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1.
Paul Levy 《Transformation Groups》2009,14(2):417-461
We generalize the basic results of Vinberg’s θ-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify
the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated
to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded
Lie algebra (in zero or odd positive characteristic), confirming a conjecture of Popov in this case. 相似文献
2.
The action of an affine algebraic group G on an algebraic variety V can be
differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf
of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a
homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety
V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully
and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way
that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the
start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ
allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting
cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing
form. 相似文献
3.
We say that a Lie algebra g is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras C n ? u(n), n ≥ 1, are rigid. On the other hand, a Lie algebra which surjects onto the three-dimensional Heisenberg algebra is not rigid. For Lie algebras of dimension ≤ 3 and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity. 相似文献
4.
Lie rings that admit an automorphism of order 4 with few fixed points are considered. For a Lie ring (algebra) L admitting
an automorphism of order 4 with a finite number m of fixed points (with a finite-dimensional subalgebra of fized points of
dimension m), it is proved that the subring 4L (algebra L) contains an ideal M with a subring of m-bounded index in the additive
group of M (a subalgebra of m-bounded codimension), which is nilpotent of class bounded by some constant. It is also shown
that, under the same premise, the factor-ring 4L/M (factor-algebra L/M) contains a subring of m-bounded index in the additive
group of 4L/M (a subalgebra of m-bounded codimension), which is nilpotent of class ≤2. Moreover, L has a subring of m-bounded
index in the additive group of L (a subalgebra of m-bounded codimension), which is soluble of derived length bounded by a
constant.
Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000.
Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 41–78, January–February, 1996. 相似文献
5.
Joseph P. Brennan 《代数通讯》2013,41(15):6409-6415
The commuting variety of pairs of symmetric commuting matrices over an algebraically closed field of characteristic zero is normal. This is shown through the direct computation of the singular locus of the variety which is known [3] to be a complete intersection. 相似文献
6.
Karl-Hermann Neeb 《manuscripta mathematica》2001,104(3):359-381
In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition
which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting
relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie
groups which are strong K?hler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras,
such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the
representation of the almost reductive algebra on the locally nilpotent ideal.
Received: 2 August 2000 / Revised version: 10 January 2001 相似文献
7.
For a reductive Lie algbera over an algbraically closed field of charasteristic zero, we consider a Borel subgroup B of its adjoint group, a Cartan subalgebra contained in the Lie algebra of B and the closure X of its orbit under B in the Grassmannian. The variety X plays an important role in the study of the commuting variety. In this note, we prove that X is Gorenstein with rational singularities. 相似文献
8.
J.-H. Eschenburg A. Kollross R. Tribuzy 《Differential Geometry and its Applications》2009,27(6):691-695
Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal. 相似文献
9.
10.
A. P. Petravchuk 《Ukrainian Mathematical Journal》1999,51(5):707-715
We consider a Lie algebraL over an arbitrary field that is decomposable into the sumL=A+B of an almost Abelian subalgebraA and a subalgebraB finite-dimensional over its center. We prove that this algebra is almost solvable, i.e., it contains a solvable ideal of
finite codimension. In particular, the sum of the Abelian and almost Abelian Lie algebras is an almost solvable Lie algebra.
Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 636–644, May, 1999. 相似文献
11.
本文主要研究了强半单的n-李代数的表示,证明了强半单的n-李代数的表示(V,ρ)可转化为一个约化李代数Lρ(V)的表示,并证明了不变线性形等其它相关性质. 相似文献
12.
《代数通讯》2013,41(7):2219-2229
ABSTRACT In this article, we focus on the result of V.F.R. Jones which says that the partition algebra is the algebra of all transformations commuting with the action of the symmetric group on tensor products of its permutation representation. In particular, we restrict the action of the symmetric group to the action of the alternating group. In this context, we compute a basis for the centralizer algebra and show when the centralizer is isomorphic to the partition algebra. 相似文献
13.
V. M. Petrogradsky 《代数通讯》2013,41(3):918-928
We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X). 相似文献
14.
Jennifer R. Daniel 《代数通讯》2013,41(5):1758-1788
In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties. 相似文献
15.
Xiaomin Tang 《Linear and Multilinear Algebra》2018,66(2):250-259
In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra. 相似文献
16.
Selene Sánchez-Flores 《代数通讯》2013,41(9):3410-3434
We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops. 相似文献
17.
Gorenstein rings occur in a multitude of different guises: as rings of invariants, as coordinate rings of certain determinantal varieties and symmetric semigroup curves, as representatives of linkage classes, and so on. In an attempt to unify this motley collection of examples (at least for finite projective dimension) one seeks a generic free resolution whose specializations yield all examples of given embedding codimension. The present paper describes a resolution for codimension four, not generic, but general enough to encompass many diverse examples. The structure of this resolution is intimately related to the differential, graded, commutative algebra that it supports, and to the deformation theory of codimension four Gorenstein algebras. These ideas are brought together in the determination of the singular locus of certain codimension four Gorenstein varieties. More generally they suggest a classification of codimension four Gorenstein rings that begins to impose some order on the examples.Research supported in part by University of Kansas General Research Allocation # 3093-XO-0038Research supported in part by a University of Tennessee Summer Faculty Development Grant 相似文献
18.
Roberta Basili 《代数通讯》2017,45(4):1533-1541
It is known that the variety of the pairs of n×n commuting upper triangular matrices is not a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n>m. We also show that m<18 and that m could be found by determining the dimension of the variety of the pairs of commuting strictly upper triangular matrices. Then, we define an embedding of any commuting variety into a grassmannian of subspaces of codimension 2. 相似文献
19.
设G是三维实李代数so(3)的复化李代数,A=C[T_1~(±1),t_2~(±2)]为复数域上的多项式环.设L(t_1,t_2,1)=G(?)_cA,d_1,d_2为L(t_1,t_2,1)的度导子.最近我们研究了李代数L(t_1,t_2,1)的自同构群结构.研究扭的Multi-loop代数L(t_1,t_2,1)(?)(Cd_1(?)Cd_2)的导子以及triple导子结构. 相似文献
20.
Dmitri A. Shmel'kin 《Geometriae Dedicata》1998,72(2):189-216
In order to describe explicitly the algebra of invariants for a non-connected reductive subgroup G GL(V) we apply the method of strata. For this we describe codimension 1 strata of the quotient V//G and study the normality property of their closures. We find some criteria for k[V]G to be polynomial or a hypersurface. Then we apply these results to complete the classification [Sh] of nonconnected simple groups G such that k[V]G is polynomial. 相似文献