Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative.     

Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

Let M be a type I von Neumann algebra with the center Z, and a faithful normal semi-finite trace τ. Consider the algebra L(M, τ) of all τ-measurable operators with respect to M and let S ₀(M, τ) be the subalgebra of τ-compact operators in L(M, τ). We prove that any Z-linear derivation of S ₀(M, τ) is spatial and generated by an element from L(M, τ).      

Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators

Consider a von Neumann algebra M with a faithful normal semifinite trace τ. We prove that each order bounded sequence of τ-compact operators includes a subsequence whose arithmetic averages converge in τ. We also prove a noncommutative analog of Pratt’s lemma for L ₁(M, τ). The results are new even for the algebra M = B(H) of bounded linear operators with the canonical trace τ = tr on a Hilbert space H. We apply the main result to L _p (M, τ) with 0 < p ≤ 1 and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of τ-compactness of the dominant.     

Quasi-elementary <Emphasis Type="Italic">H</Emphasis>-Azumaya Algebras Arising from Generalized (Anti) Yetter-Drinfeld Modules

Let H be a Hopf algebra with bijective antipode, α, β ∈ Aut _Hopf (H) and M a finite dimensional (α, β)-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H.     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

On the structure of modules over wild hereditary algebras Dedicated to Daniel Simson on the occasion of his sixtieth birthday

 Let H be a connected finite dimensional wild hereditary path-algebra over an arbitrary field K and the Auslander-Reiten translation on H-mod, the category of finite dimensional H-modules. Let X be a finite dimensional H-module. We prove unexpected new results on the structure of the shifted modules , for , their minimal projective and injective resolutions, and the Auslander-Reiten components of the one-point extensions and coextensions of H by the modules . Received: 5 November 2001     

The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ≅ U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H. Presented by D. Passman     

On locally <Emphasis Type="Italic">A</Emphasis>-convex <Emphasis Type="Italic">H</Emphasis>*-algebras

We endow any proper A-convex H*-algebra (E, τ) with a locally pre-C*-topology. The latter is equivalent to that introduced by the pre C*-norm given by Ptàk function when (E, τ) is a Q-algebra. We also prove that the algebra of complex numbers is the unique proper locally A-convex H*-algebra which is barrelled and Q-algebra.      

A Certain Regular Property of the Method I Construction and Packing Measure

Let τ be a premeasure on a complete separable metric space and let τ＊ be the Method I measure constructed from τ. We give conditions on T such that τ＊ has a regularity as follows： Every τ＊-measurable set has measure equivalent to the supremum of premeasures of its compact subsets. Then we prove that the packing measure has this regularity if and only if the corresponding packing premeasure is locally finite.     

Minimality and unique ergodicity of homogeneous actions

Consider a connected Lie groupG, a lattice Γ inG, a connected subgroupH ofG, and the adjoint representation Ad ofG on its Lie algebra g. Suppose that Ad(H) splits into a semidirect product of a reductive subgroup and the unipotent radical. We prove that the minimality of the leftH-action onG/Γ then implies its unique ergodicity. Simultaneously, we suggest a reduction of the study of finite ergodic measures for an arbitrary action (G/Γ,H), where the subgroupH∈G is connected and Γ∈G is discrete, to the case of an Abelian subgroupH. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 293–301, August, 1999.     

Embedding finite graphs into graphs colored with infinitely many colors

We prove that for any cardinalτ and for any finite graphH there is a graphG such that for any coloring of the pairs of vertices ofG withτ colors there is always a copy ofH which is an induced subgraph ofG so that both the edges of the copy and the edges of the complement of the copy are monochromatic. Research supported by Hungarian National Science Foundation OTKA grant 1805.     

Hom-configurations and noncrossing partitions

We study maximal Hom-free sets in the τ[2]-orbit category C(Q) of the bounded derived category for the path algebra associated to a Dynkin quiver Q, where τ denotes the Auslander–Reiten translation and [2] denotes the square of the shift functor. We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom _≤0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutation of maximal Hom-free sets in C(Q)\mathcal {C}(Q) and prove that the graph of these mutations is connected.     

Classification of some <Emphasis Type="Italic">τ</Emphasis>-congruence-free completely regular semigroups

Let τ be an equivalence relation on a semigroup. We introduce τ-congruence-free semigroups, extending the notion of congruence-free semigroups, and classify all completely regular semigroups which are τ-congruence-free, where τ is one of Green’s relations H,L\mathcal{H},\mathcal{L} and D\mathcal{D} respectively. Taking τ as H\mathcal{H} as well as D\mathcal{D}, this settles two open problems posed by M. Petrich and N.R. Reilly.     

Hopf algebras with rigid dualizing complexes

Let H be a Hopf algebra over a base field. If H has an ℕ-filtration such that the associated graded ring is connected graded noetherian and has enough normal elements, then H is Gorenstein. This gives a partial solution to a question of Brown and Brown-Goodearl. As a consequence, every quotient Hopf algebra of a generic quantized coordinate ring of a connected semisimple Lie group is Auslander-Gorenstein and Cohen-Macaulay. The last statement answers a question of Goodearl-Zhang.     

On the structure of graded <Emphasis Type="Italic">λ</Emphasis>-Hopf algebras

Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.     

Hypergraphs with large transversal number and with edge sizes at least four

Let H be a hypergraph on n vertices and m edges with all edges of size at least four. The transversal number τ(H) of H is the minimum number of vertices that intersect every edge. Lai and Chang [An upper bound for the transversal numbers of 4-uniform hypergraphs, J. Combin. Theory Ser. B, 1990, 50(1), 129–133] proved that τ(H) ≤ 2(n+m)/9, while Chvátal and McDiarmid [Small transversals in hypergraphs, Combinatorica, 1992, 12(1), 19–26] proved that τ(H) ≤ (n + 2m)/6. In this paper, we characterize the connected hypergraphs that achieve equality in the Lai-Chang bound and in the Chvátal-McDiarmid bound.     

A note on local automorphisms

Let H be an infinite-dimensional almost separable Hilbert space. We show that every local automorphism of ℬ(H), the algebra of all bounded linear operators on a Hilbert space H, is an automorphism.     

On weakly τ-quasinormal subgroups of finite groups

Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |Q ^G |) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H ≦ H _τG , where H _τG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let X ≦ E be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ.     

<Emphasis Type="Italic">β</Emphasis>-character Algebra and a Commuting Pair in Hopf Algebras

Let K be a field. Let H be a finite-dimensional semisimple and cosemisimple K-Hopf algebra. In this paper, we introduce a notion of β-character algebra C _β (H) for each group-like element β in H ^∗. We prove that Radford’s action of the Drinfel’d double D(H) on H _β (see Radford, J. Algebra, 270:670–695, 2003) and the right hit action of the β-character algebra C _β (H) on H _β form a commuting pair. This generalizes an earlier result of Zhu (Proc. Amer. Math. Soc., 125(10):2847–2851, 1997). A K-basis of C _β (H) is given when H is split semisimple. Finally, as an example, we explicitly construct all the simple modules for the Drinfel’d double of the unique 8-dimensional non-commutative and non-cocommutative semisimple Hopf algebra. Presented by S. Montgomery.     

大子代数中的极大幂零理想和代数同构

§ 1　Introduction and preliminariesLet Hbe a complex Hilbertspace,B(H) be the setof all linear bounded operators.A set Nofprojections on His called a nestif(1 ) Nistotally ordered by the usual oper-ator order,(2 ) Ncontains the zero operator0 and the identity operator I,and(3 ) givenany subset N0 N,the operators sup{ N:N∈N0 } and inf{ N:N∈ N0 } are both mem-bers of N.For N∈Nwith N>0 ,we define N- =sup{ N′∈N:N′相似文献