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1.
Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets. 相似文献
2.
Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q
L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z
L of L over the ring of rational integers is finitely generated. 相似文献
3.
Udo Baumgartner James Foster Jacqueline Hicks Helen Lindsay Ben Maloney Iain Raeburn 《代数通讯》2013,41(11):4135-4147
Abstract We describe the Hecke algebra ?(Γ,Γ0) of a Hecke pair (Γ,Γ0) in terms of the Hecke pair (N,Γ0) where N is a normal subgroup of Γ containing Γ0. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ?1 S = Γ/N, we show that ? (Γ,Γ0) is the twisted crossed product of ? (N,Γ0) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products. 相似文献
4.
Hiraku Nakajima 《Inventiones Mathematicae》2001,146(2):399-449
5.
Viktor Abramov 《Advances in Applied Clifford Algebras》2007,17(4):577-588
We construct a q-analog of exterior calculus with a differential d satisfying d
N
= 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found. 相似文献
6.
Claire Anantharaman-Delaroche 《Probability Theory and Related Fields》2006,135(4):520-546
We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions
of free groups and averages on spheres s2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s2n(x)) for x in noncommutative spaces Lp(A). For the Cesàro means this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende
Verfahren' theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators. 相似文献
7.
Jie Du 《manuscripta mathematica》1992,75(1):411-427
The structure of Schur algebrasS(2,r) over the integral domainZ is intensively studied from the quasi-hereditary algebra point of view. We introduce certain new bases forS(2,r) and show that the Schur algebraS(2,r) modulo any ideal in the defining sequence is still such a Schur algebra of lower degree inr. A Wedderburn-Artin decomposition ofS
K
(2,r) over a fieldK of characteristic 0 is described. Finally, we investigate the extension groups between two Weyl modules and classify the
indecomposable Weyl-filtered modules for the Schur algebrasS
Zp(2,r) withr<p
2
.
Research supported by ARC Large Grant L20.24210 相似文献
8.
U. Luther 《Integral Equations and Operator Theory》2006,54(4):541-554
We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N1, t ∈N2, or x=t, where N1,N2 are sets of measure zero. It is shown that such operators map weighted L∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel
k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors
of best weighted uniform approximation by algebraic polynomials. 相似文献
9.
《Quaestiones Mathematicae》2013,36(5):683-708
AbstractThe category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of HopfR. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist. 相似文献
10.
A commutative but not cocommutative graded Hopf algebra HN, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure
of unordered rooted trees HC, developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of
noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that
HN is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a
foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and
nonrecursive definitions of the coproduct and the antipode are derived. The relationship between HN and four other Hopf algebras is discussed. The dual of HN is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra HC of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of HN using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra HSh is obtained from HN by a quotient construction. The Hopf algebra HP of ordered trees by Foissy differs from HN in the definition of the product (noncommutative concatenation for HP and shuffle for HN) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator. 相似文献
11.
Silvia Montarani 《代数通讯》2013,41(5):1449-1467
Let Γ N : = S N ? Γ N be the wreath product of Γ, a finite subgroup of SL(2,C), by the symmetric group of degree N. In this article we classify all the irreducible representations of S N ? Γ N that can be extended to a representation of the associated symplectic reflection algebra H 1,k,c (Γ N ) (where k is a complex number and c a class function on the nontrivial elements of Γ) for nonzero values of k. 相似文献
12.
Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z F over F. For RA2 loops of type M(Dih(A), ?1,g 0), we prove that the loop algebra is in the variety generated by the algebra 3 which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ?1,g 0), where G is a non-Abelian group of exponent 4 having exactly 2 squares. 相似文献
13.
Ralf Holtkamp 《Archiv der Mathematik》1999,73(2):90-103
There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx1,...,xm ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case. 相似文献
14.
We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms
of uniform complete admissibility of the pair (c
0(N, X), c
0(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows,
using the uniform complete admissibility of the pairs (c
0(N, X), c
0(N, X)) and (C
0(R
+, X), C
0(R
+, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product
flows.
Received August 10, 2001; in revised form June 25, 2002 相似文献
15.
《代数通讯》2013,41(9):2957-2975
ABSTRACT Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) G is not finitely generated. 相似文献
16.
A Rota-Baxter operator of weight λ is an abstraction of both the integral operator (when λ=0) and the summation operator (when λ=1). We similarly define a differential operator of weight λ that includes both the differential operator (when λ=0) and the difference operator (when λ=1). We further consider an algebraic structure with both a differential operator of weight λ and a Rota-Baxter operator of weight λ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests. 相似文献
17.
Thao Tran 《Algebras and Representation Theory》2011,14(6):1025-1061
F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in
terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra.
In this paper, we define and prove the existence of analogous quantum F-polynomials for quantum cluster algebras. We prove some properties of quantum F-polynomials. In particular, we give a recurrence relation which can be used to compute them. Finally, we compute quantum
F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type An\mbox{A}_{n} quantum cluster algebras. 相似文献
18.
Yasutsugu Fujita 《manuscripta mathematica》2005,118(3):339-360
Let E be an elliptic curve over Q and p a prime number. Denote by Qp,∞ the Zp-extension of Q. In this paper, we show that if p≠3, then where E(Qp,∞)(2) is the 2-primary part of the group E(Qp,∞) of Qp,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q2,∞)(2) in terms of E(Q)(2); in case p≥5 (or in case p=3 and E(Q)(2)≠{O}), we show that E(Qp,∞)(2)=E(Q)(2). 相似文献
19.
20.
Let L
N
= L
MBM
(X
1, . .
.,X
N
;Y
1, . . . ,
Y
N
) be the minimum length of
a bipartite matching between two sets of points in
R
d
, where
X
1,...,X
N
, . . . and
Y
1, .
. . , Y
N
, . . . are random points
independently and uniformly distributed in [0,
1]
d
. We prove that for
d 3,
L
N
/N
1–1/d
converges with
probability one to a constant
MBM
(d) > 0 as N . 相似文献