Let gl,,（R） be the general linear Lie algebra of all n×n matrices over a unital commutative ring R with 2 invertible, dn（R） be the Cartan subalgebra of gln（R） of all diagonal matrices. The maximal subalgebr...Let gl,,（R） be the general linear Lie algebra of all n×n matrices over a unital commutative ring R with 2 invertible, dn（R） be the Cartan subalgebra of gln（R） of all diagonal matrices. The maximal subalgebras of gln（R） that contain dn（F：） are classified completely.展开更多
In this paper, we determine all maximal graded subalgebras of the general linear Lie superalgebras containing the standard Cartan subalgebras over a unital supercommutative superring with 2 invertible.
Let F be a field with char F = 2, l a maximal nilpotent subalgebra of the symplectic algebra sp（2m,F）. In this paper, we characterize linear maps of l which preserve zero Lie brackets in both directions. It is shown...Let F be a field with char F = 2, l a maximal nilpotent subalgebra of the symplectic algebra sp（2m,F）. In this paper, we characterize linear maps of l which preserve zero Lie brackets in both directions. It is shown that for m ≥ 4, a map φ of l preserves zero Lie brackets in both directions if and only if φ = ψcσT0λαφdηf, where ψc,σT0,λα,φd,ηf are the standard maps preserving zero Lie brackets in both directions.展开更多
Given a free ergodic action of a discrete abelian group G on a measure space (X, 7), the crossed product LX (X, 7)p G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about t...Given a free ergodic action of a discrete abelian group G on a measure space (X, 7), the crossed product LX (X, 7)p G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra.展开更多
Let $ \mathcal{L} $ (F ?) × α ? be the crossed product von Neumann algebra of the free group factor $ \mathcal{L} $ (F ?), associated with the left regular representation λ of the free group F ? with the set {u...Let $ \mathcal{L} $ (F ?) × α ? be the crossed product von Neumann algebra of the free group factor $ \mathcal{L} $ (F ?), associated with the left regular representation λ of the free group F ? with the set {u r : r ∈ ?} of generators, by an automorphism α defined by α(λ(u r )) = exp(2πri)λ(u r ), where ? is the rational number set. We show that $ \mathcal{L} $ (F ?) × α ? is a wΓ factor, and for each r ∈ ?, the von Neumann subalgebra $ \mathcal{A}_r $ generated in $ \mathcal{L} $ (F ?) × α ? by λ(u r ) and υ is maximal injective, where υ is the unitary implementing the automorphism α. In particular, $ \mathcal{L} $ (F ?) × α ? is a wΓ factor with a maximal abelian selfadjoint subalgebra $ \mathcal{A}_0 $ which cannot be contained in any hyperfinite type II1 subfactor of $ \mathcal{L} $ (F ?) × α ?. This gives a counterexample of Kadison’s problem in the case of wΓ factor.展开更多
Ge(2003)asked the question whether LF∞can be embedded into LF2 as a maximal subfactor.We answer it affirmatively with three different approaches,all containing the same key ingredient:the existence of maximal subgrou...Ge(2003)asked the question whether LF∞can be embedded into LF2 as a maximal subfactor.We answer it affirmatively with three different approaches,all containing the same key ingredient:the existence of maximal subgroups with infinite index.We also show that point stabilizer subgroups for every faithful,4-transitive action on an infinite set give rise to maximal von Neumann subalgebras.By combining this with the known results on constructing faithful,highly transitive actions,we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.展开更多
In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed th...In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.展开更多
基金supported by National Natural Science Foundation of China (Grant No.11171343)the Fundamental Research Funds for the Central Universities (Grant No. 2010LKSX05)
文摘Let gl,,（R） be the general linear Lie algebra of all n×n matrices over a unital commutative ring R with 2 invertible, dn（R） be the Cartan subalgebra of gln（R） of all diagonal matrices. The maximal subalgebras of gln（R） that contain dn（F：） are classified completely.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1117105511471090)the Natural Science Foundation of the Education Department of Heilongjiang Province(Grant No.12521158)
文摘In this paper, we determine all maximal graded subalgebras of the general linear Lie superalgebras containing the standard Cartan subalgebras over a unital supercommutative superring with 2 invertible.
基金Supported by the Doctor Foundation of Henan Polytechnic University (Grant No.B2010-93)the Natural Science Research Program of Education Department of Henan Province (Grant No.2011B110016)+1 种基金the Natural Science Foundation of Henan Province (Grant No. 112300410120)Applied Mathematics Provincial-level Key Discipline of Henan Province
文摘Let F be a field with char F = 2, l a maximal nilpotent subalgebra of the symplectic algebra sp（2m,F）. In this paper, we characterize linear maps of l which preserve zero Lie brackets in both directions. It is shown that for m ≥ 4, a map φ of l preserves zero Lie brackets in both directions if and only if φ = ψcσT0λαφdηf, where ψc,σT0,λα,φd,ηf are the standard maps preserving zero Lie brackets in both directions.
文摘Given a free ergodic action of a discrete abelian group G on a measure space (X, 7), the crossed product LX (X, 7)p G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra.
基金the National Natural Science Foundation of China (Grant Nos. 10201007, A0324614)the Natural Science Foundation of Shandong Province (Grant No. Y2006A03)
文摘Let $ \mathcal{L} $ (F ?) × α ? be the crossed product von Neumann algebra of the free group factor $ \mathcal{L} $ (F ?), associated with the left regular representation λ of the free group F ? with the set {u r : r ∈ ?} of generators, by an automorphism α defined by α(λ(u r )) = exp(2πri)λ(u r ), where ? is the rational number set. We show that $ \mathcal{L} $ (F ?) × α ? is a wΓ factor, and for each r ∈ ?, the von Neumann subalgebra $ \mathcal{A}_r $ generated in $ \mathcal{L} $ (F ?) × α ? by λ(u r ) and υ is maximal injective, where υ is the unitary implementing the automorphism α. In particular, $ \mathcal{L} $ (F ?) × α ? is a wΓ factor with a maximal abelian selfadjoint subalgebra $ \mathcal{A}_0 $ which cannot be contained in any hyperfinite type II1 subfactor of $ \mathcal{L} $ (F ?) × α ?. This gives a counterexample of Kadison’s problem in the case of wΓ factor.
基金the National Science Center(NCN)(Grant No.2014/14/E/ST1/00525)Institute of Mathematics,Polish Academy of Sciences(IMPAN)from the Simons Foundation(Grant No.346300)the Matching 2015-2019 Polish Ministry of Science and Higher Education(MNiSW)Fund,and the Research Foundation-Flanders-Polish Academy of Sciences(FWO-PAN).
文摘Ge(2003)asked the question whether LF∞can be embedded into LF2 as a maximal subfactor.We answer it affirmatively with three different approaches,all containing the same key ingredient:the existence of maximal subgroups with infinite index.We also show that point stabilizer subgroups for every faithful,4-transitive action on an infinite set give rise to maximal von Neumann subalgebras.By combining this with the known results on constructing faithful,highly transitive actions,we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.
基金This work is supported by the National Natural Science Foundation of China (10301004)
文摘In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.
