Monotone paths in ordered graphs

LetV _fin andE _fin resp. denote the classes of graphsG with the property that no matter how we label the vertices (edges, resp.) ofG by members of a linearly ordered set, there will exist paths of arbitrary finite lengths with monotonically increasing labels. The classesV _inf andE _inf are defined similarly by requiring the existence of an infinite path with increasing labels. We proveE _inf ⫋V _inf ⫋V _fin ⫋E _fin. Finally we consider labellings by positive integers and characterize the class corresponding toV _inf.     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

Building ofGL(m, D) and centralizers

LetG=GL(m, D) whereD is a central division algebra over a commutative nonarchimedean local fieldF. LetE/F be a field extension contained inM(m, D). We denote byI (resp.I _E) the nonextended affine building ofG (resp. of the centralizer ofE ^x inG). In this paper we prove that there exists a uniqueG _E-equivariant affine mapj _EI_EI. It is injective and its image coincides with the set ofE ^x-fixed points inI. Moreover, we prove thatj _E is compatible with the Moy-Prasad filtrations.This author's contribution was written while he was a post-doctoral student at King's College London and supported by an european TMR grant     

The monadic theory and the “next world”

Suppose thatV is a model of ZFC andU ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. LetB be the Boolean algebra of regular open subsets ofU. If the monadic theory ofU allows one to speak in some sense about a family ofκ everywhere dense and almost disjoint sets, then the second-orderV ^B-theory of ϰ is interpretable in the monadicV-theory ofU; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the real line. Letr be a Cohen real overV. The second-orderV[r]-theory of ℵ₀ is interpretable in the monadicV-theory of the real line. If CH holds inV then the second-orderV[r]-theory of the real line is interpretable in the monadicV-theory of the real line. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author thanks the United States-Israel Binational Science Foundation for supporting the research.     

Note on convergence and orders of vanishing along curves

LetX be a complex space andC a family of curves through ξ ∈X. Many conditions on the size ofC are known to be sufficient for the following statement: if a formal function f∈O _X ^{^} , ξ, converges (resp. has high order of vanishing) along all curves H∈C, then f converges (resp. has high order of vanishing) on X. We improve these known results using Gabrielov's theorem (resp. Lemma 1.4 below) on pullbacks of formal functions. Dedicated to Professor Masahisa Adachi on his 60th birthday     

Über Bewertungen,welche unter einer reduktiven Gruppe invariant sind

Summary LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toK ^U . Then we study the set of invariant valuations having some fixed restrictionv ₀, toK ^B . Ifv ₀ is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupW ^X . Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofK ^B .

Unterstützt durch den Schweizerischen Nationalfonds zur Förderung der wissenschaftlichen Forschung.     

Twisted product and cohomology

LetH be a Hopf algebra,H ₁ be a sub-Hopf algebra ofH, H ₂ be the quotient Hopt algebra ofH modularH ₁. This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex ofH has the same cohomology algebra with a twisted product of the cobar complexes ofH ₁ andH ₂. Supported by National Natural Science Foundation of China     

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.     

Characterization off-vectors of families of convex sets inR d Part I: Necessity of Eckhoff’s conditions

LetK=K ₁,...,K_n be a family ofn convex sets inR ^d . For 0≦i<n denote byf _i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.     

A reciprocity theorem for unitary representations of Lie groups

LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU ^L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining number ofU ^L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV _∞ toH.     

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

Structure of maximal commutative subalgebras of the first Weyl Algebra

A short proof is given to Dixmier's sixth problem from [9] for the first Weyl algebra which askswhether a polynomial of a generic element of the first Weyl algebra is a generic element. Anaffirmative answer to this question was given by A. Joseph [14]. In the paper we give an answer to a similar question but for an arbitrary element of the first Weyl algebra. This result is used then to clarify structure of maximal commutative subalgebras in the first Weyl algebraA ₁: for a given maximal commutative subalgebra C of the Weyl algebra A₁ (almost) all non-scalar elements of it have the sametype, more precisely, have one of the following types: (i) strongly nilpotent, (ii) weakly nilpotent, (iii) generic, (iv) generic except for a subset, K ^*a+K of elements of strongly semi-simple type where a∈C is an element of strongly semi-simple type and K^*=K/{0}, (v) generic except for a subset, K^*a+K of elements of weakly semi-simple type where a∈C is an element of weakly semi-simple type.

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Sunto Si fornisce una dimostrazione breve al sesto problema di Dixmier enunciato in [9] per la prima algebra di Weyl che chiedese un polinomio di un elemento generico della prima algebra di Weyl è un elemento generico. Una rispostaaffermativa a questo problema à stata data da A. Joseph in [14]. In questo articolo formiamo una risposta ad un quesito simile ma per un elemento arbitrario della prima algebra di Weyl. Questo risultato è usato quindi per chiarire la struttura delle sottoalgebre commutative massimali della prima algebra di WeylA ₁: per una data sottoalgebra commutativa massimale C dell'algebra di Weyl A₁ (quasi) tutti i suoi elementi non scalari hanno lo stessotipo; più precisamente, hanno uno dei seguenti tipi: (i) fortemente nilpotente, (ii) debolmente nilpotente, (iii) generico, (iv) generico eccetto che per un sottoinsieme K ^*a+K di elementi di tipo fortemente semisemplice, dove a∈C è un elemento di tipo debolmente semisemplice.

     

Adapted algebras for the Berenstein-Zelevinsky conjecture

Let G be a simply connected semisimple complex Lie group and fix a maximal unipotent subgroup U^- of G. Let q be an indeterminate and let B* denote the dual canonical basis (cf. [19]) of the quantized algebra Cq[U^-] of regular functions on U^-. Following [20], fix a Z^N_≧0-parametrization of this basis, where N = dim U^-. In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of B* q-commute if and only if they are multiplicative, i.e., their product is an element of B* up to a power of q. To any reduced decomposition w₀ of the longest element of the Weyl group of g, we associate a subalgebra A_w0, called adapted algebra, of Cq[U^-] such that (1) A_w0 is a q-polynomial algebra which equals Cq[U^-] up to localization, (2) A_w0 is spanned by a subset of B*, (3) the Berenstein–Zelevinsky conjecture is true on A_w0. Then we test the conjecture when one element belongs to the q-center of Cq[U^-].     

LatticeW algebras and quantum groups

We present Feigin's construction [Lectures given in Landau Institute] of latticeW algebras and give some simple results: lattice Virasoro andW ₃ algebras. For the simplest caseg=sl(2), we introduce the wholeU _q(2)) quantum group on this lattice. We find the simplest two-dimensional module as well as the exchange relations and define the lattice Virasoro algebra as the algebra of invariants ofU _q(sl(2)). Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine groupU _q(ñ₊). We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 132–147, July, 1994.     

Differential operators on homogeneous spaces. III

Summary LetG be a semi-simple complex algebraic group with Lie algebra and flag varietyX=G/B. For each primitive idealJ with trivial central character in the enveloping algebraU() we define a characteristic variety in the cotangent bundle ofX, which projects under the Springer resolution mapT ^* X onto the closure of a nilpotent orbit. We prove that this characteristic variety is theG-saturation of the characteristic variety of a highest weight module with annihilatorJ. We conjecture that it is irreducible forG=SL _n . Our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of Weyl group representations, as achieved by A. Joseph. The presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space, and to study more generally characteristic varieties of Harish-Chandra modules.Partially supported by N.S.F. research grant     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH ₁ andH ₂ be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h₁ + Adg ? h₂, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H ₁,rankG/H ₂). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H ₁ xG/H ₂) ^G for the diagonal action ofG. Ifk = ? andG _a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = g_a + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.     

A sum-product estimate in algebraic division algebras

LetA be a finite subset of some normed division algebra over ℝ with cardinality ⋎A⋎. We prove that either the sum set or the product set ofA has cardinality ⋎A⋎^1+δ for some δ>0. Partially supported by NSA grant No. MDA 904-03-1-0045.