Remarks on Grassmannian supermanifolds

Author:
Oscar Adolfo Sánchez-Valenzuela

Journal:
Trans. Amer. Math. Soc. **307** (1988), 597-614

MSC:
Primary 58A50; Secondary 14M15

DOI:
https://doi.org/10.1090/S0002-9947-1988-0940219-3

MathSciNet review:
940219

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies some aspects of a particular class of examples of supermanifolds; the *supergrassmannians*, introduced in [**Manin**]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [**Batchelor**]. The proof depends on the computation of the transition functions of the *supercotangent bundle* as defined in a previous work [**OASV 2**]. Finally, it is shown that there is a natural *supergroup action* involved (in the sense of [**OASV 3**]) and hence, the supergrassmannians may be regarded as examples of *superhomogeneous spaces*--a notion first introduced in [**Kostant**]. The corresponding *Lie superalgebra* action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.

**[Batchelor]**Marjorie Batchelor,*The structure of supermanifolds*, Trans. Amer. Math. Soc.**253**(1979), 329–338. MR**536951**, https://doi.org/10.1090/S0002-9947-1979-0536951-0**[Boyer and Gitler]**Charles P. Boyer and Samuel Gitler,*The theory of 𝐺^{∞}-supermanifolds*, Trans. Amer. Math. Soc.**285**(1984), no. 1, 241–267. MR**748840**, https://doi.org/10.1090/S0002-9947-1984-0748840-9**[Corwin, Ne'eman and Sternberg]**L. Corwin, Y. Ne’eman, and S. Sternberg,*Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry)*, Rev. Modern Phys.**47**(1975), 573–603. MR**0438925**, https://doi.org/10.1103/RevModPhys.47.573**[Guillemin and Sternberg]**V. Guillemin and S. Sternberg,*Lecture notes on conformal geometry*, courses delivered by the authors at Harvard and M.I.T., 1985-1986.**[Jadczyk and Pilch]**A. Jadczyk and K. Pilch,*Superspaces and supersymmetries*, Comm. Math. Phys.**78**(1980/81), no. 3, 373–390. MR**603500****[Kostant]**Bertram Kostant,*Graded manifolds, graded Lie theory, and prequantization*, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Springer, Berlin, 1977, pp. 177–306. Lecture Notes in Math., Vol. 570. MR**0580292****[Leĭtes]**D. A. Leĭtes,*Introduction to the theory of supermanifolds*, Uspekhi Mat. Nauk**35**(1980), no. 1(211), 3–57, 255 (Russian). MR**565567****[Manin]**Yu. I. Manin,*Holomorphic supergeometry and Yang-Mills superfields*, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 3–80 (Russian). MR**760997****[Rogers]**Alice Rogers,*A global theory of supermanifolds*, J. Math. Phys.**21**(1980), no. 6, 1352–1365. MR**574696**, https://doi.org/10.1063/1.524585**[Rothstein 1]**Mitchell J. Rothstein,*Deformations of complex supermanifolds*, Proc. Amer. Math. Soc.**95**(1985), no. 2, 255–260. MR**801334**, https://doi.org/10.1090/S0002-9939-1985-0801334-0**[Rothstein 2]**Mitchell J. Rothstein,*The axioms of supermanifolds and a new structure arising from them*, Trans. Amer. Math. Soc.**297**(1986), no. 1, 159–180. MR**849473**, https://doi.org/10.1090/S0002-9947-1986-0849473-8**[Rothstein 3]**-, private communication.**[OASV 1]**O. A. Sánchez Valenzuela,*Matrix computations in linear superalgebra*, Linear Algebra Appl. (to appear).**[OASV 2]**-,*On supervector bundles*(preprint).**[OASV 3]**Oscar Adolfo Sánchez-Valenzuela,*Linear supergroup actions. I. On the defining properties*, Trans. Amer. Math. Soc.**307**(1988), no. 2, 569–595. MR**940218**, https://doi.org/10.1090/S0002-9947-1988-0940218-1

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58A50,
14M15

Retrieve articles in all journals with MSC: 58A50, 14M15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0940219-3

Article copyright:
© Copyright 1988
American Mathematical Society