Decomposition Of Lie Algebra Research Articles

THE LEVI DECOMPOSITION OF THE LIE ALGEBRA M_2 (R)⋊gl_2 (R)

The idea of the Lie algebra is studied in this research. The decomposition between Levi sub-algebra and the radical can be used to define the finite dimensional Lie algebra. The Levi decomposition is the name for this type of decomposition. The goal of this study is to obtain a Levi decomposition of the Lie algebra of dimension 8. We compute its Levi sub-algebra and the radical of Lie algebra with respect to its basis to achieve this goal. We use literature studies on the Levi decomposition and Lie algebra in Dagli result to produce the radical and Levi sub-algebra. It has been shown that can be decomposed in the terms of the Levi sub-algebra and its radical. In this resulst, it has been given by direct computations and we obtained that the explicit formula of Levi decomposition of the affine Lie algebra whose basis is is written by with is is the Levi sub-algebra of .

On decompositions and transitive actions of nilpotent Lie groups

The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.

Orthogonal decompositions of Lie algebras over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

Abstract Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak {g}}} = {{\mathfrak {h}}} \oplus {{\mathfrak {q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal {H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak {q}}},{{\mathfrak {q}}}] \oplus i{{\mathfrak {q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak {h}}} \oplus i{{\mathfrak {q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

Strange expectations and the Winnie-the-Pooh problem

Motivated by the study of simultaneous cores, we give three proofs (in varying levels of generality) that the expected norm of a weight in a highest weight representation Vλ of a complex simple Lie algebra g is 1h+1〈λ,λ+2ρ〉. First, we argue directly using the polynomial method and the Weyl character formula. Second, we relate this problem to the “Winnie-the-Pooh problem” regarding orthogonal decompositions of Lie algebras; although this approach offers the most explanatory power by interpreting the quantity 〈λ,λ+2ρ〉 as the eigenvalue for the Casimir element on Vλ, it applies only to Cartan types other than A and C. Third, we use the combinatorics of semistandard tableaux to obtain the result in type A. We conclude with computations of many combinatorial cumulants.

Decompositions of algebras and post-associative algebra structures

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

Fricke Lie algebras and the genus zero property in Moonshine

We give a new, simpler proof that the canonical actions of finite groups on Fricke-type Monstrous Lie algebras yield genus zero functions in generalized Monstrous Moonshine, using a Borcherds–Kac–Moody Lie algebra decomposition due to Jurisich. We describe a compatibility condition, arising from the no-ghost theorem in bosonic string theory, that yields the genus zero property. We give evidence for and against the conjecture that such a compatibility for symmetries of the Monster Lie algebra gives a characterization of the Monster group.

Symplectic spreads, planar functions, and mutually unbiased bases

In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras $sl_n(\mathbb{C})$ obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are also studied. We show that the automorphism group of a complete set of MUBs is isomorphic to the automorphism group of the corresponding orthogonal decomposition of the Lie algebra $sl_n(\mathbb{C})$. In the case of symplectic spreads this automorphism group is determined by the automorphism group of the spread. By using the new notion of pseudo-planar functions over fields of characteristic two we give new explicit constructions of complete sets of MUBs.

Decompositions of quasigraded Lie algebras, non-skew-symmetric classical [formula omitted]-matrices and generalized Gaudin models

We construct a special family of quasigraded Lie algebras that generalize loop algebras in different gradings and admit Adler–Kostant–Symes decomposition into a sum of two subalgebras. We analyze the special cases when the constructed Lie algebras admit additionally other types of Adler–Kostant–Symes decompositions. Based on the proposed Lie algebras and their decompositions we explicitly construct several new classes of non-skew-symmetric classical r-matrices r(u,v) with spectral parameters. Using them we obtain new types of the generalized quantum and classical Gaudin spin chains.

On a microscopic representation of space-time

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and 'reduce' the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) $\longrightarrow$ usp(4) $\longrightarrow$ su(2)$\times$u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space $H_{5}$ with SO(5,1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into $H_{2}\oplus H_{3}$ with SO(2,1) and SO(3,1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra $H$ and subsequent coordinate restrictions.

Infinite Lie Algebras Generated by Supersymmetric Hypermatrices

In this paper we study the structure and properties of complex infinite supersymmetric hypermatrices generated by a semisimple basis, exponential sets of hypermatrices, hypermatrix Lie algebra and elements of the group of complex matrices of order two and determinant one. We study the hypermatrix Lie algebra generated by the polygons on analytic torus of genus g. By using new algebraic tools, namely cubic hypermatrices we study the algebraic structures associated with the hypermatrices of certain Lie algebras e.g. $\{sl2; f, \infty\}$; $\{sl2; \infty, \infty\}$ and $\{SL2; f, \infty\}$; $\{SL2; \infty, \infty\}$ and we construct generators of infinite periodic hypermatrix Lie algebraic structures which have classical Lie algebra decomposition; specifically a set of Lie algebras composed of hypermatrices. We study the exponential of a complex analytic Lie algebra, rotations of hypermatrices, and relations between hypermatrix groups, hypermatrix Lie Algebra, Fourier hypermatrices and the Laurent hypermatrix. Finally, as an application we will show that there is an isomorphism of the hypermatrix Lie algebra associated with a set of polygons on the torus of genus g and analytic functions associated with a countable set of solutions of a meromorphic function on the torus. In conclusion we will present a Riemann type isomorphism theorem for hypermatrices on a torus and the convoluted complex plane, generated by holomorphic functions, based on the equivalent relations of the geometry and the algebra of the torus of dimension three and genus g.

Supersymmetric Hypermatrix Lie Algebra and Hypermatrix Groups Generated by the Dihedral Set $D_3$

This work is an investigation into the structure and properties of supersymmetric hypermatrix Lie algebra generated by elements of the dihedral group $D_3$. It is based on previous work on the subject of supersymmetric Lie algebra (Schreiber, 2012).In preview work I used several new algebraic tools; namely cubic hypermatrices (including special arrangements of such hypermatrices) and I obtained an algebraic structure associated with the basis of the Lie algebra $sl_2$, and I showed that the basis elements $sl_2$ are generators of infinite periodic hypermatrix Lie algebraic structures with semisimple sub-algebras. The generated algebra has been shown to be an extended Lie hypermatrix algebra that has a classical Lie algebra decomposition composed of hypermatrices with periodic properties. The generators of higher dimensional Lie algebra were shown to be special supersymmetric, anti-symmetric and certain skew-symmetric hypermatrices. The present work takes a different look at the structure of periodic hypermatrix Lie algebra by using elements generating the classical dihedral group $D_3$. Using cubic dihedral symmetric hypermatrices (type: even-even, odd-odd, even-odd odd-even permutation) to generate Lie hypermatrix algebra I show that the extended dihedral algebra is a Lie hypermatrix algebras with special hypermatrix group properties, semisimple, symmetric, skew-symmetric, anti-symmetric, and anti-clockwise symmetric properties.

Supersymmetric Lie Algebra

This work is an investigation into the structure and properties of Lie hypermatrix algebra generated by a semisimple basis. By using new algebraic tools; namely cubic hypermatrices I obtain an algebraic structure associated with the basis of a semisimple Lie algebra, and I show that the semisimple Lie basis is a generator of infinite periodic semisimple hypermatrix structures, that has a classical Lie algebra decomposition (Bourbaki, 1980; Humphreys, 1972; Serre, 1987); specifically a set of Lie algebras composed of hypermatrices. The generators of higher dimensional semisimple Lie algebra are shown to be special supersymmetric, anti-symmetric and certain skew-symmetric hypermatrices.

Hypermatrix factors for string and membrane junctions

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n) and Sp(n) are, respectively, tensor, antisymmetric and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The -grading of the exceptional Lie algebras provides such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of 3-junctions of strings and membranes. Generalizations are also considered.

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker–Campbell–Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.

3-graded decompositions of exceptional Lie algebras g and group realizations of gev, g0 and ged, III: G=E8

In the articles [4] and [7], we completed the determination of group realizations gev and g0 of 2-graded decompositions g=g-2⊕g-1⊕g0⊕g1⊕g2 of exceptional Lie algebras g for the universal exceptional Lie groups. In the present article, which is a continuation of [5] and [8], we determine group realizations of subalgebras gev, g0 and ged of 3-graded decompositions of exceptional Lie algebras g for the universal exceptional Lie groups of type E8.

Subgrid interaction and micro-randomness – Novel invariance requirements in infinitesimal gradient elasticity

We present a micromechanically motivated form of the curvature energy in infinitesimal isotropic gradient elasticity. The basis is a homogenization/averaging scheme using a micro-randomness assumption imposed on a directional higher gradient interaction term. These directional interaction terms are matrix-valued allowing to apply the standard orthogonal Cartan Lie-algebra decomposition. Averaging over all (subgrid) directions leads to three quadratic curvature terms, which are conformally invariant when neglecting volumetric effects. Restricted to rotational inhomogeneities we motivate thereby a symmetric couple stress tensor in the infinitesimal indeterminate couple stress model of Koiter–Mindlin–Toupin-type. Relations are established to a novel conformally invariant linear Cosserat model.

Bases, filtrations and module decompositions of free Lie algebras

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.

Mutually unbiased bases and orthogonal decompositions of Lie algebras

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of $\mu$ MUBs in $\K^n$ gives rise to a collection of $\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also $\dag$-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for $n\le 5$ an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for $n=6$ there are at most three monomial MUBs.

Orthogonal decompositions and idempotent configurations in semisimple associative algebras

This is an introductory survey on the theories of orthogonal decompositions of associative algebras, balanced systems of idempotents, H-bijections of groups and H R -isomorphisms of group rings. These new theories have grown up from the investigations of orthogonal decompositions of simple Lie algebras into a sum of Cartan subalgebras carried out by A. I. Kostrikin with collaborators.