The Experts below are selected from a list of 29064 Experts worldwide ranked by ideXlab platform
Martha Precup - One of the best experts on this subject based on the ideXlab platform.
-
The singular locus of Semisimple Hessenberg varieties
Journal of Algebra, 2019Co-Authors: Erik Insko, Martha PrecupAbstract:Abstract Although regular Semisimple Hessenberg varieties are smooth and irreducible, Semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of Semisimple Hessenberg varieties corresponding to the standard Hessenberg space in all Lie types. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a Semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all Semisimple Hessenberg varieties.
-
The singular locus of Semisimple Hessenberg varieties
arXiv: Algebraic Geometry, 2017Co-Authors: Erik Insko, Martha PrecupAbstract:Although regular Semisimple Hessenberg varieties are smooth and irreducible, Semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of Semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a Semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all Semisimple Hessenberg varieties.
E. B. Vinberg - One of the best experts on this subject based on the ideXlab platform.
-
Cyclic elements in Semisimple lie algebras
Transformation Groups, 2013Co-Authors: A. G. Elashvili, E. B. VinbergAbstract:We develop a theory of cyclic elements in Semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular Semisimple. In particular, we classfiy all nilpotents giving rise to Semisimple and regular Semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups.
-
Real Semisimple Lie Groups
Lie Groups and Algebraic Groups, 1990Co-Authors: Arkadij L. Onishchik, E. B. VinbergAbstract:Our study of real Semisimple Lie groups and algebras is based on the theory of complex Semisimple Lie groups developed in Ch. 4. This is possible because the complexification of a real Semisimple Lie algebra is also Semisimple (see 1.4.7). However, the correspondence between real and complex Semisimple Lie algebras established with the help of the complexification is not one-to-one; any complex Semisimple Lie group has at least two non-isomorphic real forms. As it turns out, to describe the real forms of a given complex Semisimple Lie algebra g is the same as to classify the involutive automorphisms of g up to conjugacy in Aut g. This classification is easily obtained from the results of 4.4. The global classification of real Semisimple Lie groups makes use of the so-called Cartan decomposition of these groups which also plays an important role in various applications of the Lie group theory.
Erik Insko - One of the best experts on this subject based on the ideXlab platform.
-
The singular locus of Semisimple Hessenberg varieties
Journal of Algebra, 2019Co-Authors: Erik Insko, Martha PrecupAbstract:Abstract Although regular Semisimple Hessenberg varieties are smooth and irreducible, Semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of Semisimple Hessenberg varieties corresponding to the standard Hessenberg space in all Lie types. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a Semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all Semisimple Hessenberg varieties.
-
The singular locus of Semisimple Hessenberg varieties
arXiv: Algebraic Geometry, 2017Co-Authors: Erik Insko, Martha PrecupAbstract:Although regular Semisimple Hessenberg varieties are smooth and irreducible, Semisimple Hessenberg varieties are not necessarily smooth in general. In this paper we determine the irreducible components of Semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, which constitute the singular locus. We conclude with an example of a Semisimple Hessenberg variety corresponding to another Hessenberg space which is singular and irreducible, showing that results of this nature do not hold for all Semisimple Hessenberg varieties.
Thorsten Holm - One of the best experts on this subject based on the ideXlab platform.
-
The Structure of Semisimple Algebras: The Artin–Wedderburn Theorem
Springer Undergraduate Mathematics Series, 2018Co-Authors: Karin Erdmann, Thorsten HolmAbstract:We prove the fundamental Artin-Wedderburn Theorem, which completely classifies Semisimple algebras. We have seen before that finite direct products of matrix algebras are Semisimple algebras. When the coefficient field is algebraically closed the Artin-Wedderburn theorem shows that every Semisimple algebra is isomorphic to such an algebra. In general, a Semisimple algebra is isomorphic to a direct product of matrix algebras but where matrix blocks over division algebras may occur.
-
Semisimple Group Algebras and Maschke’s Theorem
Springer Undergraduate Mathematics Series, 2018Co-Authors: Karin Erdmann, Thorsten HolmAbstract:We investigate when a group algebra of a finite group is Semisimple, this is answered by Maschke’s theorem. Namely, a group algebra of a finite group over a field is Semisimple if and only the characteristic of the coefficient field does not divide the order of the group. We prove this theorem, and we discuss connections between the Artin-Wedderburn decomposition of a Semisimple group algebra and properties of the group.
-
Semisimple Modules and Semisimple Algebras
Springer Undergraduate Mathematics Series, 2018Co-Authors: Karin Erdmann, Thorsten HolmAbstract:Since simple modules are the ‘building blocks’ for arbitrary finite-dimensional modules, one would like to understand how modules are built up from simple modules. In this chapter we study modules which are direct sums of simple modules, this leads to the theory of Semisimple modules. For some algebras, every module is a direct sum of simple modules. These algebras are said to be Semisimple, and they have important applications. In this chapter, as an exception, we deal with arbitrary direct sums of modules. We view this as an opportunity to prove a result which does not need finiteness assumptions. The only new tool necessary is Zorn’s lemma.
Roman Avdeev - One of the best experts on this subject based on the ideXlab platform.
-
Normalizers of solvable spherical subgroups
Mathematical Notes, 2013Co-Authors: Roman AvdeevAbstract:For an arbitrary connected solvable spherical subgroup H of a connected Semisimple algebraic group G, we compute the group NG(H), the normalizer of H in G. Therebywe complete a classification of all (not necessarily connected) solvable spherical subgroups in Semisimple algebraic groups.
-
Normalizers of solvable spherical subgroups
Mathematical Notes, 2013Co-Authors: Roman AvdeevAbstract:For an arbitrary connected solvable spherical subgroup H of a connected Semisimple algebraic group G we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in Semisimple algebraic groups.Comment: v2: 12 pages, small correction
