215,196 research outputs found
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
Elemi Lie Elmélet
No full textA Lie elmélet, a Lie-csoportok és Lie-algebrák elmélete, valamint
azok alkalmazásai, a matematika egy alapvető része. A második világháború
óta nagyon sok kutatás zajlott a matematika ezen területén. Azóta
kiderült, hogy a Lie elmélet számos területet, köztük például a parciális differenciálást,
a csoport és gyűrűelméletet, a számelméletet és a fizikát is érinti.
Szakdolgozatomban néhány mátrix Lie-csoportot vizsgálok meg. Ezen
csoportoknak fontos geometriai, és algebrai tulajdonságaik vannak. Egyik
fontos jellemzőjük az, hogy a tér mozgáscsopotjait szemléltetik. Az (1.3)-as
fejezetben láthatjuk például, hogy az SO(2) csoport elemei a sík origó körüli elforgatásait szemléltetik, ebből az következik, hogy az SO(2) csoport
nem más, mint az origó központú egység sugarú kör, azaz S1.
Az ortogonális csoportok mellett vizsgálom még az unitér csoportokat is,
melyek érdekessége, hogy elemeik nem valós, hanem komplex elemű mátrixok.
Valamint szemügyre veszem a szimplektikus csoportokat is, melyek
érdekessége, hogy elemeik kvaternió elemű mátrixok.
A Lie-csoportok eleminek és csoportműveleteinek elemzése után megkeresem
azok egységelembeli érintő vektorait és érintő terét. Ez azért fontos,
mert az exponenciális leképezés egy Lie-csoport egységelembeli érintő terét
homomorf módon leképezi a Lie-csoportra. Tehát az exponenciális leképezés
homomorfizmus a Lie-csoport és annak egységbeli érintő tere között.
A Lie-csoport egység elemre vonatkozó érintő tere azonban nem csak az
exponenciális leképezés miatt fontos. Egy Lie-csoport egységelembeli érintő
tere vektorteret alkot az R felett. Értelmezzünk egy [U, V ] = UV − V U
Lie zárójel műveletet ezen vektortér elemeire vonatkozóan. Ha ezen vektortér
a műveletre nézve zárt, akkor a Lie-csoport egységelembeli érintő tere a
rajta értelmezett Lie zárójel művelettel együtt, a Lie-csoport Lie-algebráját
alkotja.gjMatematikaBs
Almost nilpotent Lie algebras
No full textThroughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised
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
C-Ideals of Lie Algebras.
No full textA subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B \cap C \leq B_L, where B_L is the largest ideal of L contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal
Braided Lie bialgebras associated to Kac-Moody algebras
No full textBraided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra sl3
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
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
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
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