Indecomposable Components Research Articles

Set-decomposition of normal rectifiable $G$-chains via an abstract decomposition principle

We introduce the notion of set-decomposition of a normal G -flat chain A in \mathbb{R}^{n} as a sequence A_{j}=A\,\text{\Large{$\llcorner$}}S_{j} associated to a Borel partition S_{j} of \mathbb{R}^{n} such that \mathbb{N}(A)=\sum \mathbb{N}(A_{j}) . We show that any normal rectifiable G -flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their “measure theoretic” connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer.As opposed to previous results, we do not assume that G is boundedly compact. Therefore, we cannot rely on the compactness of sequences of chains with uniformly bounded \N -norms. We deduce instead the result from a new abstract decomposition principle.As in earlier proofs, a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h -mass to replace integrality.

Branching problem for tensoring two Verma modules and its application to differential symmetry breaking operators

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of [Formula: see text] occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras [Formula: see text].

Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation

AbstractFor any $$\Lambda >0$$ Λ > 0 , let $$\mathcal {M}_{n,\Lambda }$$ M n , Λ denote the space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in Euclidean space $$\mathbb {R}^{n+m}$$ R n + m with uniformly bounded 2-dilation $$\Lambda $$ Λ of their graphic functions. In this paper, we show that this is a natural class to extend structural results known for codimension one. In particular, we prove that any tangent cone C of $$M\in \mathcal {M}_{n,\Lambda }$$ M ∈ M n , Λ at infinity has multiplicity one. This enables us to get a Neumann–Poincaré inequality on stationary indecomposable components of C. A corollary is a Liouville theorem for M. For small $$\Lambda >1$$ Λ > 1 (we can take any $$\Lambda <\sqrt{2}$$ Λ < 2 ), we prove that (i) for $$n\le 7$$ n ≤ 7 , M is flat; (ii) for $$n>8$$ n > 8 and a non-flat M, any tangent cone of M at infinity is a multiplicity one quasi-cylindrical minimal cone in $$\mathbb {R}^{n+m}$$ R n + m whose singular set has dimension $$\le n-7$$ ≤ n - 7 .

Pointed Hopf algebras of discrete corepresentation type

We classify pointed Hopf algebras of discrete corepresentation type over an algebraically closed field K with characteristic zero. For such algebras H, we explicitly determine the algebra structure up to isomorphism for the link indecomposable component B containing the unit. It turns out that H is a crossed product of B and a certain group algebra.

Decomposition of integral metric currents

In the setting of complete metric spaces, we prove that integral currents can be decomposed as a sum of indecomposable components. In the special case of one-dimensional integral currents, we also show that the indecomposable ones are exactly those associated with injective Lipschitz curves or injective Lipschitz loops, therefore extending Federer's characterisation to metric spaces. Moreover, some applications of our main results will be discussed.

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration \({\cal X}\) is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of \({\cal X}\) comes from a certain Cartesian decomposition. It is proved that if the coherent configuration \({\cal X}\) is thick, then there is a unique maximal Cartesian decomposition of \({\cal X}\), i.e., there is exactly one internal tensor decomposition of \({\cal X}\) into indecomposable components. In particular, this implies an analog of the Krull-Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.

CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY

Let (V, ω) be an orthosymplectic ℤ2-graded vector space and let 𝔤:= 𝔤𝔬𝔰𝔭 (V, ω) denote the Lie superalgebra of similitudes of (V, ω). It is known that as a 𝔤-module, the space (V ) of superpolynomials on V is completely reducible, unless dim \( {V}_{\overline{\mathrm{o}}} \) and dim \( {V}_{\overline{1}} \) are positive even integers and dim \( {V}_{\overline{\mathrm{O}}}\le \dim\ {V}_{\overline{1}} \). When (V ) is not a completely reducible 𝔤-module, we construct a natural basis \( {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} \) of “Capelli operators” for the algebra (V ) 𝔤 of 𝔤 -invariant superpolynomial superdifferential operators on V , where the index set 𝒯 is the set of integer partitions of length at most two. We compute the action of the operators \( {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} \) on maximal indecomposable components of (V ) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where (V ) is completely reducible, the eigenvalues of a subfamily of the {D⋋} are not given by specializing the Knop–Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. This is in contrast with what occurs for previously studied Capelli operators. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall–Ramanujan hypergeometric identity.We also transcend our results on the eigenvalues of Capelli operators to the Deligne category Rep (Ot). More precisely, we define categorical Capelli operators \( {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} \) that induce morphisms of indecomposable components of symmetric powers of Vt, where Vt is the generating object of Rep (Ot). We obtain formulas for the eigenvalue polynomials associated to the \( {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} \) that are analogous to our results for the operators \( {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} \).

A Clebsch-Gordan decomposition in positive characteristic

Let K be an algebraically closed field and G=SL2(K). Let E be the natural module and SrE the rth symmetric power. We consider here, for r,s≥0, the tensor product of SrE and the dual of SsE. In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when K is a field of positive characteristic. We show that each indecomposable component occurs with multiplicity one and identify which modules occur for given r and s.

Decomposing Gorenstein rings as connected sums

In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know whether it decomposes as a connected sum and if so, what are its components. We answer these questions in the Artinian case and investigate conditions on the ring which force it to be indecomposable as a connected sum. We further give a characterization for Gorenstein Artin local rings to be decomposable as connected sums, and as a consequence, obtain results about its Poincaré series and minimal number of generators of its defining ideal. Finally, we show that the indecomposable components appearing in the connected sum decomposition are unique up to isomorphism.

Generators for decompositions of tensor products of modules associated with standard Jordan partitions

ABSTRACTIf K is a field of finite characteristic p, G is a cyclic group of order q = pα, U and W are indecomposable KG-modules with dim U = m and dim W = n, and λ(m,n,p) is a standard Jordan partition of mn, we describe how to find a generator for each of the indecomposable components of the KG-module U ⊗ W.

Diamond cone for [formula omitted

In the present paper, we define the diamond cone for the Lie superalgebra spo(2m,1), considering the (covariant) tensor representation of spo(2m,1). The diamond cone is no more indecomposable. Nevertheless, we give a basis for each indecomposable component, using quasistandard Young tableaux for spo(2m,1).We realize a bijection between the set of semistandard tableaux with shape λ and the set of quasistandard tableaux with shape μ≤λ. This gives the compatibility of the diamond cone with the natural stratification of the reduced shape algebra.

Computing the extensions of preinjective and preprojective Kronecker modules

Let I, I′ be preinjective Kronecker modules (i.e. all their indecomposable components are preinjective). We describe the modules M for which there exists an exact sequence 0→I′→M→I→0 by explicit, easy to check numerical conditions, resulting in an algorithm (linear in the number of indecomposable components) for the decision problem. We also propose a method to generate all extensions of I′ by I and we give a different proof for a theorem in [13] providing numerical criteria in terms of Kronecker invariants for the existence of a monomorphism f:I′→I. All these results apply dually to preprojective modules as well.

Automorphisms of real Lie algebras of dimension five or less

The Lie algebra version of the Krull–Schmidt theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form.

Characterization of Stability Region for General Autonomous Nonlinear Dynamical Systems

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets.

Generators for decompositions of tensor products of modules

If K is a field of finite characteristic p, G is a cyclic group of order q = pα, U and W are indecomposable KG-modules, and p ≥ dim U + dim W − 1, we describe how to find a generator for each of the indecomposable components of the KG-module \({U \otimes W}\).

On preprojective short exact sequences in the Kronecker case

Let P , P ′ be preprojective Kronecker modules (i.e., all their indecomposable components are preprojective). We give a numerical criterion in terms of the so-called Kronecker invariants for the existence of an epimorphism g : P → P ′ . As an application, we describe the possible middle terms in certain preprojective short exact sequences. We also prove that the possible middle terms in preprojective short exact sequences do not depend on the base field k . All the results above can be dualized for preinjective modules.

Diamond cone for [formula omitted

The diamond cone S red for a semi simple Lie algebra g is a quotient of the shape algebra S of g . If n is the nilpotent factor of the Iwasawa decomposition of g , we get an indecomposable n -module. If g = sl ( m ) or sp ( 2 m ) , particular basis in S red , were defined, using the notion of quasi standard Young tableaux. In the present paper, we define the diamond cone for the Lie superalgebra sl ( m / 1 ) , starting with the covariant tensor representations of sl ( m / 1 ) . The diamond cone is no more indecomposable, but we give basis for each its indecomposable component, using quasi standard Young tableaux for sl ( m / 1 ) .

A decomposability criterion for elementary theories

We prove that each elementary theory has a unique decomposition into indecomposable components and formulate a decomposability criterion.

Bipartite operator decomposition of graphs and the reconstruction conjecture

Abstract We consider a binary operation • on the set of triads ( G , A , B ) , where G is a graph and ( A , B ) is the partition of the set of the set V ( G ) . The operation • of multiplication of a triad and a graph is defined and the properties of the introduced operations are described. We study in detail the subcase, when the triads have the form ( G , A , B ) , where G is a bipartite graph and ( A , B ) is its bipartition. For this case the decomposition theorem stating that any graph except the described family of exceptions can be uniquely decomposed into indecomposable components with respect to the operation • is proved. Using this theorem, we proved the following. Let the graph G have a module M with associated partition ( A , B , M ) , where A ∼ M and B ≁ M , such that G [ A ∪ B ] is a bipartite graph with bipartition ( A , B ) . Then the graph G is reconstructible.

Varieties of modules for [formula omitted

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C , that is the set of pairs of ( n × n ) -matrices ( A , B ) such that A 2 = B 2 = [ A , B ] = 0 , is equidimensional. C can be identified with the ‘variety of n-dimensional modules’ for Z / 2 Z × Z / 2 Z , or equivalently, for k [ X , Y ] / ( X 2 , Y 2 ) . On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e 2 = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z / 2 Z × Z / 2 Z -modules.