共查询到20条相似文献,搜索用时 125 毫秒
1.
Christopher Kennedy 《Algebras and Representation Theory》2010,13(3):359-381
Deep matrix algebras were originally created by Cuntz (Comm. Math. Phys. 57:173–185, 1977) and McCrimmon (2006). Further study of the associative case was done by the author in Kennedy (2004) and Kennedy (Algebr. Represent. Theory 9:525–537, 2006). In this paper, the associative algebra
DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) based on a set X over a field
\mathbbK{\mathbb{K}} and various of its subalgebras are studied for the purpose of determining the structure of the associated Lie algebra
\mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}) and its subalgebras. Several key examples of deep matrix Lie algebras are constructed. These are shown to be either simple
or nearly simple depending on the cardinality of the set X. Cartan subalgebras are constructed and two of the key Lie algebras are then decomposed with respect to the adjoint action
of these subalgebras. In the process, an infinite dimensional analogue to
\mathfraksl2(\mathbbK)\mathfrak{sl}_2({\mathbb{K}}) is naturally realized as a key subalgebra in deep matrix Lie algebras. 相似文献
2.
Kassandra Averill Ann Johnston Ryan Northrup Robert Silversmith Aaron Luttman 《Central European Journal of Mathematics》2012,10(2):646-655
It has been shown that any Banach algebra satisfying ‖f
2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras
of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not
been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued
Lipschitz functions. Denote by Lip(X,
\mathbbF\mathbb{F}) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where
\mathbbF\mathbb{F} = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X,
\mathbbF\mathbb{F}) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values
in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by
generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While
the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions
on such an algebra for it to hold and to guarantee the existence of the Shilov boundary. 相似文献
3.
A variety ${\mathbb{V}}${\mathbb{V}} is var-relatively universal if it contains a subvariety
\mathbbW{\mathbb{W}} such that the class of all homomorphisms that do not factorize through any algebra in
\mathbbW{\mathbb{W}} is algebraically universal. And
\mathbbV{\mathbb{V}} has an algebraically universal α-expansion
a\mathbbV{\alpha\mathbb{V}} if adding α nullary operations to all algebras in
\mathbbV{\mathbb{V}} gives rise to a class
a\mathbbV{\alpha\mathbb{V}} of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety
\mathbbV{\mathbb{V}} has an algebraically universal α-expansion
a\mathbbV{\alpha\mathbb{V}} . This note contains a more general result that proves this conjecture. 相似文献
4.
Erin W. Chambers Vin de Silva Jeff Erickson Robert Ghrist 《Discrete and Computational Geometry》2010,44(1):75-90
Fix a finite set of points in Euclidean n-space
\mathbbEn\mathbb{E}^{n}
, thought of as a point-cloud sampling of a certain domain
D ì \mathbbEnD\subset\mathbb{E}^{n}
. The Vietoris–Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed
but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris–Rips complex to
\mathbbEn\mathbb{E}^{n}
that has as its image a more accurate n-dimensional approximation to the homotopy type of D. 相似文献
5.
Nikolaos Bournaveas Timothy Candy 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):67-78
It is known from Czubak (Anal PDE 3(2):151–174, 2010) that the space–time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in
Hs(\mathbbR2){H^s(\mathbb{R}^2)} for ${s>\frac{1}{4}}${s>\frac{1}{4}}. Here we prove local well-posedness for arbitrary initial data in
Hs(\mathbbR2){H^s(\mathbb{R}^2)} with ${s>\frac{1}{4}}${s>\frac{1}{4}} in the Lorenz gauge. 相似文献
6.
A. S. Morozov 《Algebra and Logic》2012,51(1):66-88
It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) . 相似文献
7.
Sorin G. Gal 《Complex Analysis and Operator Theory》2012,6(2):515-527
Attaching to a compact disk
[`(\mathbbDr)]{\overline{\mathbb{D}_{r}}} in the quaternion field
\mathbbH{\mathbb{H}} and to some analytic function in Weierstrass sense on
[`(\mathbbDr)]{\overline{\mathbb{D}_{r}}} the so-called q-Bernstein operators with q ≥ 1, Voronovskaja-type results with quantitative upper estimates are proved. As applications, the exact orders of approximation
in
[`(\mathbbDr)]{\overline{\mathbb{D}_{r}}} for these operators, namely
\frac1n{\frac{1}{n}} if q = 1 and
\frac1qn{\frac{1}{q^{n}}} if q > 1, are obtained. The results extend those in the case of approximation of analytic functions of a complex variable in disks
by q-Bernstein operators of complex variable in Gal (Mediterr J Math 5(3):253–272, 2008) and complete the upper estimates obtained for q-Bernstein operators of quaternionic variable in Gal (Approximation by Complex Bernstein and Convolution-Type Operators, 2009; Adv Appl Clifford Alg, doi:, 2011). 相似文献
8.
Alexander Premet 《Inventiones Mathematicae》2010,181(2):395-420
Let ${\mathfrak{g}}Let
\mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field
\mathbbK\mathbb{K} of characteristic 0. Let
\mathfrakg\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of
\mathfrakg{\mathfrak{g}} and
\mathfrakg\Bbbk=\mathfrakg\mathbbZ?\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where
\Bbbk\Bbbk is the algebraic closure of
\mathbbFp{\mathbb{F}}_{p}. Let
G\BbbkG_{\Bbbk} be a simple, simply connected algebraic
\Bbbk\Bbbk-group with
\operatornameLie(G\Bbbk)=\mathfrakg\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra
U(\mathfrakg\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for
\mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions
\Bbbk (\mathfrakg\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield
\Bbbk(\mathfrakg\Bbbk)G\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions
\mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield
\mathbbK(\mathfrakg)\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if
\mathfrakg{\mathfrak{g}} is of type B
n
, n≥3, D
n
, n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the
above types. In other words, if
\mathfrakg{\mathfrak{g}} is of type B
n
, n≥3, D
n
, n≥4, E6, E7, E8 or F4, then the Lie field of
\mathfrakg{\mathfrak{g}} is more complicated than expected. 相似文献
9.
Brad C. Johnson Thomas M. Sellke 《Methodology and Computing in Applied Probability》2010,12(1):139-154
Suppose an urn contains m distinct balls, numbered 1,...,m, and let τ denote the number of i.i.d. samples required to observe all of the balls in the urn. We generalize the partial fraction expansion
type arguments used by Pólya (Z Angew Math Mech 10:96–97, 1930) for approximating
\mathbbE(t)\mathbb{E}(\tau) in the case of fixed sample sizes to obtain an approximation of
\mathbbE(t)\mathbb{E}(\tau) when the sample sizes are i.i.d. random variables. The approximation agrees with that of Sellke (Ann Appl Probab 5(1):294–309,
1995), who made use of Wald’s equation and a Markov chain coupling argument. We also derive a new approximation of
\mathbbV(t)\mathbb{V}(\tau), provide an (improved) bound on the error in these approximations, derive a recurrence for
\mathbbE(t)\mathbb{E}(\tau), give a new large deviation type result for tail probabilities, and look at some special cases. 相似文献
10.
Let μ be a Poisson random measure, let
\mathbbF\mathbb{F} be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let
\mathbbG\mathbb{G} be the initial enlargement of
\mathbbF\mathbb{F} with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the
\mathbbG\mathbb{G}-compensator relative to the
\mathbbF\mathbb{F}-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the
filtration is enlarged from
\mathbbF\mathbb{F} to
\mathbbG\mathbb{G}. In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable
\mathbbF\mathbb{F}-martingale is a
\mathbbG\mathbb{G}-semimartingale that belongs to the normed space S1\mathcal{S}^{1} relative to
\mathbbG\mathbb{G}. 相似文献
11.
Christopher Kennedy 《Algebras and Representation Theory》2011,14(6):1187-1202
This paper continues the study of associative and Lie deep matrix algebras,
DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and
\mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras,
BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and
\mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal
lattices. In particular,
\mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra
\mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct
product of
\mathfraksln{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on
\mathfraksl2\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of
\mathfraksl2{\mathfrak{{sl}_2}}) and
\mathfrakbld{\mathfrak{bld}}. 相似文献
12.
For any category of interest ℂ we define a general category of groups with operations
\mathbbCG, \mathbbC\hookrightarrow\mathbbCG\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}, and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of
\mathbbCG\mathbb{C_G}. The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings
of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product
\textUSGA(A)\ltimes A{\text{USGA}}(A)\ltimes A is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative,
alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always
exist actors. 相似文献
13.
Boris Širola 《Central European Journal of Mathematics》2011,9(6):1317-1332
Let
\mathbbK\mathbb{K} be a field, G a reductive algebraic
\mathbbK\mathbb{K}-group, and G
1 ≤ G a reductive subgroup. For G
1 ≤ G, the corresponding groups of
\mathbbK\mathbb{K}-points, we study the normalizer N = N
G
(G
1). In particular, for a standard embedding of the odd orthogonal group G
1 = SO(m,
\mathbbK\mathbb{K}) in G = SL(m,
\mathbbK\mathbb{K}) we have N ≅ G
1 ⋊ μ
m
(
\mathbbK\mathbb{K}), the semidirect product of G
1 by the group of m-th roots of unity in
\mathbbK\mathbb{K}. The normalizers of the even orthogonal and symplectic subgroup of SL(2n,
\mathbbK\mathbb{K}) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof
in the odd orthogonal case to be completed here. Also, for G = GL(m,
\mathbbK\mathbb{K}) and G
1 = O(m,
\mathbbK\mathbb{K}) we have N ≅ G
1 ⋊
\mathbbK\mathbb{K}
×. In both of these cases, N is a self-normalizing subgroup of G. 相似文献
14.
Let K be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras
whose group of group-likes is
\mathbbS4\mathbb{S}_4. We also describe all pointed Hopf algebras over
\mathbbS5\mathbb{S}_5 whose infinitesimal braiding is associated to the rack of transpositions. 相似文献
15.
Juan A. Aledo Victorino Lozano José A. Pastor 《Mediterranean Journal of Mathematics》2010,7(3):263-270
We prove that the only compact surfaces of positive constant Gaussian curvature in
\mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in
\mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant
angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive
constant Gaussian curvature in
\mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in
\mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds
are attained, the surface is again a piece of a rotational complete surface. 相似文献
16.
Raymond Mortini 《Monatshefte für Mathematik》2011,36(4):299-311
We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra
A(\mathbbD){A(\mathbb{D})}, many of its subalgebras, and the algebra H
∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable
rank of
A(\mathbbD){A(\mathbb{D})} and H
∞ is bigger than 2. 相似文献
17.
An integral coefficient matrix determines an integral arrangement of hyperplanes in
\mathbbRm{\mathbb{R}^m} . After modulo q reduction ${(q \in {\mathbb{Z}_{ >0 }})}${(q \in {\mathbb{Z}_{ >0 }})} , the same matrix determines an arrangement Aq{\mathcal{A}_q} of “hyperplanes” in
\mathbbZmq{\mathbb{Z}^m_q} . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of Aq{\mathcal{A}_q} in
\mathbbZmq{\mathbb{Z}^m_q} is a quasi-polynomial in ${q \in {\mathbb{Z}_{ >0 }}}${q \in {\mathbb{Z}_{ >0 }}} . Moreover, they proved in the central case that the intersection lattice of Aq{\mathcal{A}_q} is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement
[^(B)]m[0,a]{\hat{\mathcal{B}}_m^{[0,a]}} of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results. 相似文献
18.
Let
\mathbbF\mathbb{F} be a field of characteristic 0, and let G be an additive subgroup of
\mathbbF\mathbb{F}. We define a class of infinite-dimensional Lie algebras
\mathbbF\mathbb{F}-basis {L
μ, V
μ, W
μ | μ ∈ G}, which are very closely related to W-algebras. In this paper, the second cohomology group of is determined. 相似文献
19.
We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function
f : V ? \mathbbC{f : V \rightarrow \mathbb{C}} on a finite-dimensional vector space V over a finite field
\mathbbF{\mathbb{F}} has large Gowers uniformity norm ||f||Us+1(V){{\parallel{f}\parallel_{U^{s+1}(V)}}} , then there exists a (non-classical) polynomial
P: V ? \mathbbT{P: V \rightarrow \mathbb{T}} of degree at most s such that f correlates with the phase e(P) = e
2πiP
. This conjecture had already been established in the “high characteristic case”, when the characteristic of
\mathbbF{\mathbb{F}} is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work
of Green and the first author [22] and of Kaufman and Lovett [28]. 相似文献
20.
Taras Radul 《Applied Categorical Structures》2011,19(4):729-739
Let
\mathbbF\mathbb{F} be a monad in the category Comp. We build for each
\mathbbF\mathbb{F}-algebra a convexity in general sense (see van de Vel 1993). We investigate properties of such convexities and apply them to prove that the multiplication map of the order-preserving
functional monad is soft. 相似文献