Sum Of Algebras Research Articles

Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra

In this paper we study non-nilpotent non-Lie Leibniz F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document}-algebras with one-dimensional derived subalgebra, where F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document} is a field with char(F)≠2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {char}}(\\mathbb {F}) \ e 2$$\\end{document}. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document}, where n=dimFLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=\\dim _\\mathbb {F}L_n$$\\end{document}. This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), which is only valid when F=C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}=\\mathbb {C}$$\\end{document}. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document}. Eventually, we solve the coquecigrue problem for Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_n$$\\end{document} by integrating it into a Lie rack.

Exactly solvable Hamiltonian fragments obtained from a direct sum of Lie algebras.

Exactly solvable Hamiltonians are useful in the study of quantum many-body systems using quantum computers. In the variational quantum eigensolver, a decomposition of the target Hamiltonian into exactly solvable fragments can be used for the evaluation of the energies via repeated quantum measurements. In this work, we apply more general classes of exactly solvable qubit Hamiltonians than previously considered to address the Hamiltonian measurement problem. The most general exactly solvable Hamiltonians we use are defined by the condition that within each simultaneous eigenspace of a set of Pauli symmetries, the Hamiltonian acts effectively as an element of a direct sum of so(N) Lie algebras and can, therefore, be measured using a combination of unitaries in the associated Lie group, Clifford unitaries, and mid-circuit measurements. The application of such Hamiltonians to decomposing molecular electronic Hamiltonians via graph partitioning techniques shows a reduction in the total number of measurements required to estimate the expectation value compared to previously used exactly solvable qubit Hamiltonians.

Classification of flat Lorentzian nilpotent Lie algebras

AbstractWe give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo‐Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left‐invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if denotes the ‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are and the semidirect products and , defined by some particular derivations . In all those cases we also find the equivalence classes of flat Lorentzian products.

On a new sum of equality algebras

This paper explores the construction of ordinal sums in the class of equality algebras. Although the ordinal sum of linear equality algebras remains linear, this property might not hold for representable equality algebras in general. We present (prime) deductive systems for the ordinal sum of each family of equality algebras, which help us find a necessary and sufficient condition under which an ordinal sum of equality algebras is representable.

On derivations of Leibniz algebras

<p>Leibniz algebras are non-antisymmetric generalizations of Lie algebras. In this paper, we investigate the properties of complete Leibniz algebras under certain conditions on their extensions. Additionally, we explore the properties of derivations and direct sums of Leibniz algebras, proving several results analogous to those in Lie algebras.</p>

Multiparameter colored partition category and the product of the reduced Kronecker coefficients

We introduce and study a multiparameter colored partition category CPar(x) by extending the construction of the partition category, over an algebraically closed field k of characteristic zero and for a multiparameter x∈kr. The morphism spaces in CPar(x) have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group Cr. We show that the endomorphism spaces of CPar(x) and additive Karoubi envelope of CPar(x) are generically semisimple. The category CPar(x) is rigid symmetric strict monoidal and we give a presentation of CPar(x) as a monoidal category. The path algebra of CPar(x) admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups G(r,n). We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of CPar(x) in two ways. Among other things, this gives a formula for the product of the reduced Kronecker coefficients in terms of the Littlewood–Richardson coefficients for G(r,n) and certain Kronecker coefficients for the wreath product (Cr×Cr)≀Sn. For r=1, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson–Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.

Generalisations of Hecke algebras from loop braid groups

We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we %introduce consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (non-functor) Burau representation and the (functor) Deguchi {\em et al}-Kauffman--Saleur-Rittenberg representations here called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t \in k$ the loop-Hecke parameter. We prove the following: 1) $LH_n$ is finite dimensional over a field. 2) The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 3) Over $k=\mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. 4) We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 5) The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. \item For $t^2 \neq 1$ then $LH_n \cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.

Classification of generalized Einstein metrics on three-dimensional Lie groups

AbstractWe develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor, and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric, and the divergence operator. The resulting expression is polynomial and homogeneous of degree 2 in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.

Bounded version of approximate module character amenability of Banach algebras

The bounded version of approximate module character amenability of Banach algebras is introduced and studied. This new concept is characterized by several different concepts such as bounded approximate module character means. Moreover, this new concept is investigated for second dual, unitization, tensor product and lp-direct sums of Banach algebras.

Bounded approximate version of module character contractibility of Banach algebras

The (bounded) approximate version of module character contractibility of Banach algebras is introduced and studied. This new concept is characterized by several different concepts such as bounded approximate module character diagonals. Moreover, this new concept is investigated for second dual, unitization, tensor product and lp-direct sums of Banach algebras.

The structure of symmetric tensor powers of composition algebras

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.

Rigidity results for Lie algebras admitting a post-Lie algebra structure

We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. We show that if [Formula: see text] is semisimple and [Formula: see text] is arbitrary, then we have rigidity in the sense that [Formula: see text] and [Formula: see text] must be isomorphic. The proof uses a result on the decomposition of a Lie algebra [Formula: see text] as the direct vector space sum of two semisimple subalgebras. We show that [Formula: see text] must be semisimple and hence isomorphic to the direct Lie algebra sum [Formula: see text]. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras [Formula: see text]. We prove additional existence results for pairs [Formula: see text], where [Formula: see text] is complete, and for pairs, where [Formula: see text] is reductive with [Formula: see text]-dimensional center and [Formula: see text] is solvable or nilpotent.

On Parafree Leibniz Algebras

The parafree Leibniz algebras are a special class of Leibniz algebras which have many properties with a free Leibniz algebra. In this note, we introduce the structure of parafree Leibniz algebras. We survey the
 important results in parafree Leibniz algebras which are analogs of corresponding results in parafree Lie algebras. We first investigate some properties of subalgebras and quotient algebras of parafree Leibniz algebras. Then, we describe the direct sum of parafree Leibniz algebras. We show that the direct sum of two parafree Leibniz algebras is a Leibniz algebra. Furthermore, we prove that the direct sum of two parafree Leibniz algebras is again parafree.

On left φ-essential Connes-amenability of Banach algebras

In this paper, we introduce and study the notion of left [Formula: see text]-essential Connes amenable for dual Banach algebras. We investigate the hereditary properties of this new concept and we give some results for [Formula: see text]-Lau product and module extension. For unital dual Banach algebras, we show that left [Formula: see text]-essential Connes-amenability and left [Formula: see text]-Connes amenability are equivalent. Finally, with various examples, we examined this concept for upper triangular matrix algebras and [Formula: see text]-direct sum of Banach algebras.

The structure of algebraic Baer ∗-algebras

The purpose of this note is to describe when a general complex algebraic $^*$-algebra is pre-$C^*$-normed, and to investigate their structure when the $^*$-algebras are Baer $^*$-rings in addition to algebraicity. As a main result we prove the following theorem for complex algebraic Baer $^*$-algebras: every $^*$-algebra of this kind can be decomposed as a direct sum $M\oplus B$, where $M$ is a finite dimensional Baer $^*$-algebra and $B$ is a commutative algebraic Baer $^*$-algebra. The summand $M$ is $^*$-isomorphic to a finite direct sum of full complex matrix algebras of size at least $2\times2$. The commutative summand $B$ is $^*$-isomorphic to the linear span of the characteristic functions of the clopen sets in a Stonean topological space. As an application we show that a group $G$ is finite exactly when the complex group algebra $\mathbb{C}[G]$ is an algebraic Baer $^*$-algebra.

On the structure of quasi-topology and BED property for direct sum of Banach algebras

On the structure of quasi-topology and BED property for direct sum of Banach algebras

Meet infinite distributivity for congruence lattices of direct sums of algebras

Abstract We study meet infinite distributivity (MID) in congruence lattices of direct sums of algebras. The main result of this note is that in a congruence distributive variety the MID of congruence lattices is preserved by direct sums of algebras, it means that the congruence lattice of a direct sum of algebras is MID if and only if a congruence lattice of every constituent algebra is MID.

4d higgsed network calculus and elliptic DIM algebra

Supersymmetric gauge theories of certain class possess a large hidden nonperturbative symmetry described by the Ding-Iohara-Miki (DIM) algebra which can be used to compute their partition functions and correlators very efficiently. We lift the DIM-algebraic approach developed to study holomorphic blocks of 3d linear quiver gauge theories one dimension higher. We employ an algebraic construction in which the underlying trigonometric DIM algebra is elliptically deformed, and an alternative geometric approach motivated by topological string theory. We demonstrate the equivalence of these two methods, and motivated by this, prove that elliptic DIM algebra is isomorphic to the direct sum of a trigonometric DIM algebra and an additional Heisenberg algebra.

On complete Leibniz algebras

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

Diagonal decompositions of shapes and their algebras

AbstractThe formal approach to shapes and their algebras, as it appears in shape grammar theory, has been reviewed. It starts with geometric elements and their partial algebras, continues to shapes, their algebras, and boundaries, as well as algebras that calculate with shapes and their boundaries. There is a number of new concepts introduced along the way. These include diagonal decompositions and their algebras which simplify calculations with shapes, b-paired diagonal decompositions which extend calculations with shapes and their boundaries from diagonal shapes only to all shapes, andm-order boundaries which extend the concept of shape boundaries and allow for calculations with multiple representations of shapes. It also shows that algebras of shapes are infinite direct sums of diagonal algebras.