dx.doi.org/10.1142/s1793525317500285.

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We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.

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25

Adams, Jeffrey. "Coadjoint orbits and reductive dual pairs." Advances in Mathematics 63, no. 2 (February 1987): 138–51. http://dx.doi.org/10.1016/0001-8708(87)90050-8.

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26

Deltour, G. "Kirwan polyhedron of holomorphic coadjoint orbits." Transformation Groups 17, no. 2 (December 24, 2011): 351–92. http://dx.doi.org/10.1007/s00031-011-9164-3.

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27

Kirillov, A. A. "On the combinatorics of coadjoint orbits." Functional Analysis and Its Applications 27, no. 1 (January 1993): 62–64. http://dx.doi.org/10.1007/bf01768672.

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28

Xue, Ting. "Nilpotent coadjoint orbits in small characteristic." Journal of Algebra 397 (January 2014): 111–40. http://dx.doi.org/10.1016/j.jalgebra.2013.09.001.

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29

Witten, Edward. "Coadjoint orbits of the Virasoro group." Communications in Mathematical Physics 114, no. 1 (March 1988): 1–53. http://dx.doi.org/10.1007/bf01218287.

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30

Haller, Stefan, and Cornelia Vizman. "Nonlinear flag manifolds as coadjoint orbits." Annals of Global Analysis and Geometry 58, no. 4 (September 8, 2020): 385–413. http://dx.doi.org/10.1007/s10455-020-09725-6.

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Abstract A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

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31

DELIUS, GUSTAV W., PETER VAN NIEUWENHUIZEN, and V. G. J. RODGERS. "THE METHOD OF COADJOINT ORBITS: AN ALGORITHM FOR THE CONSTRUCTION OF INVARIANT ACTIONS." International Journal of Modern Physics A 05, no. 20 (October 20, 1990): 3943–83. http://dx.doi.org/10.1142/s0217751x90001690.

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The method of coadjoint orbits produces for any infinite dimensional Lie (super) algebra A with nontrivial central charge an action for scalar (super) fields which has at least the symmetry A. In this article, we try to make this method accessible to a larger audience by analyzing several examples in more detail than in the literature. After working through the Kac-Moody and Virasoro cases, we apply the method to the super Virasoro algebra and reobtain the supersymmetric extension of Polyakov's local nonpolynomial action for two-dimensional quantum gravity. As in the Virasoro case this action corresponds to the coadjoint orbit of a pure central extension. We further consider the actions corresponding to the other orbits of the super Virasoro algebra. As a new result we construct the actions for the N = 2 super Virasoro algebra.

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32

NASRIN, SALMA. "CORWIN–GREENLEAF MULTIPLICITY FUNCTIONS FOR HERMITIAN SYMMETRIC SPACES AND MULTIPLICITY-ONE THEOREM IN THE ORBIT METHOD." International Journal of Mathematics 21, no. 03 (March 2010): 279–96. http://dx.doi.org/10.1142/s0129167x10006021.

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Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.For this, we study the Corwin–Greenleaf multiplicity function [Formula: see text] for Hermitian symmetric spaces G/K. First, we prove that [Formula: see text] for any G-coadjoint orbit [Formula: see text] and any K-coadjoint orbit [Formula: see text] if [Formula: see text]. Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.Second, we find a necessary and sufficient condition for [Formula: see text] by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

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33

Rodgers, Vincent G. J., and Takeshi Yasuda. "From Diffeomorphisms to Dark Energy?" Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2467–74. http://dx.doi.org/10.1142/s0217732303012702.

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There are two physical actions that have a natural setting in terms of the coadjoint representation of the algebra of diffeomorphisms and of affine Lie algebras. One is the usual geometric action that comes from coadjoint orbits. The other action lives on the phase space that is transverse to the orbits and are called transverse actions, where Yang-Mills theory in two dimensions is an example. Here we show that the transverse action associated with the Virasoro algebra might contain clues for a theory for dark energy. These actions might also suggests a mechanism for symmetry changing.

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34

Penna, Vittorio, and Mauro Spera. "On coadjoint orbits of rotational perfect fluids." Journal of Mathematical Physics 33, no. 3 (March 1992): 901–9. http://dx.doi.org/10.1063/1.529741.

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35

Li, Zongyi. "The Mackey obstruction and the coadjoint orbits." Transactions of the American Mathematical Society 346, no. 2 (February 1, 1994): 693–705. http://dx.doi.org/10.1090/s0002-9947-1994-1276935-9.

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36

Beltiţă, Ingrid, and Daniel Beltiţă. "Coadjoint orbits of stepwise square integrable representations." Proceedings of the American Mathematical Society 144, no. 3 (June 30, 2015): 1343–50. http://dx.doi.org/10.1090/proc/12761.

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37

Alekseev, Anton, and Samson L. Shatashvili. "Coadjoint Orbits, Cocycles and Gravitational Wess–Zumino." Reviews in Mathematical Physics 30, no. 06 (July 2018): 1840001. http://dx.doi.org/10.1142/s0129055x18400019.

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About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group [Formula: see text]. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group [Formula: see text]. In the case of [Formula: see text] being a central extension, we construct Wess–Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov–Wiegmann (PW) formula with boundary term given by the 2-cocycle which defines the central extension. In particular, we obtain a PW type formula for Polyakov’s gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess–Zumino terms obtained from geometric actions, and that in this case, the path integral is expressed in terms of the 2-cocycle which defines the central extension. In memory of our teacher Ludwig Faddeev

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38

Hurtubise, J. C. "Finite-dimensional coadjoint orbits in loop algebras." Letters in Mathematical Physics 30, no. 2 (February 1994): 99–104. http://dx.doi.org/10.1007/bf00939697.

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39

Neeb, Karl-Hermann, and Christoph Zellner. "Oscillator algebras with semi-equicontinuous coadjoint orbits." Differential Geometry and its Applications 31, no. 2 (April 2013): 268–83. http://dx.doi.org/10.1016/j.difgeo.2012.10.010.

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40

PARADAN, P. E., and M. VERGNE. "ADMISSIBLE COADJOINT ORBITS FOR COMPACT LIE GROUPS." Transformation Groups 23, no. 3 (December 1, 2017): 875–92. http://dx.doi.org/10.1007/s00031-017-9457-2.

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41

BALOG, J., L. FEHÉR, and L. PALLA. "COADJOINT ORBITS OF THE VIRASORO ALGEBRA AND THE GLOBAL LIOUVILLE EQUATION." International Journal of Modern Physics A 13, no. 02 (January 20, 1998): 315–62. http://dx.doi.org/10.1142/s0217751x98000147.

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The classification of the coadjoint orbits of the Virasoro algebra is reviewed and then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well known that the Liouville equation for a smooth, real field φ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field g∈ SL (2,R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field Q=κg22 where κ≠0 is a constant, what we call the global Liouville equation. Corresponding to the winding number of the SL(2,R) WZNW model, there is a topological invariant in the reduced theory, given by the number of zeros of Q over a period. By the substitution Q=± exp (-φ/2), the Liouville theory for a smooth φ is recovered in the trivial topological sector. The nontrivial topological sectors can be viewed as singular sectors of the Liouville theory that contain blowingup solutions in terms of φ. Since the global Liouville equation is conformally invariant, its solutions can be described by explicitly listing those solutions for which the stress–energy tensor belongs to a set of representatives of the Virasoro coadjoint orbits chosen by convention. This direct method permits to study the "coadjoint orbit content" of the topological sectors as well as the behavior of the energy in the sectors. The analysis confirms that the trivial topological sector contains special orbits with hyperbolic monodromy and shows that the energy is bounded from below in this sector only.

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42

Kurniadi, E. "Representasi Unitar Tak Tereduksi Grup Lie Dari Aljabar Lie Filiform Real Berdimensi 5." JURNAL ILMIAH MATEMATIKA DAN TERAPAN 17, no. 1 (June 25, 2020): 100–108. http://dx.doi.org/10.22487/2540766x.2020.v17.i1.15185.

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In this paper, we study a harmonic analysis of a Lie group of a real filiform Lie algebra of dimension 5. Particularly, we study its irreducible unitary representation (IUR) and contruct this IUR corresponds to its coadjoint orbits through coadjoint actions of its group to its dual space. Using induced representation of a 1-dimensional representation of its subgroup we obtain its IUR of its Lie group

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43

Alldridge, Alexander, Joachim Hilgert, and Tilmann Wurzbacher. "SUPERORBITS." Journal of the Institute of Mathematics of Jussieu 17, no. 5 (July 20, 2016): 1065–120. http://dx.doi.org/10.1017/s147474801600030x.

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We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov–Kostant–Souriau type. Applying a family version of Kirillov’s orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.

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44

SELIVANOV, K. G. "GEOMETRY AND PHYSICS ON w∞ ORBITS." Modern Physics Letters A 08, no. 12 (April 20, 1993): 1139–51. http://dx.doi.org/10.1142/s0217732393002622.

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We apply the coadjoint orbit technique to the group of area preserving diffeomorphisms (APD) of a 2D manifold, particularly to the APD of the semi-infinite cylinder which is identified with w∞. The geometrical action obtained is relevant to both w gravity and 2D turbulence. For the latter we describe the Hamiltonian, which appears to be given by the Schwinger mass term, and discuss some possible developments within our approach. Next we show that the set of highest weight orbits of w∞ splits into subsets, each of which consists of highest weight orbits of wN for a given N. We specify the general APD geometric action to an orbit of wN and describe an appropriate set of observables, thus getting an action and observables for wN gravity. We compute also the Ricci form on the wN orbits, what gives us the critical central charge of the wN string, which appears to be the same as the one of the WN string.

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45

Robinson, P. L. "Mp c Structures, Coadjoint Orbits and Admissible Characters." Bulletin of the London Mathematical Society 24, no. 3 (May 1992): 289–92. http://dx.doi.org/10.1112/blms/24.3.289.

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46

Paradan, Paul-Emile. "The Fourier Transform of Semi-Simple Coadjoint Orbits." Journal of Functional Analysis 163, no. 1 (April 1999): 152–79. http://dx.doi.org/10.1006/jfan.1998.3381.

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47

Goldin, R. F., and A. L. Mare. "Cohomology of symplectic reductions of generic coadjoint orbits." Proceedings of the American Mathematical Society 132, no. 10 (June 2, 2004): 3069–74. http://dx.doi.org/10.1090/s0002-9939-04-07443-x.

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48

Rosensteel, George. "Symplectic manifolds, coadjoint orbits, and mean field theory." International Journal of Theoretical Physics 25, no. 5 (May 1986): 553–59. http://dx.doi.org/10.1007/bf00668789.

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49

Panov, A. N. "Involutions in S n and associated coadjoint orbits." Journal of Mathematical Sciences 151, no. 3 (June 2008): 3018–31. http://dx.doi.org/10.1007/s10958-008-9016-4.

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50

Ignatev, M. V., and A. N. Panov. "Coadjoint orbits of the group UT(7, K)." Journal of Mathematical Sciences 156, no. 2 (January 2009): 292–312. http://dx.doi.org/10.1007/s10958-008-9267-0.

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