Linear supergroup actions. I. On the defining properties
Author: Oscar Adolfo Sánchez-Valenzuela
Journal: Trans. Amer. Math. Soc. 307 (1988), 569-595
MSC: Primary 58A50
MathSciNet review: 940218
Abstract: This paper studies the notions of linearity and bilinearity in the category of supermanifolds. Following the work begun by [OASV2], we deal with supermanifoldifications of supervector spaces. The -module operations are defined componentwise. The linearity and bilinearity properties are stated by requiring commutativity of some appropriate diagrams of supermanifold morphisms. It is proved that both linear and bilinear supermanifold morphisms are completely determined by their underlying continuous maps, which in turn have to be linear (resp., bilinear) in the usual sense. It is observed that whereas linear supermanifold morphisms are vector bundle maps, bilinear supermanifold morphisms are not. A natural generalization of the bilinear evaluation map is given and some applications pointing toward the notions of linear supergroup actions and adjoint and coadjoint actions are briefly discussed.
- [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
- [OASV1] O. A. Sánchez Valenzuela, Matrix computations in linear superalgebra, Linear Algebra Appl. (to appear).
- [OASV2] -, On supervector bundles (preprint).
- B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin and New York, 1977. MR 0580292 (58:28326)
- D. Leĭtes, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1-64. MR 565567 (81j:58003)
- Yu. Manin, Holomorphic supergeometry and Yang-Mills superfields (preprint). MR 760997 (86e:32038)
- O. A. Sánchez Valenzuela, Matrix computations in linear superalgebra, Linear Algebra Appl. (to appear).
- -, On supervector bundles (preprint).
Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A50
Retrieve articles in all journals with MSC: 58A50