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1.
Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C
* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C
*. 相似文献
2.
Teng Xia Ju 《数学学报(英文版)》2009,25(5):831-844
In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^* are semisimple, then gl.dim(A#σH)=gl.dim(A), where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim(C α H)=gl.dim(C). 相似文献
3.
We mainly study the global dimension of ω-smash coproducts. We show that if H is a Hopf algebra with a bijective antipode S H , and C ω ? H denotes the ω-smash coproduct, then gl.dim(C ω ? H) ≤ gl.dim(C) + gl.dim(H), where gl.dim(H) denotes the global dimension of H as a coalgebra. 相似文献
4.
Yasuji Takeuchi 《代数通讯》2013,41(5):1657-1664
Let C be a coalgebra and let M be a right C-comodule. In this article we investigate the dimension of the space f of comodule morphisms from C to M. Our main result (Theorem 4) states that dim(∫M)≤dim(M), for every left co-Frobeuius coalgebra (with some additional properties) and for every finite dimensional right comodule M. In particular, taking M = ki we obtain 相似文献
5.
CHEN Xiaowu HUANG Hualin & ZHANG Pu Department of Mathematics University of Science Technology of China Hefei China USTC Shanghai Institute for Advanced Studies Shanghai China Mathematical Section the Abdus Salam ICTP Strada Costiera Trieste Italy Department of Mathematics Shanghai Jiao Tong University Shanghai China 《中国科学A辑(英文版)》2006,49(1):9-26
6.
We study certain comodule structures on spaces of linear morphisms between H-comodules, where H is a Hopf algebra over the field k. We apply the results to show that H has non-zero integrals if and only if there exists a non-zero finite dimensional injective right H-comodule. Using this approach, we prove an extension of a result of Sullivan, by showing that if H is involutory and has non-zero integrals, and there exists an injective indecomposable right comodule whose dimension is not a multiple of char(k), then H is cosemisimple. Also we prove without using character theory that if H is cosemisimple and M is an absolutely irreducible right H-comodule, then char(k) does not divide dim(M). 相似文献
7.
Let K be a field. Let H be a finite-dimensional semisimple and cosemisimple K-Hopf algebra. In this paper, we introduce a notion of β-character algebra C
β
(H) for each group-like element β in H
∗. We prove that Radford’s action of the Drinfel’d double D(H) on H
β
(see Radford, J. Algebra, 270:670–695, 2003) and the right hit action of the β-character algebra C
β
(H) on H
β
form a commuting pair. This generalizes an earlier result of Zhu (Proc. Amer. Math. Soc., 125(10):2847–2851, 1997). A K-basis of C
β
(H) is given when H is split semisimple. Finally, as an example, we explicitly construct all the simple modules for the Drinfel’d double of the
unique 8-dimensional non-commutative and non-cocommutative semisimple Hopf algebra.
Presented by S. Montgomery. 相似文献
8.
András Máthé 《Israel Journal of Mathematics》2008,164(1):285-302
We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H
s
) and (ℝ, B, H
t
) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H
d
is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss.
To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d.
We also prove this statement in a more general form: If A ⊂ ℝn is Borel and ƒ: A → ℝm is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A).
Partially supported by the Hungarian Scientific Research Fund grant no. T 49786. 相似文献
9.
William H. Graves 《Discrete Mathematics》1974,10(1):75-92
The cohomology groups Hn(M, C) are studied, where C is the incidence coalgebra of a locally finite partially ordered set P and where M is a C-C comodule depending on a convex coarsening of the given partial order on P. The case where P is a geometric lattice and the convex coarsening is just the equality relation is emphasized. 相似文献
10.
To any right comodule coalgebra C over a Hopf algebra H we associate a left H-comodule algebra A. Under certain conditions, in particular in the case where H has nonzero integrals, we show that the category of right C, H-comodules is isomorphic to a certain subcategory of the category of Doi–Hopf modules associated to A. As an application, we investigate the connection between C and the smash coproduct C ? H being right semiperfect. 相似文献
11.
For a quasi-Hopf algebra H, a left H-comodule algebra and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules C ?(H) in terms of modules. We will also show that for an H-bicomodule algebra and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules. 相似文献
12.
We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair
of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H
α⋈β
G ≅H
α′⋈β′
(G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H
α⋈β
G ≅ H
α′⋈β′
G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications
two Schreier type classification theorems for bicrossed products of groups are given. 相似文献
13.
Fang LI Gong Xiang LIU 《数学学报(英文版)》2006,22(4):1027-1046
There are at least two kinds of generalization of Hopf algebra, i.e. pre-Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let L = (L, ×, I, a, l, r) be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = (L, ×, a) and used it to characterize almost bialgebra and pre-Hopf algebra in Comm. in Algebra, 32(2): 397-441 (2004). Our aim in this paper is to generalize tensor category L = (L, ×, I, a, l, r) by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll^-1 = rr^-1 = id into regular natural transformations lll= l, rrr = r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized (bialgebras) Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed. 相似文献
14.
Sonia Natale 《Israel Journal of Mathematics》2007,162(1):253-273
We describe the exponent of a group-theoretical fusion category C = C(G, ω, F, α) associated to a finite group G in terms of group cohomology. We show that the exponent of C divides both e(ω)expG and (expG)2, where e(ω) is the cohomological order of the 3-cocycle ω. In particular, expC divides (dim C)2.
This work was partially supported by CONICET, Fundación Antorchas, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC). 相似文献
15.
D. Simson 《Ukrainian Mathematical Journal》2009,61(6):964-987
We study the class of coalgebras C of fc-tame comodule type introduced by the author. With any basic computable K-coalgebra C and a bipartite vector v = (v′|v″) ∈ K
0(C) × K
0(C), we associate a bimodule matrix problem Mat
v
C
(ℍ), an additive Roiter bocs B
C
v
, an affine algebraic K-variety Comod
C
v
, and an algebraic group action G
C
v
× Comod
C
v
→ Comod
C
v
. We study the fc-tame comodule type and the fc-wild comodule type of C by means of Mat
v
C
(ℍ), the category rep
K
(B
C
v
) of K-linear representations of B
C
v
, and geometry of G
C
v
-orbits of Comod
C
v
. For computable coalgebras C over an algebraically closed field K, we give an alternative proof of the fc-tame-wild dichotomy theorem. A characterization of fc-tameness of C is given in terms of geometry of G
C
v
-orbits of Comod
v
. In particular, we show that C is fc-tame of discrete comodule type if and only if the number of G
C
v
-orbits in Comod
C
v
is finite for every v = (v′|v″) ∈ K
0(C) × K
0(C). 相似文献
16.
Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $
\bar C
$
\bar C
(X) of C(X) such that the pair ($
\bar C
$
\bar C
(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $
\bar C
$
\bar C
(X) coincides with the space USCC
F
(X,
相似文献
17.
I. E. Wijayanti 《Journal of Mathematical Sciences》2007,142(2):1895-1898
The wedge product of subcoalgebras of a coalgebra can be used to define coprime coalgebras. On the other hand, coprime elements
in the big lattice of preradicals in module categories also lead to the definition of coprime modules. Considering a coalgebra
C as a module over its dual algebra C*, this yields another notion of coprimeness for coalgebras. Under special conditions, the two definitions coincide.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 45–49, 2005. 相似文献
18.
Sergio Vessella 《数学学报(英文版)》2005,21(2):351-380
Let Γ be a portion of a C
1,α boundary of an n-dimensional domain D. Let u be a solution
to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We
prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous
result we prove that if u (x, t) = O (|x|k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically
equal to zero.
This work is partially supported by MURST, Grant No. MM01111258 相似文献
19.
Representation-Directed Incidence Coalgebras of Intervally Finite Posets and the Tame-Wild Dichotomy
Daniel Simson 《代数通讯》2013,41(7):2764-2784
Incidence coalgebras C = K □ I of intervally finite posets I that are representation-directed are characterized in the article, and the posets I with this property are described. In particular, it is shown that the coalgebra C = K □ I is representation-directed if and only if the Euler quadratic form q C : ?(I) → ? of C is weakly positive. Every such a coalgebra C is tame of discrete comodule type and gl. dimC ≤ 2. As a consequence, we get a characterization of the incidence coalgebras C = K □ I that are left pure semisimple in the sense that every left C-comodule is a direct sum of finite dimensional subcomodules. It is shown that every such coalgebra C = K □ I is representation-directed and gl. dimC ≤ 2. Finally, the tame-wild dichotomy theorem is proved, for the coalgebras K □ I that are right semiperfect. 相似文献