On the Dimension of the Congruence Centralizer

Doklady Mathematics - Let A be a nonsingular complex (n × n) matrix. The congruence centralizer of A is the collection $$\mathcal{L}$$ of matrices X satisfying the relation $$X{\kern 1pt}...     

Quantum algebras with representation ring of $$ \mathfrak{s}\mathfrak{l}_2 $$ type

A reducible representation of the Temperley-Lieb algebra is constructed on a tensor product of n-dimensional spaces. As a centralizer of this action, we obtain a quantum algebra (quasi-triangular Hopf algebra) with the representation ring that is equivalent to the representation ring of the Lie algebra. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 167–177.     

完全分配CSL代数上的中心化子

设L是可分Hilbert空间上的完全分配交换子空间格,A是Alg L的子代数并且包含Alg L的全体有限秩算子.主要结果是:(1)A上的中心化子是拟空间的;(2)Alg L上的Jordan中心化子是中心化子;(3)当L是套时,Alg L上的Lie中心化子可表示成一个中心化子与一个可加泛函之和的形式,该泛函作用在形如AB-BA的算子上为零.     

On commuting polynomial automorphisms of {\mathbb{C}}^{k} , k ≥ 3

We characterize the polynomial automorphisms of ${\mathbb{C}}^3We characterize the polynomial automorphisms of , which commute with a regular automorphism. We use their meromorphic extension to and consider their dynamics on the hyperplane at infinity. We conjecture the additional hypothesis under which the same characterization is true in all dimensions. We give a partial answer to a question of S. Smale that in our context can be formulated as follows: can any polynomial automorphism of be the uniform limit on compact sets of polynomial automorphisms with trivial centralizer (i.e. )? Partially supported by Progetto MURST di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse. Supported by Istituto Nazionale Alta Matematica, “F. Severi”, Roma and G.N.S.A.G.A., Roma.     

Local Duality for Modules over Noetherian Commutative Rings

Some applications of the general theorem on the existence of local duality for modules over Noetherian commutative rings are given. Let be a Noetherian commutative ring, let be a set of maximal ideals in , and let . Then the category of Artinian modules is dual to the category of Noetherian modules. Several structural results are proved, including the theorem on the structure of Artinian modules over principal ideal domains. For rings of special kinds, double centralizer theorems are proved. Bibliography: 5 titles.     

SPECTRAL PROPERTY OF CERTAIN MORAN MEASURES IN $\mathbb{R}^{n}$ R n

Acta Mathematica Hungarica - Let $$R_k = {\rm diag}(r_{k,1}, r_{k,2}, \ldots, r_{k,n})$$ be the $$n \times n$$ expanding integer diagonal matrix, and let $$D_k\subset \mathbb Z^{n}$$ . We show that...     

Compactness and local compactness for {\mathbb{R}}-trees

We establish (geometric) criteria for an -tree to be compact and to be locally compact. It follows that locally compact -trees are separable. Received: 10 September 2007     

Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in $$\mathbb {R}^3$$ R 3 Using Regularization and Blowup

Journal of Nonlinear Science - Two-fold singularities in a piecewise smooth (PWS) dynamical system in $$\mathbb {R}^3$$ have long been the subject of intensive investigation. The interest stems...     

Algebras in which non-scalar elements have small centralizers

We describe algebras in which the centralizer of every non-scalar element is equal to the subalgebra generated by this element, and finite-dimensional algebras (over perfect fields) in which the centralizer of every non-scalar element is commutative.     

On critical nets in $${\mathbb {R}}^k$$ R k

Geometriae Dedicata - Critical nets in $${\mathbb {R}}^k$$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length...     

On automorphisms of A-groups

Let G be an A-group (i.e. a group in which xx ^α  = x ^α x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that, in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r. Received: 23 March 2008, Revised: 20 May 2008     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

When is a centralizer near-ring isomorphic to a matrix near-ring?

Necessary conditions are found for a centralizer near-ring M_A(G) to be isomorphic to a matrix near-ring, where G is a finite group which is cyclic as an M_A(G)-module There are centralizer near-rings which are matrix near-rings. A class of such near-rings is exhibited. Examples of centralizer near-rings which are not matrix near-rings are given.     

Dark $$f(\mathcal{R},\varphi,\chi)$$ universe with Noether symmetry

Theoretical and Mathematical Physics - Using the Noether symmetry approach, we investigate $$f( \mathcal{R} , \varphi ,\chi)$$ theories of gravity, where $$ \mathcal{R} $$ is the scalar curvature,...     

A Note on the Tight Spherical 7-Design in {\mathbb R}^{23} and 5-Design in {\mathbb R}^{7*}

In this note we prove the uniqueness of the tight spherical 7-design in consisting of 4600 vectors and with automorphism group 2 × Co₂ as well as the uniqueness of the tight spherical 5-design in on 112 vectors and with automorphism group 2 × Sp₆(2).To the memory of Jaap Seidel     

The Cartan matrix of a centralizer algebra

The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension.     

On the radical of Brauer algebras

The radical of the Brauer algebra is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of . The proof is by direct approach for x  =  0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable -modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley–Lieb algebra—realised inside —acting on it.

“Ahi quanto a dir che sia è cosa dura lo radical dell’algebra di Brauer pur se’l pensier già muove a congettura” N. Barbecue, “Scholia”

Partially supported by the European RTN “LieGrits”, contract no. MRTN-CT-2003-505078, and by the Italian PRIN 2005 “Moduli e teorie di Lie”.     

A representation of bivariate extreme value distributions via norms on \mathbb{R}^{2}

It is known that a bivariate extreme value distribution (EVD) with reverse exponential margins can be represented as , , where is a suitable norm on . We prove in this paper the converse implication, i.e., given an arbitrary norm on , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , we obtain a characterization of the class of EVDs in terms of compact and convex subsets of .