Pair Of Algebras Research Articles

Derivation double Lie algebras

We study classical [Formula: see text]-matrices [Formula: see text] for Lie algebras [Formula: see text] such that [Formula: see text] is also a derivation of [Formula: see text]. This yields derivation double Lie algebras [Formula: see text]. The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

Normal connections on three-dimensional manifolds with solvable transformation group

The purpose of the work is the classification of three-dimensional homogeneous spaces, allowing a normal connection, description of invariant affine connections on those spaces together with their curvature and torsion tensors, holonomy algebras. We consider only the case, when Lie group is solvable. The local classification of homogeneous spaces is equivalent to the description of the effective pairs of Lie algebras. We study the holonomy algebras of homogeneous spaces and find when the invariant connection is normal. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character.

The Drinfel’d double versus the Heisenberg double for Hom-Hopf algebras

Let ([Formula: see text], [Formula: see text]) be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel’d double [Formula: see text] in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid’s bicrossproduct for Hopf algebras (see [S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995)]) and another one is to introduce the notion of dual pairs of Hom-Hopf algebras. Then we study the relation between the Drinfel’d double [Formula: see text] and Heisenberg double [Formula: see text], generalizing the main result in [J. H. Lu, On the Drinfel’d double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763–776]. The examples given in the paper are especially, not obtained from the usual Hopf algebras.

On Hom-Prealternative Bialgebras

The aim of this paper is to study phase spaces of Hom-alternative algebras. We introduce notions of Hom-prealternative algebra and Hom-prealternative bialgebra. Bimodules and matched pairs of Hom-prealternative algebra are also considered. Furthermore, we show that the notion of phase space of a Hom-alternative algebra is equivalent to the notion of Hom-prealternative bialgebra. The coboundary Hom-prealternative bialgebra and Hom-PA equation are also described.

Some upper bounds on the dimension of the Schur multiplierof a pair of nilpotent Lie algebras

Let $(L,N)$ be a pair of Lie algebras where $N$ is an ideal of the finite dimensional nilpotent Lie algebra $L$. Some upper bounds on the dimension of the Schur multiplier of $(L,N)$ are obtained without considering the existence of a complement for $N$. These results are applied to derive a new bound on the dimension of the Schur multiplier of a nilpotent Lie algebra.

Symmetry criteria for quantum simulability of effective interactions

What can one do with a given tunable quantum device? We provide complete symmetry criteria deciding whether some effective target interaction(s) can be simulated by a set of given interactions. Symmetries lead to a better understanding of simulation and permit a reasoning beyond the limitations of the usual explicit Lie closure. Conserved quantities induced by symmetries pave the way to a resource theory for simulability. On a general level, one can now decide equality for any pair of compact Lie algebras just given by their generators without determining the algebras explicitly. Several physical examples are illustrated, including entanglement invariants, the relation to unitary gate membership problems, as well as the central-spin model.

A visit to maximal non-ACCP subrings

In this paper, we characterize maximal non-ACCP subrings R of a domain S in case (R, S) is a residually algebraic pair and R is semilocal. In this paper, we also consider a K-algebra S, a nonzero proper ideal I of S and a subring D of the field K and we determine necessary and sufficient conditions in order that D + I is a maximal non-ACCP subring of S. This gives an example of a maximal non-ACCP subring R of a domain S such that (R, S) is a normal pair and R is not semilocal.

On the restriction of Zuckerman’s derived functor modules A q (λ) to reductive subgroups

In this article, we study the restriction of Zuckerman's derived functor $({\frak g},K)$-modules $A_{\frak q}}(\lambda)$ to ${\frak g}'$ for symmetric pairs of reductive Lie algebras $({\frak g},{\frak g}')$. When the restriction decomposes into irreducible $({\frak g}',K')$-modules, we give an upper bound for the branching law. In particular, we prove that each $({\frak g}',K')$-module occurring in the restriction is isomorphic to a submodule of $A_{\frak q}'}(\lambda')$ for a parabolic subalgebra ${\frak q}'$ of ${\frak g}'$, and determine their associated varieties. For the proof, we realize $A_{\frak q}}(\lambda)$ on complex partial flag varieties by using ${\mathcal D}$-modules.

Finite-dimensional representations of the elliptic modular double

We investigate the kernel space of an integral operator M(g) depending on the “spin” g and describing an elliptic Fourier transformation. The operator M(g) is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with the parameters η and τ, Imτ > 0, Imη > 0. For two-dimensional lattices g = nη + mτ/2 and g = 1/2 + nη + mτ/2 with incommensurate 1, 2η, τand integers n,m > 0, the operator M(g) has a finite-dimensional kernel that consists of the products of theta functions with two different modular parameters and is invariant under the action of generators of the elliptic modular double.

Lie foliations transversely modeled on nilpotent Lie algebras

In this paper, we study on the following two problems of realization for Lie foliations. (1) Which pair of Lie algebras $$(\mathfrak {g},\mathfrak {h})$$ can be realized as a Lie $$\mathfrak {g}$$ -foliation in a closed manifold with the structure Lie algebra $$\mathfrak {h}$$ ? (2) Which pair $$(\mathfrak {g},m)$$ can be realized as a Lie $$\mathfrak {g}$$ -flow in a closed manifold with the structure Lie algebra $${\mathbb {R}}^m$$ ? We give a complete answer to (1) in the case where $$\mathfrak {g}$$ is a nilpotent Lie algebra and give a complete answer to (2) in the case where $$\mathfrak {g}$$ is a nilpotent Lie algebra which has a rational structure.

Action of derived automorphisms on infinity-morphisms

In this paper we investigate how to simultaneously change homotopy algebras of a certain type and a corresponding infinity morphism between them, and show that this can be done in a homotopically unique way. More precisely, for a reduced cooperad C, given \Omega(C)-algebras V and W and an infinity-morphism U from V to W, for any derivation \phi of \Omega(C) we produce new \Omega(C)-algebras V' and W' and a new infinity-morphism U' between them, that are unique up to homotopy. Operads play the central role in answering this question, in particular a 2-colored operad Cyl(C) that governs pairs of homotopy algebras and infinity-morphisms between them.

Cohomology jump loci of differential graded Lie algebras

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

A construction of derived equivalent pairs of symmetric algebras

Recently, Hu and Xi have exhibited derived equivalent endomorphism rings arising from (relative) almost split sequences as well as AR-triangles in triangulated categories. We present a broader class of triangles (in algebraic triangulated categories) for which the endomorphism rings of different terms are derived equivalent. We then study applications involving 0-Calabi-Yau triangulated categories. In particular, applying our results in the category of perfect complexes over a symmetric algebra gives a nice way of producing pairs of derived equivalent symmetric algebras. Included in the examples we work out are some of the algebras of dihedral type with two or three simple modules. We also apply our results to stable categories of Cohen-Macaulay modules over odd-dimensional Gorenstein hypersurfaces having an isolated singularity.

Characterization of relative n-isoclinism of a pair of Lie algebras

In 1994, Moneyhun introduced and studied the concept of isoclinism in Lie algebras. Moghaddam and parvaneh in 2009 introduced and gave some properties of isoclinism of a pair of Lie algebras. In this paper we introduce the concept of relative n-isoclinism between two pairs of Lie algebras. They form equivalence classes and we show that each equivalent class contains an n-stem pair of Lie algebras. We also show that for a relative n-isoclinism family of Lie algebras C say, consists of (M, L) such that L is finitely generated and [M, n L] is finite dimension, then there exists an n-stem pair of Lie algebras (R, S) ∊ C such that ; for all (M, L) ∊ C}.

A Note on Padé Approximants Pairs as Limits of Algebraic Polynomials Pairs of Weighted Best Approximation in Orlicz Spaces

Fil: Levis, Fabian Eduardo. Universidad Nacional de Rio Cuarto. Facultad de Ciencias Exactas, Fisico-Quimicas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

Abstract structure of unitary oracles for quantum algorithms

We show that a pair of complementary dagger-Frobenius algebras, equipped with a self-conjugate comonoid homomorphism onto one of the algebras, produce a nontrivial unitary morphism on the product of the algebras. This gives an abstract understanding of the structure of an oracle in a quantum computation, and we apply this understanding to develop a new algorithm for the deterministic identification of group homomorphisms into abelian groups. We also discuss an application to the categorical theory of signal-flow networks.

The decategorification of bordered Khovanov homology

In [A type D structure in Khovanov homology, preprint (2013), arXiv:1304.0463; A type A structure in Khovanov homology, preprint (2013), arXiv:1304.0465] the author constructed a package which describes how to decompose the Khovanov homology of a link ℒ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for ℒ is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the types A and D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to I. Petkova's decategorification of bordered Floer homology, [The decategorification of bordered Heegaard-Floer homology, preprint (2012), arXiv:1212.4529v1], and shows how they recover the Jones polynomial. We also use the decategorifications to compare this approach to tangle decompositions with M. Khovanov's from [A functor-valued invariant of tangles, Algebr. Geom. Topol.2 (2002) 665–741].

The logarithms of Dehn twists

Let Σ be an oriented connected compact surface of genus g (≥ 1) with 1 boundary component. Choose a basepoint ∗ ∈ ∂Σ. We denote π := π1(Σ, ∗) and H := H1(Σ;Q). The simple loop going around the boundary in the opposite direction defines an element ζ ∈ π. Any simple closed curve C ⊂ Σ defines the right handed Dehn twist tC along C as an element of the mapping class group of the surface Σ relative to the boundary ∂Σ. The classical formula says the action |tC | of the Dehn twist tC on the homology group H is given by |tC | = 1H − [C] ⊗ [C] ∈ Hom(H,H), where [C] ∈ H is the homology class of C with a fixed orientation, and we identify H ⊗H = Hom(H,H), Y ⊗ Z → (X → (X · Y )Z), by the Poincare duality. Our result generalizes this formula to the action of tC on the completed group ring Qπ, where the completion is induced by the augmentation ideal Iπ ⊂ Qπ. Massuyeau [10] introduced the notion of a symplectic expansion of the group π, which provides an isomorphism of pairs of complete Hopf algebras

Distinguished nilpotent orbits, Kostant pairs and normalizers of Lie algebras

A pair of Lie algebras (g,g1) will be called a Kostant pair if g is semisimple, g1 is reductive in g and the restriction of the Killing form Bg to g1 is nondegenerate. We study the class of such (nonsymmetric) pairs and obtain some useful and new structural results. We study the structure of the normalizers Ng(g1), and as a consequence we obtain some corresponding worthy results about algebraic groups. In particular we consider an interesting case when g1 is a distinguished sl2-subalgebra of g. Combined with the research due to V.L. Popov we observe that the notions of self-normalizing (reductive) subalgebras of a semisimple Lie algebra and projective self-dual algebraic subvarieties of the usual nilpotent cones are closely related.

Categorification and Heisenberg doubles arising from towers of algebras

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doubles and show that if induction and restriction satisfy Mackey-like isomorphisms then the Fock space representation of the Heisenberg double has a natural categorification. This unifies the existing categorifications of the polynomial representation of the Weyl algebra and the Fock space representation of the Heisenberg algebra. Second, we develop in detail the theory applied to the tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenberg algebras. As an application of a generalized Stone–von Neumann Theorem, we give a new proof of the fact that the ring of quasisymmetric functions is free over the ring of symmetric functions.