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.     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

A Class of Multiplier Hopf Algebras

We compute the Drinfel’d double for the bicrossproduct multiplier Hopf algebra A = k[G] ⋊ K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] ⋊ K(H) if there is a factor-reversing group isomorphism. Presented by A. Verschoren.     

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. (Received 15 November 2000)     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Homogeneous Metrics with Nonnegative Curvature

Given compact Lie groups H⊂G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.     

Lie subalgebras in a certain operator Lie algebra with involution

We show in a certain Lie*-algebra, the connections between the Lie subalgebra G ⁺:= G + G* + [G, G*], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgebras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion: Let A ⊂ B(X) be a space of self-adjoint operators and ℒ:= A ⊕ iA the corresponding complex Lie*-algebra. G ⁺ = G + G* + [G, G*] and G are two LM-decomposable Lie subalgebras of ℒ with the decomposition G ⁺ = R(G ⁺) + S, G = R _G + S _G , and R _G ⊆ R(G ⁺). Then G ⁺ is ideally finite iff R _G ⁺:= R _G + R* _G ^* + [R _G , R _G ^*] is a quasisolvable Lie subalgebra, S _G ⁺:= S _G + S _G ^* + [S _G , S _G ^*] is an ideally finite semisimple Lie subalgebra, and [R _G , S _G ] = [R _G ^*, S _G ] = {0}.     

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.     

On irreducible algebras of conformal endomorphisms over a linear algebraic group

We study the algebra of conformal endomorphisms Cend ^G,G _n of a finitely generated free module M _n over the coordinate Hopf algebra H of a linear algebraic group G. It is shown that a conformal subalgebra of Cend _n acting irreducibly on M _n generates an essential left ideal of Cend ^G,G _n if enriched with operators of multiplication on elements of H. In particular, we describe such subalgebras for the case where G is finite. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Reciprocal Polynomials and p-Group Cohomology

Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ?-algebra R _G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring $H^\ast(G, \mathbb{F}_p)Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ℤ-algebra R _G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring H^*(G, \mathbbF_p)H^\ast(G, \mathbb{F}_p) of G has the same spectrum as the ring of invariants of R _G mod p (R_G ?_\mathbbZ \mathbbF_p)^G(R_G \otimes_{\mathbb{Z}} \mathbb{F}_p)^G where the action of G is induced by conjugation.     

Central invariants of H-module algebras

Let H be a Hopf algebra over a field k:, and A an H-module algebra, with subalgebra of H-invariants denoted by A^H . When (H, R) is quasitriangular and A is quantum commutative with respect to (H,R), (e.g. quantum planes, graded commutative superalgebras), then A^H ? center of A = Z(A). In this paper we are mainly concerned with actions of H for which A^H ? Z(A). We show that under this hypothesis there exists strong relations between the ideal structures of A^H A and A#H.

We demonstrate the theorems by constructing an example of a quantum commutative A, so that A/A^H is H ^?-Galois. This is done by giving (C G)^? G = Z_n × Z_n , a nontrivial quasitriangular structure and defining an action of it on a localization of the quantum plane.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module _E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if _R M is an arbitrary unitary left module and M ⁺ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M ⁺) under the canonical ring homomorphism R → E(M ⁺). Then it holds that if _E(M+) N is a pure submodule in _E(M+) M ⁺, then _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.     

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σ _g∈G R _g be any associative algebra over a field such that the subspacesR _g satisfyR _g R _h ⊆R _gh . We prove that ifR ₁ satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withR ^H satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.     

The higher lower central series

The Baer invariants Γ _n (G) of a group are central extensions of the elementsγ _n (G) of the lower central series. We show that the inclusionsγ _n +1 ⊂γ _n can be lifted to functor morphisms Γ _n+1→Γ _n and a canonical Lie algebra, analogous to Lazard’s Lie algebra, can be constructed which is explicitly computable. This is applied in various ways.     

On amenable groups of automorphisms on Von Neumann algebras

LetG ⊂ Aut ℳ be a countable group, ℳ a Von Neumann algebra. LetE be a set of pure states on ℳ such thatG*E ⊂E, S ^G be the set ofG invariant states on ℳ andS _E ^G =S ^G ∩w* cl coE. We investigate in this paper some geometric properties for the setS _E ^G which turn out to be equivalent to amenability for the groupG. For example, we show thatS _E ^G ⊂ ℳ_* (S _E ^G has the WRNP) implies that ℳ contains minimal projections (ê containsfinite G invariant orbits) hold true, for all ℳ iffG is amenable. Furthermore we show that ifG is amenable thenS ^G ∩M _* ^⊥ contains a big set, thus improving results obtained by Ching Chou in [2]. These results imply that no action of an amenable countable groupG on an arbitraryW* algebra ℳ iss — strongly ergodic. Moreover cardS ^G ∩M _* ^⊥ ≧2 ^c (see M. Choda [4], K. Schmidt [21] and compare with A. Connes and B. Weiss [5]). The author gratefully acknowledges the support of an Izaak Walton Killam Memorial Senior Fellowship.     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.