Dimensional Lie Algebra Research Articles

Jordan–Kronecker Invariants for Lie Algebras of Small Dimensions

In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space 𝔤 proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan–Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras).

The Polynomial Modules over Quantum Group Uq(sl3)

Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.

On the classification of 3-dimensional complex hom-Lie algebras

Let hom-Ln(C) be the algebraic set of complex n-dimensional hom-Lie algebras. The group GL(n,C) acts on it via change of basis. We classify hom-Lie structures with nilpotent twisting map on 3-dimensional complex Lie algebras, up to isomorphism, and we give the classification of orbit closures in such family. For this purpose, we introduce some invariants of hom-Lie algebras to study both problems. The ideas and techniques presented here can be easily extrapolated to study similar problems on other linear algebraic structures.

Dual Lie bialgebra structures of the twisted Heisenberg–Virasoro type

In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg–Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg–Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.

Odd‐dimensional counterparts of abelian complex and hypercomplex structures

AbstractWe introduce the notion of abelian almost contact structures on an odd‐dimensional real Lie algebra . We investigate correspondences with even‐dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when carries a compatible inner product. The classification of 5‐dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3‐contact structures on real Lie algebras of dimension , obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3‐contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.

Equivalent constructions of nilpotent quadratic Lie algebras

The double extension and the T⁎-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

Contact seaweeds

A ($2k+1$)$-$dimensional contact Lie algebra is one which admits a one-form $\varphi$ such that $\varphi \wedge (d\varphi)^k\ne0$. Such algebras have index one, but this is not generally a sufficient condition. Here we show that index-one type-A seaweed algebras are necessarily contact. Examples, together with a method for their explicit construction, are provided.

Classification of a family of 4-dimensional anti-commutative algebras and their automorphisms

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field K of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of K by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by K.

Levi Decomposition of Frobenius Lie Algebra of Dimension 6

In this paper, we study notion of the Lie algebra of dimension 6. The finite dimensional Lie algebra can be expressed in terms of decomposition between Levi subalgebra and the maximal solvable ideal. This form of decomposition is called Levi decomposition. The work aims to obtain Levi decomposition of Frobenius Lie algebra of dimension 6. To achieve this aim, we compute Levi subalgebra and the maximal solvable ideal (radical) of with respect to its basis. To obtain Levi subalgebra and the maximal solvable ideal, we apply literature reviews about Lie algebra and decomposition Levi in Dagli result. For future research, decomposition Levi for higher dimension of Frobenius Lie algebra is still an open problem.

Almost Kaehlerian and Hermitian Structures on Four Dimensional Indecomposable Lie Algebras

It is known that from a given almost Hermitian structure on a simply connected Liegroup, one can obtain left-invariant almost Hermitian structure on its Lie algebra.In this work, we consider Mubarakzyanov’s classification of four-dimensional realLie algebras and evaluate the existence of almost Hermitian structures on four dimensional decomposable real Lie algebras. In particular, we focus on almost Kaehlerian and Hermitian structures on these Lie algebras.

2nd order approximate Noether and Lie symmetries of Gibbons–Maeda–Garfinkle–Horowitz–Strominger charged black hole in the Einstein frame

In this paper, approximate Noether and Lie symmetries of 2nd order for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) charged black hole in the Einstein frame are analyzed comprehensively. To explore the approximate Noether symmetries of 2nd order, Noether symmetries of Minkowski spacetime are used which forms a 17 dimensional Lie algebra. It is observed that no new approximate Noether symmetry is obtained at 1st and 2nd order. To examine the 1st and 2nd order approximate Lie symmetries of the GMGHS black hole spacetime, 35 Lie symmetries (exact) of the Minkowski spacetime are used which forms an algebra sl(6, R). It is shown that no new approximate Lie symmetry exists at 1st and 2nd order and only exact 35 symmetries are recouped as trivial approximate Lie symmetries at both orders. Furthermore, no energy rescaling factor is seen in this spacetime.

BAZI LİE CEBİRLERİ ÜZERİNDE HEMEN-HEMEN KONTAKT YAPILAR

In this manuscript, we show that there are no almost contact structures with parallel characteristic vector field on certain 7 dimensional Lie algebras over the real field.

New irreducible non-weight Virasoro modules from tensor products

The tensor product non-weight Virasoro modules are studied in this paper, where and (with and V being an irreducible module over a certain finite dimensional Lie algebra related to the Virasoro algebra) are irreducible Virasoro modules, respectively, defined in Liu and Zhao [Generalized polynomial modules over the Virasoro algebra. Proc Amer Math Soc. 2016;144:5103–5112] and Lü and Zhao [Irreducible Virasoro modules from irreducible Weyl modules. J Algebra. 2014;414:271–287]. The necessary and sufficient condition for to be irreducible is obtained. Then we determine the necessary and sufficient condition for two such irreducible modules to be isomorphic. At last, we show that the irreducible -modules are new for m>1.

Universal enveloping algebra of a pair of compatible Lie brackets

By applying the Poincaré—Birkhoff—Witt property and the Gröbner—Shirshov bases technique, we find the linear basis of the associative universal enveloping algebra in the sense of Ginzburg and Kapranov of a pair of compatible Lie brackets. We state that the growth rate of this universal enveloping over [Formula: see text]-dimensional compatible Lie algebra equals [Formula: see text].

The Virasoro-Like Algebra of a Frobenius Manifold

Abstract For an arbitrary calibrated Frobenius manifold, we construct an infinite dimensional Lie algebra, called the Virasoro-like algebra, which is a deformation of the Virasoro algebra of the Frobenius manifold. By using the Virasoro-like algebra we give a family of quadratic PDEs that are satisfied by the genus-zero free energy of the Frobenius manifold. We also derive, under the semisimplicity assumption, the Virasoro constraints for the corresponding abstract Hodge partition function.

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ell ,k-ell ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension.

The structure of almost Abelian Lie algebras

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media — cosmology, crystallography, etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin, etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and this paper is the first step in addressing this wide and diverse class of groups and algebras. This paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.

THE NON-DEGENERACY OF THE SKEW-SYMMETRIC BILINEAR FORM OF THE FINITE DIMENSIONAL REAL FROBENIUS LIE ALGEBRA

A Frobenius Lie algebra is recognized as the Lie algebra whose stabilizer at a Frobenius functional is trivial. This condition is equivalent to the existence of a skew-symmetric bilinear form which is non-degenerate. On the other hand, the Lie algebra is Frobenius as well if its orbit on the dual vector space is open. In this paper, we study the skew-symmetric bilinear form of finite dimensional Frobenius Lie algebra corresponding to its Frobenius functional. The work aims to prove that a Lie algebra of dimension is Frobenius if and only if the -th derivation of the Frobenius functional is not equal to zero. Indeed, this condition implies that the skew-symmetric bilinear form is non-degenerate and vice versa. In addition, some properties of Frobenius functionals are obtained. Furthermore, the computations are given using the coadjoint orbits and the structure matrix. As a discussion, we can investigate these results in the algebra case whether giving rise to a left-invariant K hler structure of a Frobenius Lie group or not.

Examples of Almost Para-Contact Metric Structures on 5-dimensions

In this study, the classes of several almost paracontact metric structures on 5 dimensional nilpotent Lie algebras are determined. It is also shown that there are no $\eta-$ Einstein structures on 5 dimensional nilpotent Lie algebras.