Computation with linear algebraic groups / Willem Adriaan de Graaf
- Author:
- De Graaf, Willem A.
- Published:
- Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
- Physical Description:
- xiv, 327 pages ; 24 cm.
- Series:
- Contents:
- Machine generated contents note: 1.Closed Sets in Affine Space -- 1.1.Closed sets in affine space -- 1.1.1.Affine space and polynomial maps -- 1.1.2.Closed sets -- 1.1.3.Closed sets and ideals -- 1.1.4.Coordinate ring and regular maps -- 1.1.5.Irreducible closed sets -- 1.1.6.Products of closed sets -- 1.2.Tangent space -- 1.2.1.Dual numbers and tangent vectors -- 1.2.2.Differentials -- 1.2.3.Tangent space of a product of closed sets -- 1.3.Dimension -- 1.3.1.Specializations and generic points -- 1.4.Dominant maps -- 1.5.Grobner bases -- 1.6.Elimination -- 1.7.Dimension of ideals -- 1.8.Ideal quotients -- 1.9.Radical of an ideal -- 1.10.Notes -- 2.Lie Algebras -- 2.1.Basic constructions -- 2.1.1.Algebras -- 2.1.2.Homomorphisms and representations -- 2.1.3.Structure constants -- 2.1.4.Centre, centralizer and normalizer -- 2.2.Jordan decomposition of a linear transformation -- 2.3.Derivations -- 2.4.Nilpotency -- 2.4.1.Engel's theorem -- 2.4.2.Nilradicals -- 2.5.Cartan subalgebras -- 2.5.1.Primary decomposition -- 2.5.2.Fitting decomposition -- 2.5.3.Cartan subalgebras of Lie algebras -- 2.5.4.Weights and roots -- 2.6.Solvability -- 2.6.1.Lie's theorem -- 2.6.2.Cartan's criterion for solvability -- 2.6.3.Computing the solvable radical -- 2.6.4.Finding a non-nilpotent element -- 2.6.5.Computing the nilradical -- 2.7.Semisimple Lie algebras -- 2.7.1.Derivations and the Jordan decomposition -- 2.7.2.Levi's theorem -- 2.8.Root systems -- 2.8.1.Cartan matrices -- 2.8.2.Root systems -- 2.8.3.The Weyl group -- 2.9.Classification of simple Lie algebras -- 2.9.1.Representations of sl(2, k) -- 2.9.2.From Lie algebra to root system -- 2.9.3.Canonical generators and isomorphisms -- 2.9.4.Structure constants of semisimple Lie algebras -- 2.10.Universal enveloping algebras -- 2.10.1.Poincare-Birkhoff-Witt theorem -- 2.10.2.Left ideals of universal enveloping algebras -- 2.10.3.Integral forms -- 2.11.Representations of semisimple Lie algebras -- 2.11.1.Highest weight modules -- 2.11.2.Classification of irreducible modules -- 2.11.3.Path model -- 2.11.4.Constructing irreducible modules -- 2.12.Reductive Lie algebras -- 2.13.The Jacobson-Morozov theorem -- 2.14.Notes -- 3.Linear Algebraic Groups: Basic Constructions -- 3.1.Definition and first properties -- 3.2.Connected components -- 3.3.Semidirect products -- 3.4.The Lie algebra of an algebraic group -- 3.5.Subgroups and subalgebras -- 3.6.Examples -- 3.7.Morphisms and representations -- 3.8.Adjoint representation -- 3.9.Characters and diagonalizable groups -- 3.10.Jordan decomposition -- 3.10.1.In the Lie algebra of an algebraic group -- 3.10.2.In an algebraic group -- 3.11.The unipotent radical -- 3.12.Algebraic groups acting on closed sets -- 3.13.Specification of an algebraic group -- 3.14.Notes -- 4.Algebraic Groups and Their Lie Algebras -- 4.1.G(δ) -- 4.2.The Lie correspondence -- 4.3.Algebraic Lie algebras -- 4.3.1.The algebraic hull -- 4.3.2.Unipotent groups -- 4.3.3.The structure of algebraic Lie algebras -- 4.4.Computing the algebraic hull -- 4.5.Computing defining polynomials for an algebraic group from its Lie algebra -- 4.5.1.Unipotent case -- 4.5.2.Diagonalizable case -- 4.5.3.General case -- 4.6.The algebraic group generated by a given set -- 4.6.1.Unipotent case -- 4.6.2.Semisimple case -- 4.6.3.The algorithm -- 4.7.Generators, centralizers and normalizers -- 4.7.1.Generating sets -- 4.7.2.The centralizer of an algebraic subgroup -- 4.7.3.The normalizer of an algebraic subgroup -- 4.8.Orbit closures -- 4.9.Notes -- 5.Semisimple Algebraic Groups -- 5.1.Groups defined by certain generators and relations -- 5.2.Chevalley groups -- 5.2.1.Admissible lattices -- 5.2.2.Chevalley's commutator formula -- 5.2.3.Chevalley groups -- 5.2.4.Bruhat decomposition -- 5.2.5.Presentation of G -- 5.3.Semisimple algebraic groups -- 5.4.Root data -- 5.5.Irreducible representations in positive characteristic -- 5.6.Multiplication in a semisimple algebraic group -- 5.7.From matrix to word -- 5.8.Reductive algebraic groups -- 5.9.Regular subalgebras of semisimple Lie algebras -- 5.10.Notes -- 6.Generators of Arithmetic Groups -- 6.1.Brief introduction to arithmetic groups -- 6.2.Algorithms for lattices -- 6.3.Arithmetic subgroups of unipotent groups -- 6.3.1.L-groups -- 6.3.2.The derived representation -- 6.3.3.The algorithm -- 6.4.Arithmetic subgroups of tori -- 6.4.1.Unit groups of orders in toral matrix algebras -- 6.4.2.Generators of arithmetic subgroups of tori -- 6.5.Notes -- 7.Invariants of Algebraic Groups -- 7.1.The Derksen ideal -- 7.2.The invariant field -- 7.3.Computing invariants of reductive groups -- 7.3.1.Reynolds operator -- 7.3.2.Computing generators of the invariant ring -- 7.4.The nullcone -- 7.4.1.Characteristics -- 7.4.2.Stratification of the nullcone -- 7.4.3.Computing characteristics of the strata -- 7.5.Notes -- 8.Nilpotent Orbits -- 8.1.Weighted Dynkin diagrams -- 8.2.Computing representatives and weighted Dynkin diagrams -- 8.3.θ-groups -- 8.3.1.Cyclic gradings of g -- 8.3.2.The nilpotent orbits of a θ-group -- 8.3.3.Carrier algebras -- 8.4.Listing the nilpotent orbits of a θ-group -- 8.4.1.Using sl2-triples -- 8.4.2.Using carrier algebras -- 8.5.Closures of nilpotent orbits -- 8.5.1.Deciding emptiness -- 8.5.2.Traversing a W0-orbit -- 8.6.Notes.
- Subject(s):
- ISBN:
- 9781498722902 hardcover
1498722903 hardcover - Bibliography Note:
- Includes bibliographical references and indexes.
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