弱 Hopf 超代数 wsldq(m|n)

该文依据弱Hopf代数的定义给出弱Hopf超代数的定义. 进而利用弱反极取代Hopf代数中反极的方法, 构造一类不是Hopf超代数的弱Hopf超代数wsl^d_q(m|n), 并给出了wsl^d_q(m|n)的PBW基.     

Orthosymplectic Quantum Function Superalgebras OSPq（2l＋ 1｜2n）

By the R-matrix of orthosymplectic quantum superalgebra U _q (osp(2l+1|2n)) in the vector representation, we establish the corresponding quantum Hopf superalgebra OSP _q (2l + 1|2n). Furthermore, it is shown that OSP _q (2l + 1|2n) is coquasitriangular.     

Cocycles on a finite dimensional quotient of uq (sl(2))

We prove that certain algebra quotients of Hopf algebras are twisted Hopf algebras. On the other handu_q (sl(2)) is a crossed product of a central subalgebra with a quotient [Ubar], when q is a root of 1. Using the cocycle involved in this crossed product we construct non-trivial complex cocycles τ and we find the isomorphism classes of the corresponding twisted Hopf algebras _τ [Ubar]. These provide complex projective representations of [Ubar] which are not ordinary representations.     

Skew derivations andU q (sl(2))

This note first describes the basic properties of the skew derivations on the polynomial ringk[X]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U _q(sl(2)), has a unique action on C[X], where “action” means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU _q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it. Both authors were supported by the NSF, S. Montgomery by grant DMS 87-00641, and S. P. Smith by DMS 87-02447.     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Lax Operator for the Quantised Orthosymplectic Superalgebra Uq[osp(m|n)]

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U _q [osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U _q [osp(m|n)] module. The case W = V is studied as an example. Presented by A. Verschoren.     

A reduction method for atypical representations of classical Lie superalgebras

We introduce a reduction method for studying representations of classical Lie superalgebras with atypical central character. We show that the atypical quotient of universal enveloping algebra has a non-trivial Jacobson radical. The factor by this radical has a new center, which is calculated for sl(1|n) and psl(2|2). Using this center we obtain new character formulae, generalization of Borel-Weil-Bott and Beilinson-Bernstein localization theorems.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

EXISTENCE OF FAST-DECAY GROUND STATE OF P-LAPLACIAN EQUATION

Using the shooting argument and an approximating method, this paper is concerned with the existence of fast-decay ground state of p-Laplacian equation: Δ_pu + f(u) = 0, in Rⁿ, where f(u) behaves just like f(u) = u^q – u^s, as s > q >np/(n – p) – 1.     

Vanishing exponential integrability for functions whose gradients belong to L(log(e+L))

If the gradient of u(x) is nth power locally integrable on Euclidean n-space, then the integral average over a ball B of the exponential of a constant multiple of |u(x)−u_B|^n/(n−1), u_B=average of u over B, tends to 1 as the radius of B shrinks to zero—for quasi almost all center points. This refines a result of N. Trudinger (1967). We prove here a similar result for the class of gradients in Lⁿ(log(e+L))^α, 0?α?n−1. The results depend on a capacitary strong-type inequality for these spaces.     

On two-sided and asymptotic estimates for the norms of embedding operators of into L q (dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L _q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ_{1, n, β} , β > 0, of the boundary value problems (?d ²/dx ²) ⁿ u(x) = λ|x| ^β?1, x ∈ (?1, 1), u ^(k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}.     

Cantor spectrum for the almost Mathieu equation

For a dense G_δ of pairs (λ, α) in R², we prove that the operator (Hu)(n) = u(n + 1) + u(n ?1) + λ cos(2παn + θ) u(n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case $\text{α =}\text{p}\text{q}$ of which we mention: (a) If qθ is not an integral multiple of π, then all gaps are open, and (b) If q is even and θ is chosen suitably, then the middle gap is closed for all λ.     

Rank varieties for Hopf algebras

We construct rank varieties for the Drinfeld double of the Taft algebra Λ_n and for u_q(sl₂). For the Drinfeld double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of Λ-modules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext^∗(M,M) is finitely generated over the cohomology ring of the Drinfeld double for any finitely generated module M.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

An inequality for polynomials with a prescribed zero

Let p_n(z)=∑_(k-0)~n a_kz~k be a polynomial of degree n such that |p_n(z)|≤M for |z|≤1. It is well.known that for 0≤u相似文献   

Quantum superdeterminants for OSP q (1-2n)

It is shown that there exists a quantum superdeterminant sdet _q T for the quantum super group OSP _q (1|2n). It is also shown that the quantum superdeterminant sdet _q T is a group-like element and central, and that the square of sdet _q T for OSP _q (1|2n) is equal to 1.     

Module categories over pointed Hopf algebras

We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups u_q(\mathfraksl₂){u_q(\mathfrak{sl}_2)}.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems

We prove that the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) where 0<α?1, 0<β?1, and p and q are nonnegative continuous functions has a nonnegative entire radial solution satisfying lim|x|→∞u(x)=lim|x|→∞v(x)=∞ if and only if the functions p and q satisfy