Lie Color Algebras Research Articles

Integrable Z22-graded extensions of the Liouville and Sinh–Gordon theories

Abstract In this paper we present a general framework to construct integrable Z 2 2 -graded extensions of classical, two-dimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and Sinh–Gordon models; they are based on Z 2 2 -graded color Lie algebras and their fields satisfy a parabosonic statistics. The mathematical tools here introduced are the Z 2 2 -graded covariant extensions of the Lax pair formalism and of the Polyakov’s soldering procedure. The Z 2 2 -graded Sinh–Gordon model is derived from an affine Z 2 2 -graded color Lie algebra, mimicking a procedure originally introduced by Babelon-Bonora to derive the ordinary Sinh–Gordon model. The color Lie algebras under considerations are: the 6-generator Z 2 2 -graded sl 2, the Z 2 2 -graded affine s l 2 ^ algebra with two central extensions, the Z 2 2 -graded Virasoro algebra obtained from a Hamiltonian reduction.

A connection between Uq(sl(3)) and Z2×Z2-graded special linear Lie colour algebras via Klein operators

We provide an explicit connection between the quantum group Uq(sl(3)) in the limit q → −1 and a Z2×Z2-graded Lie colour algebra. This is realised through an algebra embedding assisted by a Klein operator. This provides a proof of concept of an explicit method that could lead to constructing more general Z2×Z2-graded Lie colour algebras, and their representations.

Gravity = Yang–Mills

This essay’s title is justified by discussing a class of Yang–Mills-type theories of which standard Yang–Mills theories are special cases but which is broad enough to include gravity as a double field theory. We use the framework of homotopy algebras, where conventional Yang–Mills theory is the tensor product K⊗g of a ‘kinematic’ algebra K with a color Lie algebra g. The larger class of Yang–Mills-type theories are given by the tensor product of K with more general Lie-type algebras, of which K itself is an example, up to anomalies that can be canceled for the tensor product with a second copy K¯. Gravity is then given by K⊗K¯.

Enhancing Lie color algebras

Lie color algebras generalize Lie superalgebras. We adapt a construction of Bahturin and Pagon to create enhanced Lie color algebras. This construction offers a much-needed method for constructing simple Lie color algebras. To illustrate its applicability, we demonstrate how to enhance any simple Lie superalgebra and obtain a simple Lie color algebra. Additionally, we show that if a Lie color algebra has a nonzero determinant, this property extends to its enhancement. This ensures its universal enveloping algebra is semiprime. Extra conditions on the grading group provide a primeness criterion.

Cohomology and formal deformations of n -Hom–Lie color algebras

UDC 512.5 The aim of this paper is to provide a cohomology of n -Hom–Lie color algebras, in particular, a cohomology governing one-parameter formal deformations. Then we also study formal deformations of the n -Hom–Lie color algebras and introduce the notion of Nijenhuis operator on a n -Hom–Lie color algebra, which may give rise to infinitesimally trivial ( n - 1 ) -order deformations. Furthermore, in connection with Nijenhuis operators, we introduce and discuss the notion of product structure on n -Hom–Lie color algebras.

Construction of color Lie algebras from homomorphisms of modules of Lie algebras

A close relation between categories of color Lie algebras and sets of pairs of homomorphisms of modules of Lie algebras is discussed, which is applicable to construct color Lie algebras from Lie algebras and their modules. It is also shown that such construction yields equivalent categories of color Lie algebras under an equivalent relation between symmetric bicharacters. As an application we construct and classify all simple color Lie algebras from the three dimensional simple complex Lie algebra and its finite-dimensional simple modules.

Generalized Lie Triple Derivations of Lie Color Algebras and Their Subalgebras

Consider a Lie color algebra, denoted by L. Our aim in this paper is to study the Lie triple derivations TDer(L) and generalized Lie triple derivations GTDer(L) of Lie color algebras. We discuss the centroids, quasi centroids and central triple derivations of Lie color algebras, where we show the relationship of triple centroids, triple quasi centroids and central triple derivation with Lie triple derivations and generalized Lie triple derivations of Lie color algebras L. A classification of Lie triple derivations algebra of all perfect Lie color algebras is given, where we prove that for a perfect and centerless Lie color algebra, TDer(L)=Der(L) and TDer(Der(L))=Inn(Der(L)).

N-derivations of lie color algebras

The aim of this article is to discuss the n-derivation algebras of Lie color algebras. It is proved that, if the base ring contains 1/n-1, L is a perfect Lie color algebra with zero center, then every triple derivation of L is a derivation, and every n-derivation of the derivation algebra nDer(L)) is an inner derivation.

The Quantum Spaces of Certain Graded Algebras Related to $\mathfrak {sl}(2,\Bbbk )$

Inspired by the work of Le Bruyn and Smith in (Proc. Amer. Math. Soc. 118(3): 725–730, 1993) and the work of Shelton and Vancliff in (Comm. Alg. 30(5): 2535-2552, 2002), we analyze certain graded algebras related to the Lie algebra $\mathfrak {sl}(2,\Bbbk)$ using geometric techniques in the spirit of Artin, Tate and Van den Bergh. In particular, we discuss the point schemes and line schemes of certain quadratic quantum $\mathbb {P}^{3}$ s associated to the Lie superalgebra $\mathfrak {sl}(1|1)$ , to a quantized enveloping algebra, $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ , of $\mathfrak {sl}(2,\Bbbk)$ , and to a color Lie algebra $\mathfrak {sl}_k(2,\Bbbk)$ , respectively. The geometry we consider identifies certain normal elements in the universal enveloping algebra of $\mathfrak {sl}(1|1)$ and in $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ .

Associating geometry to the Lie superalgebra 𝔰𝔩(1|1) and to the color Lie algebra 𝔰𝔩^{𝔠}₂(\Bbbk)

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

On generalized Jordan prederivations and generalized prederivations of Lie color algebras

In this paper, the concepts of (generalized) (θ,φ)-prederivations and (generalized) Jordan (θ,φ)-prederivations on a Lie color algebra are introduced. It is proved that Jordan (θ,φ)-prederivations (resp. generalized Jordan (θ,φ)-prederivations) are (θ,φ)-prederivations (resp. generalized (θ,φ)-prederivations) on a Lie color algebra under some conditions. In particular, Jordan θ-prederivations are θ-prederivations on a Lie color algebra.

Poisson color algebras of arbitrary degree

ABSTRACTA Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g0, where g0 is some element of the grading group G such that g0 = 0 or 4g0≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.

$T^*$-extensions and abelian extensions of hom-Lie color algebras

We study hom-Nijenhuis operators, $T^ \ast$-extensions and abelian extensions of hom-Lie color algebras. We show that the infinitesimal deformation generated by a hom-Nijenhuis operator is trivial. Many properties of a hom-Lie color algebra can be lifted to its $T^ \ast$-extensions such as nilpotency, solvability and decomposition. It is proved that every finite-dimensional nilpotent quadratic hom-Lie color algebra over an algebraically closed field of characteristic not 2 is isometric to a $T^ \ast$-extension of a nilpotent Lie color algebra. Moreover, we introduce abelian extensions of hom-Lie color algebras and show that there is a representation and a 2-cocycle, associated to any abelian extension.

On split regular Hom-Lie color algebras

We introduce the class of split regular Hom-Lie color algebras as the natural generalization of split Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Lie color algebra $L$ is of the form $L = U + \sum\limits_{[j] \in \Lambda/\sim}I_{[j]}$ with $U$ a subspace of the abelian graded subalgebra $H$ and any $I_{[j]}$, a well described ideal of $L$, satisfying $[I_{[j]}, I_{[k]}] = 0$ if $[j]\neq [k]$. Under certain conditions, in the case of $L$ being of maximal length, the simplicity of the algebra is characterized.

PBW deformations of quantum symmetric algebras and their group extensions

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

Milnor–Moore categories and monadic decomposition

In this paper monoidal Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg–Moore categories. To this aim we introduce the notion of Milnor–Moore category as a monoidal category for which a Milnor–Moore type Theorem holds. We also show how to lift the property of being a Milnor–Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.

Constructions and Cohomology of Hom–Lie Color Algebras

The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α k -generalized derivations and provide examples.

Color cyclic homology and Steinberg Lie color algebras

The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ɛ-cyclic homology group HC 1(R, ɛ) of the Γ-graded associative algebra R (which could be seemed as the generalization of cyclic homology group and the ℤ/2ℤ-graded version of cyclic homology that was introduced by Kassel) to calculate the universal central extension of Steinberg Lie color algebras.

Gradings and symmetries on Heisenberg type algebras

We describe the fine (group) gradings on the Heisenberg algebras, on the Heisenberg superalgebras and on the twisted Heisenberg algebras. We compute the Weyl groups of these gradings. Also the results obtained respect to Heisenberg superalgebras are applied to the study of Heisenberg Lie color algebras.

Cohomology and deformations of 3-Lie colour algebras

In this paper, we introduce the concept of -Lie colour algebras which is a generalization of -Lie algebras. The cohomology theory of 3-Lie colour algebras is developed. The deformations of 3-Lie colour algebras are considered via the cohomology theory. We also study the abelian extensions of 3-Lie colour algebras in details.