Reduced words and a length function for G(e,1,n)

In the first part of this paper we study normal forms of elements of the imprimitive complex reflection group G(e,1,n). This allows to prove a conjecture of Broué on basis elements and the canonical symmetrizing form of the associated cyclotomic Hecke algebra. Secondly we introduce a root system for G(e,1,n) and study the associated length function. This has many properties in common with the usual length function for finite Weyl groups.     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

Conservative algebras of 2-dimensional algebras,II

In 1990 Kantor defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n>1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of W(2). Also similar problems are solved for the algebra W₂ of all commutative algebras on the 2-dimensional vector space and for the algebra S₂ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.     

On Ramsey Numbers for Trees Versus Wheels of Five or Six Vertices

 For given two graphs G dan H, the Ramsey number R(G,H) is the smallest positive integer n such that every graph F of order n must contain G or the complement of F must contain H. In [12], the Ramsey numbers for the combination between a star S _n and a wheel W _m for m=4,5 were shown, namely, R(S _n ,W ₄)=2n−1 for odd n and n≥3, otherwise R(S _n ,W ₄)=2n+1, and R(S _n ,W ₅)=3n−2 for n≥3. In this paper, we shall study the Ramsey number R(G,W _m ) for G any tree T _n . We show that if T _n is not a star then the Ramsey number R(T _n ,W ₄)=2n−1 for n≥4 and R(T _n ,W ₅)=3n−2 for n≥3. We also list some open problems. Received: October, 2001 Final version received: July 11, 2002 RID="*" ID="*" This work was supported by the QUE Project, Department of Mathematics ITB Indonesia Acknowledgments. We would like to thank the referees for several helpful comments.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.     

The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ? _G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra W _G =S(?^*)⊗∧?^*, ? _G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ _G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W _G →? _G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology H _G (B)→ℋ _G (B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?) ^G ≅U(?) ^G between the ring of invariant polynomials and the ring of Casimir elements. Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999     

A modified BLM approach to quantum affine {\mathfrak{gl}_n}

We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)} _A,j , for affine quantum Schur algebras S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set {A(j)} _A,j for an associated algebra [^(K)]_\vartriangle(n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators E^\vartriangle_h,h+1(0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A(j) in [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras \mathfrak H_\vartriangle(n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for \mathfrak H_\vartriangle(n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace \mathfrak A_\vartriangle(n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by {A(j)} _A,j contains the quantum enveloping algebra U_\vartriangle(n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n,r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace \mathfrak A_\vartriangle(n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of \mathfrakgl_n{\mathfrak{gl}_n} . We conjecture that \mathfrak A_\vartriangle(n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} .     

Modular <Emphasis Type="Italic">W</Emphasis> and <Emphasis Type="Italic">H</Emphasis> Type Lie Algebras

The modular Witt algebra W(p, n) and H(p, 2n) are defined on the polynomial rings Z_p[x₁,...,x_n] and Z_p[X₁,...,x_n, y₁,...,y_n] respectively. We generalize Z_p[x₁,...,xn] and Zp[x₁,...,xn, y₁,...,y_n], so we get the generalized W-type and H-type modular Lie algebras. We find all the derivations of W(p, 1).AMS Subject Classification: Primary 17B40, 17B56.     

Clifford Analysis Techniques for Spherical PDE

For a ? R\alpha \in \mathbf{R}, the class of a-\alpha -order spherical harmonic functions in an open set W í\Omega \subseteq S^n-1\mathbf{S}^{n-1}, H^a(W)H^{\alpha }(\Omega ) is defined as the C²-C^{2}-solutions of D_au=0\Delta _{\alpha }u=0; where D_a=D_s+a(n+a-2)\Delta _{\alpha }=\Delta _{s}+\alpha (n+\alpha -2) is the spherical Laplace--Beltrami operator of order a\alpha and D_s\Delta _{s} is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in H^a(W)H^{\alpha }(\Omega ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Topological singularities in W S,P ( S N , S 1)

We are interested in the location of the singularities of maps u ∈ W ^s,p (S ^N , S ¹) when 1 ≤ sp and 1 < sp < 2. To this end, we consider the distributional Jacobian. We show that the range of this operator on W ^s,p (S ^N , S ¹) is the closure in W ^s−2,p ⋂ W ^−1,sp of the set of N − 2-currents defined as the integration on smooth oriented N − 2-dimensional boundaryless submanifolds.     

The limiting case of Haglund's (q,t)-Schröder theorem and an involution formula

As a generalization of Haglund's statistic on Dyck paths [Conjectured statistics for the q,t-Catalan numbers, Adv. Math. 175 (2) (2003) 319–334; A positivity result in the theory of Macdonald polynomials, Proc. Nat. Acad. Sci. 98 (2001) 4313–4316], Egge et al. introduced the (q,t)-Schröder polynomial S_n,d(q,t), which evaluates to the Schröder number when q=t=1 [A Schröder generalization of Haglund's statistic on Catalan paths, Electron. J. Combin. 10 (2003) 21pp (Research Paper 16, electronic)]. In their paper, S_n,d(q,t) was conjectured to be equal to the coefficient of a hook shape on the Schur function expansion of the symmetric function e_n, which Haiman [Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] has shown to have a representation-theoretic interpretation. This conjecture was recently proved by Haglund [A proof of the q,t-Schröder conjecture, Internat. Math. Res. Not. (11) (2004) 525–560]. However, because that proof makes heavy use of symmetric function identities and plethystic machinery, the combinatorics behind it is not understood. Therefore, it is worthwhile to study it combinatorially. This paper investigates the limiting case of the (q,t)-Schröder Theorem and obtains interesting results by looking at some special cases.     

Deddens Algebras and Shift

We consider Deddens algebras associated to operators of the form S−λI, where S is the unilateral shift and λ is a complex number. We show that such an algebra properly contains the commutant of S and that it is always weakly dense in L(H){{\mathcal L}({\mathcal H})}. Yet, it contains no rank one operators, unless λ = 0, in which case it equals L(H){{\mathcal L}({\mathcal H})}.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

Quasiconvex functions, SO(n) and two elastic wells

We use W^1,∞ approximations of minimizing sequences to study the growth of some quasiconvex functions near their zero sets. We show that for SO(n), the quasiconvexification of the distance function dist²(·, SO(n)) can be bounded below by the distance function itself. In certain cases of the incompatible two elastic well structure, we establish a similar result. We also prove that for small Lipschitz perturbations of SO(n) and of the two well structure, the Young measure limits of gradients supported on these perturbed sets are Dirac masses.     

N(2,2,0)代数的E-反演半群

本文研究了N(2,2,0)代数(S,*,△,0)的E-反演半群.利用N(2,2,0)代数的幂等元,弱逆元,中间单位元的性质和同宇关系,得到了N(2,2,0)代数的半群(S,*)构成E-反演半群的条件及元素α的右伴随非零零因子唯一,且为α的弱逆元等结论,这些结果进一步刻画了N(2,2,0)代数的结构.     

A generalized numerical range: the range of a constrained sesquilinear form

Let M_n denote the algebra of all nxn complex matrices. For a given q?C with ∣Q∣≤1, we define and denote the q-numerical range of A?M_n by

W_q (A)={x ^? Ay:x,y?C ⁿ , _{x ^? x?y ^? y=1,x ^? y=q} }

The q-numerical radius is then given by r_q (A)=sup{∣z∣:z?W _q (A)}. When q=1,W _q (A) and r _q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W _q (A) is the inner q-numerical radius defined by [rtilde] _q (A)=inf{∣z∣:z?W _q (A)}

In this paper, we describe some basic properties of W _q (A), extending known results on the classical numerical range. We also study the properties of r_q considered as a norm (seminorm if q=0) on M_n .Finally, we characterize those linear operators L on M_n that leave W_q ,r_q of [rtilde]_q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.