Some properties of semifinite tracial subalgebras

We obtained some characterizations of Jensen type inequalities in tracial subalgebras and gave some characterizations of subdiagonal algebras of semifinite von Neumann algebras.     

Polynomial detection of matrix subalgebras

The double Capelli polynomial of total degree is

It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree is a polynomial identity for . (Here, is a field and is the algebra of matrices over .) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree is a polynomial identity for any proper -subalgebra of . Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.

     

PBW-bases of coideal subalgebras and a freeness theorem

Let be a character Hopf algebra. Every right coideal subalgebra U that contains the coradical has a PBW-basis which can be extended up to a PBW-basis of If additionally U is a bosonization of an invariant with respect to the left adjoint action subalgebra, then is a free left (and right) U-module with a free PBW-basis over U. These results remain valid if is a braided Hopf algebra generated by a categorically ordered subset of primitive elements. If the ground field is algebraically closed, the results are still true provided that is a pointed Hopf algebra with commutative coradical and is generated over the coradical by a direct sum of finite-dimensional Yetter-Drinfeld submodules of skew primitive elements.

     

The Kac Jordan superalgebra: automorphisms and maximal subalgebras

In this note the group of automorphisms of the Kac Jordan superalgebra is described and used to classify the maximal subalgebras.

     

幂零矩阵子代数

本文较详细地描述了幂零矩阵子代数,并且获得了一些有趣的结果.     

Serial subalgebras of finitary Lie algebras

A Lie subalgebra of is said to be finitary if it consists of elements of finite rank. We show that, if acts irreducibly on , and if is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of acts irreducibly on too. When , it follows that the locally solvable radical of such is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic are hyperabelian.

     

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.     

Conjugation-uniqueness of exact Borel subalgebras

It is proved that the exact Borel subalgebras of a basic quasi-hereditary algebra are conjugate to each other. Moreover, the inner automorphism group of a basic quasi-hereditary algebra acts transitively on the set of its exact Borel subalgebras. Project supported by the National Natural Science Foundation of China (Grant No. 19771070). and partly supported by the NSF of Hainan Province (Grant No. 19702) and by the Natural Science Foundation of Education Department of Hainan province.     

Strong conjugation actions induced by flags of exact Borel subalgebras

Let A be a quasi-hereditary algebra with a strong exact Borel subalgebra. It is proved that for any standard semisimple subalgebra T there exists an exact Borel subalgebra B of A such that T is a maximal semisimple subalgebra of B. It is shown that the maximal length of flags of exact Borel subalgebras of A is the difference of the radium and the rank of Grothendic group of A plus 2. The number of conjugation-classes of exact Borel subalgebras is 1 if and only if A is basic; the number is 2 if and only if A is semisimple. For all other cases, this number is 0 or no less than 3. Furthermore, it is shown that all the exact Borel subalgebras are idempotent-conjugate to each other, that is, for any exact Borel subalgebras B and C of A, there exists an idempotent e of A, and an invertible element u of A, such that eBe = u^-1eCeu.     

辛代数的极大幂零子代数的保李括积零的线性映射

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.     

On the theta completions for maximal subalgebras

Let L be a finite dimensional Lie algebra. Then for a maximal subalgebra M of L, a 𝜃-completion for M is a subalgebra C of L such that CM and M_L?C and C∕M_L contains no non-zero ideal of L∕M_L, properly. And a 𝜃-completion C of M is said to be a strong 𝜃-completion, if C = L or there exists a subalgebra B of L such that C be maximal in B and B is not a 𝜃-completion for M. These are analogous to the concepts of 𝜃-completion and strong 𝜃-completion of a maximal subgroup of a finite group. Now, we consider the influence of these concepts on the structure of a finite dimensional Lie algebra.     

Engel Subalgebras of n-Lie Algebras

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.     

Slant joint antieigenvalues and antieigenvectors of operators in normal subalgebras

We will study the slant joint antieigenvalues and antieigenvectors of pairs of operators that belong to the same closed normal subalgebra of the algebra of bounded operators on a separable Hilbert space. This extends the slant antieigenvalue theory from single normal operators to pairs of normal operators. Our results may be viewed as extensions of the Greub-Rheinboldt inequality from two positive operators to two normal operators.     

A class of non-nilpotent 2-solvable n-Lie algebras

This article concerns a class of finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebras. It is shown that if L is a finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebra, then L can be decomposed into a semi-direct of an ideal A and an (n ? 1)-dimensional subalgebra H ₀ of L. Furthermore, H ₀ acts irreducibly on A/A ¹, and H ₀ + A ¹ is a self-normalizing maximal subalgebra of L with the core A ¹, the derived algebra of A.     

辛三代数的Frattini子代数

本文研究了辛三代数的Frattini子代数和基本辛三代数的问题.利用Frattini子代数和基本辛三代数的性质,得到了辛三代数的非嵌入定理,从而推广了李三系中关于Frattini子系的结果.