42,280 research outputs found
Some topics in K-theory
No full textIn part A (II) the uniqueness theorem for equivariant cohomology theories, (proved in part A (I)), is used to calculate the operation rings, Op(KG,KG^) and OP(KG^,KG^), (^ is I(G)-adic completion ; G is a finite group). Semi-groups, SG < KG, are introduced and Op(~SG(-), KG) is calculated in order to investigate (Op(KG), the ring of self-operations of KG. Finally Op(KG) is related to the other two rings of operations and any self-operation of KG is proved to be continuous with respect to the I(G)-adic topology.
In Part B some higher order operations in K-theory, called Massey products, are defined and proved to be the differentials in the Equivariant Kunneth Formula spectral sequence in K-theory.
In Part C the Rothenberg-Steenrod spectral sequences are used (i) to calculate the K-theory of conjugate bundles of Lie groups, (ii) to prove a small theorem on the K-theory of homogeneous spaces of Lie groups, and (iii) to calculate the homological dimension of R(H) as an R(G)-module, for an inclusion of Lie groups, H<G. As an example of (ii) the algebra K (SO(m)) and the operation ring, lim<--- K(SO(m)), are computed
Solvable Lie A-algebras.
No full textA finite-dimensional Lie algebra over a field is called an -algebra if all of its nilpotent subalgebras are abelian. This is analogous to the concept of an -group: a finite group with the property that all of its Sylow subgroups are abelian. These groups were first studied in the 1940s by Philip Hall, and are still studied today. Rather less is known about -algebras, though they have been studied and used by a number of authors. The purpose of this paper is to obtain more detailed results on the structure of solvable Lie -algebras. \par It is shown that they split over each term in their derived series. This leads to a decomposition of as where is an abelian subalgebra of and for each . It is shown that the ideals of relate nicely to this decomposition: if is an ideal of then . When is nilpotent we can locate the position of the maximal nilpotent subalgebras: if is a maximal nilpotent subalgebra of then where is a Cartan subalgebra of . \par If has a unique minimal ideal then . If, in addition, is strongly solvable the maximal nilpotent subalgebras of are and the Cartan subalgebras of (that is, the subalgebras that are complementary to .) Necessary and sufficient conditions are given for such an algebra to be an -algebra. Finally, more detailed structure results are given when the underlying field is algebraically closed
Nonabelian Cohomology of Compact Lie Groups
No full textGiven a Lie group G with finitely many components and a compact Lie group A which acts on G by automorphisms, we prove that there always exists an A-invariant maximal compact subgroup K of G, and that for every such K, the natural map H(1)(A,K) -> H(1)(A,G) is bijective. This generalizes a classical result of Serre and a recent result of the first and third named authors of the current paper.MathematicsSCI(E)0ARTICLE2231-2361
Elementary Lie Algebras and Lie A-Algebras.
No full textA finite-dimensional Lie algebra L over a field F is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. The present paper is primarily concerned with the classification of elementary Lie algebras. In particular, we provide a complete list in the case when F is algebraically closed and of characteristic different from 2,3, reduce the classification over fields of characteristic 0 to the description of elementary semisimple Lie algebras, and identify the latter in the case when F is the real field. Additionally it is shown that over fields of characteristic 0 every elementary Lie algebra is almost algebraic; in fact, if L has no non-zero semisimple ideals, then it is elementary if and only if it is an almost algebraic A-algebra
The theorem of Lie and hyperplane subalgebras of Lie algebras
No full textPoguntke D. The theorem of Lie and hyperplane subalgebras of Lie algebras. Geometriae Dedicata. 1992;43(1):83-91
Invariants of automorphic lie algebras
No full textAutomorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras.
The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms.
All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity.In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring
Automorphic Lie algebras with dihedral symmetry
No full textThe concept of automorphic Lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever–Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is sl2(C) and the poles of the automorphic Lie algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of automorphic Lie algebras with dihedral symmetry, valid for poles at exceptional and generic orbits
On upper modular subalgebras of a Lie algebra.
No full textThis paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper modular atom which is not an ideal. Finally it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a μ-algebra
Higher-dimensional automorphic lie algebras
No full textThe paper presents the complete classification of Automorphic Lie Algebras based on sln(C)sln(C) , where the symmetry group G is finite and acts on sln(C)sln(C) by inner automorphisms, sln(C)sln(C) has no trivial summands, and where the poles are in any of the exceptional G-orbits in C¯¯¯¯C¯ . A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on the one hand a powerful tool from the computational point of view; on the other, it opens new questions from an algebraic perspective (e.g. structure theory), which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that this class of Automorphic Lie Algebras associated with the TOY groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only; thus, they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring
Quasi-Kahler Chern-flat manifolds and complex 2-step nilpotent Lie algebras
No full textThe study of quasi-Kaehler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kaehler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-bi-invariant almost complex structures on Lie algebras to bi-invariant complex structures. Some natural algebraic problems are approached and some exotic examples are carefully describe
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