1,720,993 research outputs found
Etale descent of derivations
Get PDFWe study étale descent of derivations of algebras with values in a module. The algebras under consideration are twisted forms of algebras over rings, and apply to all classes of algebras, notably associative and Lie algebras, such as the multiloop algebras that appear in the construction of extended affine Lie algebras. The main result is Theorem 2.7.Fil: Neher, Erhard. University of Ottawa; CanadáFil: Pianzola, Arturo. University of Alberta; Canadá. Universidad Centro de Altos Estudios en Ciencias Exactas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
On conjugacy of Cartan subalgebras in extended affine Lie algebras
Get PDFThat finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate. For symmetrizable Kac-Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac-Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac-Moody Lie algebras.Fil: Chernousov, Vladimir. University of Alberta; Canadá. Natural Sciences and Engineering Research Council; CanadáFil: Neher, Erhard. University of Ottawa; Canadá. Natural Sciences and Engineering Research Council; CanadáFil: Pianzola, Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. University of Alberta; Canadá. Universidad Centro de Altos Estudios en Ciencia Exactas. Departamento de Matemáticas; ArgentinaFil: Yahorau, Uladzimir. University of Alberta; Canadá. Natural Sciences and Engineering Research Council; Canad
Derivations, invariant forms and the second homology group of orthosymplectic Lie superalgebras.
No full textWe develop the description of the derivation algebras of orthosymplectic Lie super-algebras over supercommutative, associative superrings containing ½ and determine conditions under which the derivation algebra can be written as a semidirect product of the inner and the outer derivations. We then describe the supersymmetric invariant forms of the elementary orthosymplectic Lie superalgebra and determine the outer derivations which are skew with respect to a given supersymmetric invariant form. Finally, we describe the universal central extension and its centre, the second homology group, of the elementary orthosymplectic Lie superalgebra. The original motivation for this comes from the theory of extended affine Lie algebras. Specialized to the orthosymplectic Lie superalgebras representing the centreless cores of extended affine Lie algebras of type B and D, the above descriptions are the necessary and sufficient building blocks for the construction of an extended affine Lie algebra of type B and D from its centreless core
Derivations, invariant forms and the second homology group of orthosymplectic Lie superalgebras.
No full textWe develop the description of the derivation algebras of orthosymplectic Lie super-algebras over supercommutative, associative superrings containing ½ and determine conditions under which the derivation algebra can be written as a semidirect product of the inner and the outer derivations. We then describe the supersymmetric invariant forms of the elementary orthosymplectic Lie superalgebra and determine the outer derivations which are skew with respect to a given supersymmetric invariant form. Finally, we describe the universal central extension and its centre, the second homology group, of the elementary orthosymplectic Lie superalgebra. The original motivation for this comes from the theory of extended affine Lie algebras. Specialized to the orthosymplectic Lie superalgebras representing the centreless cores of extended affine Lie algebras of type B and D, the above descriptions are the necessary and sufficient building blocks for the construction of an extended affine Lie algebra of type B and D from its centreless core
Trace Formulas, Invariant Bilinear Forms and Dynkin Indices of Lie Algebra Representations Over Rings
No full textThe trace form gives a connection between the representation ring and the space of invariant bilinear forms of a Lie algebra . This thesis reviews the definition of the trace of an endomorphism of a finitely generated projective module over a commutative ring . We then use this to look at the trace form of a finitely generated projective representation of a Lie algebra over and its representation ring. While doing so, we prove a few trace formulas which are useful in the theory of the Dynkin index, an invariant introduced by Dynkin in 1952 to study homomorphisms between simple Lie algebras
The Hyperbolic Formal Affine Demazure Algebra
No full textIn this thesis, we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malagón-López, Savage and Zainoulline in two directions. First, we introduce and study the notion of formal Demazure lattices of a Kac-Moody root system and show that the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra
Homogeneous Projective Varieties of Rank 2 Groups
No full textRoot systems are a fundamental concept in the theory of Lie algebra. In this thesis, we will use two different kind of graphs to represent the group generated by reflections acting on the elements of the root system. The root
systems we are interested in are those of type A2, B2 and G2. After drawing the graphs, we will study the algebraic groups corresponding to those root systems. We will use three different techniques to give a geometric description of the homogeneous spaces G/P where G is the algebraic group corresponding to the root system and P is one of its parabolic subgroup. Finally, we will make a link between the graphs and the multiplication of
basis elements in the Chow group CH(G/P)
Homogeneous Projective Varieties of Rank 2 Groups
No full textRoot systems are a fundamental concept in the theory of Lie algebra. In this thesis, we will use two different kind of graphs to represent the group generated by reflections acting on the elements of the root system. The root
systems we are interested in are those of type A2, B2 and G2. After drawing the graphs, we will study the algebraic groups corresponding to those root systems. We will use three different techniques to give a geometric description of the homogeneous spaces G/P where G is the algebraic group corresponding to the root system and P is one of its parabolic subgroup. Finally, we will make a link between the graphs and the multiplication of
basis elements in the Chow group CH(G/P)
The Hyperbolic Formal Affine Demazure Algebra
No full textIn this thesis, we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malagón-López, Savage and Zainoulline in two directions. First, we introduce and study the notion of formal Demazure lattices of a Kac-Moody root system and show that the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra
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