On the construction of $  \mathbb Z^n_2- $grassmannians as homogeneous $  \mathbb Z^n_2- $spaces

Mohammad Mohammadi; Saad Varsaie; Mohammad Mohammadi; Saad Varsaie

doi:10.3934/era.2022012

Electronic Research Archive

2022, Volume 30, Issue 1: 221-241. doi: 10.3934/era.2022012

Previous Article Next Article

Research article

On the construction of $ \mathbb Z^n_2- $grassmannians as homogeneous $ \mathbb Z^n_2- $spaces

Mohammad Mohammadi ^,,
Saad Varsaie

Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), No. 444, Prof. Yousef Sobouti Blvd. P. O. Box 45195-1159 Zanjan, Postal Code 45137-66731, Iran

Received: 23 May 2021 Revised: 05 September 2021 Accepted: 29 October 2021 Published: 24 December 2021

In this paper, we construct the $ \mathbb Z^n_2- $grassmannians by gluing of the $ \mathbb Z^n_2- $domains and give an explicit description of the action of the $ \mathbb Z^n_2- $Lie group $ GL(\overrightarrow{\textbf{m}}) $ on the $ \mathbb Z^n_2- $grassmannian $ G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}}) $ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the $ \mathbb Z^n_2- $grassmannian.
- $ \mathbb Z^n_2- $Lie group,
- Homogeneous superspace,
- $ \mathbb Z^n_2- $manifold,
- $ \mathbb Z^n_2- $ grassmannian
Citation: Mohammad Mohammadi, Saad Varsaie. On the construction of $ \mathbb Z^n_2- $grassmannians as homogeneous $ \mathbb Z^n_2- $spaces[J]. Electronic Research Archive, 2022, 30(1): 221-241. doi: 10.3934/era.2022012

Related Papers:

Abstract

In this paper, we construct the $ \mathbb Z^n_2- $grassmannians by gluing of the $ \mathbb Z^n_2- $domains and give an explicit description of the action of the $ \mathbb Z^n_2- $Lie group $ GL(\overrightarrow{\textbf{m}}) $ on the $ \mathbb Z^n_2- $grassmannian $ G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}}) $ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the $ \mathbb Z^n_2- $grassmannian.

References

[1]	T. Covolo, J. Grabowski, N. Poncin, $ \mathbb Z^n_2-$supergeometry I: manifolds and morphisms. arXiv: 1408.2755.
[2]	T. Covolo, J. Grabowski, N. Poncin, Splitting theorem for $ \mathbb Z^n_2-$supermanifolds, J. Geom. Phys., 110 (2016), 393–401. https://doi.org/10.1016/j.geomphys.2016.09.006 doi: 10.1016/j.geomphys.2016.09.006
[3]	T. Covolo, J. Grabowski, N. Poncin, The category of $ \mathbb Z^n_2-$supermanifolds, J. Math. Phys., 57 (2016), 073503. https://doi.org/10.1063/1.4955416 doi: 10.1063/1.4955416
[4]	T. Covolo, V. Ovsienko, N. Poncin, Higher trace and Berezinian of matrices over a Clifford algebra, J. Geom. Phys., 62 (2012), 2294–2319. https://doi.org/10.1016/j.geomphys.2012.07.004 doi: 10.1016/j.geomphys.2012.07.004
[5]	W. M. Yang, S. C. Jing, A New Kind of Graded Lie Algebra and Parastatistical Supersymmetry, Science in China (Series A), 44 (2001), 1167–1173. https://doi.org/10.1007/BF02877435 doi: 10.1007/BF02877435
[6]	L. Balduzzi, C. Carmeli, G. Cassinelli, Super G-spaces, Symmetry in mathematics and physics, AMS: Providence, RI, USA, (2009), 159–176. https://doi.org/10.1090/conm/490/09594
[7]	L. Balduzzi, C. Carmeli, R. Fioresi, Quotients in supergeometry, Symmetry in mathematics and physics, AMS: Providence, RI, USA, (2009), 177–187. https://doi.org/10.1090/conm/490/09595
[8]	C. Carmeli, L. Caston, R. Fioresi, Mathematical Foundations of Supersymmetry, European Mathematical Society, 2011. https://doi.org/10.4171/097
[9]	Y. I. Manin, Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 289, Springer-Verlag, Berlin, 1988, Translated from the Russian by N. Koblitz and J. R. King.
[10]	T. Covolo, S. Kwok, N. Poncin, Differential Calculus on $\mathbb Z^n_2$supermanifolds. arXiv preprint arXiv:1608.00949, 2016.
[11]	A. J. Bruce, N. Poncin, Products in the category of $ \mathbb Z^n_2-$manifolds, J. Nonlinear Math. Phys., 26 (2019), 420–453. https://doi.org/10.1080/14029251.2019.1613051 doi: 10.1080/14029251.2019.1613051
[12]	B. Jubin, A. Kotov, N. Poncin, V. Salnikov, Differential graded Lie groups and their differential graded Lie algebras, arXiv: 1906.09630.
[13]	F. Bahadorykhalily, M. Mohammadi, S. Varsaie, A class of homogeneous superspaces associated to odd involutions, Period Math. Hung., 82 (2021), 153–172. https://doi.org/10.1007/s10998-020-00364-9
[14]	O. Sanchez-Valenzuela, Remarks on grassmannian supermanifolds, Trans. Am. Math. Soc., 307 (1988), 597–614. https://doi.org/10.2307/2001190 doi: 10.2307/2001190
[15]	M. Mohammadi, S. Varsaie, Supergrassmannians as Homogeneous Superspaces, Asian-European J. Math., 14 (2021), 2150053. https://doi.org/10.1142/S1793557121500534 doi: 10.1142/S1793557121500534
[16]	M. Roshandelbana, S. Varsaie, Analytic Approach To $\nu$-Classes. arXiv: 1801.06633
[17]	A. J. Bruce, N. Poncin, Functional analytic issues in $ \mathbb Z^n_2-$Geometry, Revista de la Unión Matemática Argentina, 60 (2019), 611–636. https://doi.org/10.33044/revuma.v60n2a21 doi: 10.33044/revuma.v60n2a21
[18]	R. Fioresi, M. A. Lledo, V. S. Varadarajan, The Minkowski and conformal superspaces, J. Math. Phys., 48 (2007), 113505. https://doi.org/10.1063/1.2799262 doi: 10.1063/1.2799262
[19]	T. Covolo, S. Kwok, N. Poncin, The Frobenius Theorem for $ \mathbb Z^n_2$supermanifolds, arXiv: 1608.00961.
[20]	S. Maclane, Categories for the Working Mathematician, Springer, 1978. https://doi.org/10.1007/978-1-4757-4721-8
[21]	V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes in Mathematics, vol. 11, New York University CourantInstitute of Mathematical Sciences, New York, 2004. https://doi.org/10.1090/cln/011

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)