Quantum sℓ(2) action on a divided-power quantum plane at even roots of unity

We describe a nonstandard version of the quantum plane in which the basis is given by divided powers at an even root of unity q = e^iπ/p. It can be regarded as an extension of the “nearly commutative” algebra ?[X, Y] with XY = (?1)^pYX by nilpotents. For this quantum plane, we construct a Wess-Zumino-type de Rham complex and find its decomposition into representations of the 2p ³ -dimensional quantum group $ \bar {\mathcal{U}} $ _q s?(2) and its Lusztig extension U _q s?(2); we also define the quantum group action on the algebra of quantum differential operators on the quantum plane.     

Finite dimensional quantum Teichmüller space from the quantum torus at root of unity

Representation theory of the quantum torus Hopf algebra, when the parameter q is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a ‘multiplicity module’ tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map T between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the Hom spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map A on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that T and A satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichmüller spaces of bordered Riemann surfaces. All constructions and proofs in the present work use only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works.     

An Equivalent Form of the Dedekind Axiomand Its Application(Ⅱ)——Also on the unity of the Continuous Induction,the Mathem atical Induction and the Transfinite Induction

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＣｏｎｔｉｎｕｏｕｓＩｎｄｕｃｔｉｏｎｏｆｒｅａｌｎｕｍｂｅｒｉｓｒｅｗｒｉｔｔｅｎａｓｂｅｌｏｗｉｎｔｈｉｓｐａｐｅｒ（Ⅰ）．ＴｈｅＣｏｎｔｉｎｕｏｕｓＩｎｄｕｃｔｉｏｎ（Ａ）：ＳｕｐｐｏｓｅｔｈａｔＰｘｉｓａｐｒｏｐｏｓｉｔｉｏｎｃｏｎｃｅｒｎｉｎｇａｒｅａｌｎｕｍｂｅｒｘ．Ｉｆｔｈｅｒｅｉｓｓｏｍｅｘ０ｓｕｃｈｔｈａｔＰｘｉｓｔｒｕｅｆｏｒｅｖｅｒｙｘ＜ｘ０，ｔｈｅｎＰｘｉｓｔｒｕｅｆｏｒｅｖｅｒｙｒｅａｌｎｕｍｂｅｒｘ．ＩｆＰｘｉｓｔｒｕｅｆｏｒｅ…     

Arrangements of Semicircles on S

ＡｒａｎｇｅｍｅｎｔｓｏｆＳｅｍｉｃｉｒｃｌｅｓｏｎＳ１ＬｉＹｉｎｇｚｉ（李英姿）ＤｉｎｇＲｅｎ（丁仁）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＨｅｂｅｉＴｅａｃｈｅｒｓ’Ｕｎｉｖｅｒｓｉｔｙ，Ｓｈｉｊｉａｚｈｕａｎｇ，Ｈｅｂｅｉ，０５００１６）Ｃｏｍｍｕｎｉｃ...     

Results on Commutativity of Rings

We establish two theorems on the commutativity of rings with identity.Whileour first result is based on an iteration method,the second deals with centralconditions.     

Boundeness of a Class of Maximal Operators on H~p Spaces on Compact Lie Groups

In this paper we discuss the weak type(H^p,L^p) boundedness of a class of maximal operators T*^ψ and themaximal strong mean boundedness of a family of the operators ｛T^ψ｝ on the atomic H^p spaces on compact Lic groups.Also,we obtain the correspoding convergent results.     

A Note on Property of Cauchy Integral on Classical Domains

In[1],ZhengXueanprovedthat:letRI(n×n)befirstclassicaldomain,Unbecharacteris-ticboundaryofRI,x∈Un,f(x)∈L2(Un).Asz=rx(0≤r<1)→x,Cauchyintegral∫Unf(y)det(I-zy′)-ndyconvergetoafunctioninL2(Un).Inthispaper,wewillfacusourselfonCauchyintegralofL2onclassicaldomains[2]andgetsomeproperties.Themainresultisfollowing:Theorem　LetRbeoneofclassicaldomains,LbecharacteristicboundaryofR,f∈L2(L),H(z,ξ)beCauchykernelofRandF(z)=∫Lf(u)H(z,u)u　z∈R,(1)wheretherightisLebesgueintegral;uist…     

Some Notes on Solvability of LPDO

The purpose of this paper is to discuss how the lower order terms of a LPDO with multiple characteristics influence the solvability of the operators. We begin with the followiog operator: L(x,D)=D_1~2+x_1~2D_2~2+pD_2+qD_1+c where p,q,c are assumed be complex constants. Theorem 1 The operator L will be locally solvable for any q,c, if either |Re p|<1 or Im p=0.     

Note on a Conjecture of Gopakumar-Vafa

We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.     

Positive Operators à la Aumann-Shapley on Spaces of Functions on D-Lattices

Let L be a σ-complete D-lattice and BV the AL-space of all realvalued, null in zero, functions on L of bounded variation. We prove the existence of a continuous Aumann-Shapley operator on the closed subspace of BV generated by powers of nonatomic σ-additive positive modular measures on L. The integral representation of on a class of functions that correspond to measure games is also exhibited. Dedicated to Professor Paolo De Lucia on the occasion of his 70th birthday.     

Randomization of Q_p spaces on the unit ball of C~n

In this paper, we study the random power series of several complex variables, and give some sufficient conditions for random power series to belong to Qp and Qp,0 spaces by means of their characterizations of Carleson measure. Moreover, we show that such conditions are best possible in some sense.     

Spatiality of Derivations on the Algebra of τ-Compact Operators

This paper is devoted to derivations on the algebra S ₀(M, τ) of all τ-compact operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. The main result asserts that every t _τ -continuous derivation ${D : S_0(M, \tau) \rightarrow S_{0}(M, \tau)}$ is spatial and implemented by a τ-measurable operator affiliated with M, where t _τ denotes the measure topology on S ₀(M, τ). We also show the automatic t _τ -continuity of all derivations on S ₀(M, τ) for properly infinite von Neumann algebras M. Thus in the properly infinite case the condition of t _τ -continuity of the derivation is redundant for its spatiality.     

Comparison of topologies on ⁎-algebras of locally measurable operators

We consider the local measure topology ${t(\mathcal{M})}$ on the ?-algebra ${LS(\mathcal{M})}$ of all locally measurable operators and on the ?-algebra ${S(\mathcal{M},\tau)}$ of all τ-measurable operators affiliated with a von Neumann algebra ${\mathcal{M}}$ . If τ is a semifinite but not a finite trace on ${\mathcal{M},}$ then one can consider the τ-local measure topology t _τ l and the weak τ-local measure topology t _{w τ l} . We study relationships between the topology ${t(\mathcal{M})}$ and the topologies t _τ l , t _{w τ l} , and the (o)-topology ${t_o(\mathcal{M})}$ on ${LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^\ast=T\}}$ . We find that the topologies ${t(\mathcal{M})}$ and t _τ l (resp. ${t(\mathcal{M})}$ and t _{w τ l} ) coincide on ${S(\mathcal{M},\tau)}$ if and only if ${\mathcal{M}}$ is finite, and ${t(\mathcal{M})=t_o(\mathcal{M})}$ on ${LS_h(\mathcal{M})}$ holds if and only if ${\mathcal{M}}$ is a σ-finite and finite. Moreover, it turns out that the topology t _{τ l} (resp. t _{w τ l} ) coincides with the (o)-topology on ${S_h(\mathcal{M},\tau)}$ only for finite traces. We give necessary and sufficient conditions for the topology ${t(\mathcal{M})}$ to be locally convex (resp., normable). We show that (o)-convergence of sequences in ${LS_h(\mathcal{M})}$ and convergence in the topology ${t(\mathcal{M})}$ coincide if and only if the algebra ${\mathcal{M}}$ is an atomic and finite algebra.     

Another proof of Grothendieck’s theorem on the splitting of vector bundles on the projective line

This note contains another proof of Grothendieck‘s theorem on the splitting of vector bundles on the projective line over a field k. Actually the proof is formulated entirely in the classical terms of a lattice \(\Lambda \cong k[T]^d\), discretely embedded into the vector space \(V \cong K_\infty ^d\), where \(K_\infty \cong k((1/T))\) is the completion of the field of rational functions k(T) at the place \(\infty \) with the usual valuation.     

Convergence of random zeros on complex manifolds

We show that the zeros of random sequences of Gaussian systems of polynomials of in- creasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular,the normalized distribution of zeros of systems of m polynomials of degree N,orthonor- malized on a regular compact set K(?)C~m,almost surely converge to the equilibrium measure on K as N→∞.     

On characterizations of isometries on function spaces

The paper gives characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball B_n in C~n.     

Some Applications of Besov Spaces on Fractals

Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.