Z-graded Lie Research Articles

Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics

A Z2×Z2 -graded Lie superalgebra g is a Z2×Z2 -graded algebra with a bracket [[⋅,⋅]] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z2×Z2 -graded Lie superalgebra osp(2m1+1,2m2|2n1,2n2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z2×Z2 -graded matrix. Since the orthosymplectic Lie superalgebra osp(2m+1|2n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from osp(2m1+1,2m2|2n1,2n2) . Some special cases are of particular interest, even when one is dealing with parabosons only.

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.

ℤ2 × ℤ2-graded Lie algebras and supseralgebras and related physical models

In this paper we construct physical systems with ℤ2×ℤ2-graded symmetries. There are two different structures:ℤ2×ℤ2-graded Lie algebras andℤ2×ℤ2-graded Lie superalgebras. Physical models with the latter symmetries can be seen as generalizations of supersymmetric models. The systems described by ℤ2 × ℤ2-graded Lie algebras have not been investigated yet (up to our knowledge). We present examples of physical models invariant under each one of both kinds of graded structures.

Characterization of simple smooth modules

In this paper, we characterize simple smooth modules over some infinite-dimensional Z-graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z-graded Lie algebra g locally finitely acts on a simple g-module V, then V is a smooth g-module. These infinite-dimensional Z-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from “locally finitely” to “locally nilpotently”. We also show that these are not the case for the Heisenberg algebra.

On classical Z2×Z2-graded Lie algebras

We construct classes of Z2×Z2-graded Lie algebras corresponding to the classical Lie algebras in terms of their defining matrices. For the Z2×Z2-graded Lie algebra of type A, the construction coincides with the previously known class. For the Z2×Z2-graded Lie algebra of types B, C, and D, our construction is new and gives rise to interesting defining matrices closely related to the classical ones but undoubtedly different. We also give some examples and possible applications of parastatistics.

Restricted modules over some ℤ-graded Lie algebras

In this paper, we answer a question asked by Kaiming Zhao at the mini course “Simple representations of the Virasoro algebra” at Xiamen mathematical center in the fall term of 2021. More precisely, we prove that if one special element in the Lie algebra (including the Virasoro algebra, the Heisenberg-Virasoro algebra and affine Virasoro algebras) acts locally nilpotently on an irreducible module V, then V is a restricted -module.

Representations of twisted infinite Lie conformal superalgebras

Following the twisting rule from Ramond Lie superalgebra to Neveu-Schwarz Lie superalgebra, we introduce the twisted version T(p) of a class of Z-graded Lie conformal superalgebras, where p is a nonzero parameter. We completely classify their small rank conformal modules and finite irreducible conformal modules, which turn out to be much richer than those over the original version. As applications, we also develop the representation theory of many new finite Lie conformal superalgebras.

Another class of simple graded Lie conformal algebras that cannot be embedded into general Lie conformal algebras

In a previous paper by the authors, we obtain the first example of a finitely freely generated simple Z-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying Z-graded simple Lie conformal algebras G=⊕i=−1∞Gi satisfying the following,(1)G0≅Vir, the Virasoro conformal algebra;(2)Each Gi for i≥−1 is a Vir-module of rank one. These algebras include some Lie conformal algebras of Block type.

Classification of minimal Z2×Z2-graded Lie (super)algebras and some applications

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z2×Z2-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

Homogeneous models for Levi degenerate CR manifolds

We extend the notion of a fundamental negatively Z-graded Lie algebra mx=⊕p≤−1mxp associated to any point of a Levi nondegenerate Cauchy-Riemann (CR) manifold to the class of k-nondegenerate CR manifolds (M,D,J) for all k≥2 and call this invariant the core at x∈M. It consists of a Z-graded vector space mx=⊕p≤k−2mxp of height k−2 endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of (M,D,J) and the Levi forms of higher order. In the case of CR manifolds of hypersurface type, we propose a definition of a homogeneous model of type m, that is, a homogeneous k-nondegenerate CR manifold M=G/Go with core m associated with an appropriate Z-graded Lie algebra Lie(G)=g=⊕gp and subalgebra Lie(Go)=go=⊕gop of the nonnegative part ⊕p≥0gp. It generalizes the classical notion of Tanaka of homogeneous models for Levi nondegenerate CR manifolds and the tube over the future light cone, the unique (up to local CR diffeomorphisms) maximally homogeneous 5-dimensional 2-nondegenerate CR manifold. We investigate the basic properties of cores and models and study the 7-dimensional CR manifolds of hypersurface type from this perspective. We first classify cores of 7-dimensional 2-nondegenerate CR manifolds up to isomorphism and then construct homogeneous models for seven of these classes. We finally show that there exists a unique core and homogeneous model in the 3-nondegenerate class.

Study of Graded Algebras and General Linear Group with Lie Superalgebras and R-Algebra

Some elements of theory of Z2 graded rings, modules and algebras. Z2-graded tensor algebra, Lie superalgrbras and matrices with entries in a Z2-graded commutative ring are treated in our present paper. At last a Theorem 4.4.on the set of square matrices in the graded R-algebra , MR-[m I n] is established.
 Dhaka Univ. J. Sci. 68(1): 1-5, 2020 (January)

The Z2×Z2-graded general linear Lie superalgebra

We present a novel realization of the Z2×Z2-graded Lie superalgebra gl(m1,m2|n1,n2) inside an algebraic extension of the enveloping algebra of the Z2-graded Lie superalgebra gl(m|n), with m = m1 + m2 and n = n1 + n2. A consequence of this realization is that the representations of gl(m|n) “lift up” to representations of gl(m1,m2|n1,n2), with matrix elements differing only by a sign, which we are able to characterize concisely.

Parabosons, parafermions and representations of ℤ2 × ℤ2-graded Lie superalgebras

For a set of m parafermion operators and n paraboson operators, there are two nontrivial ways to unify them in a larger algebraic structure. One of these corresponds to the orthosymplectic Lie superalgebra . The other one is no longer a ℤ2-graded Lie superalgebra but a ℤ2 × ℤ2-graded Lie superalgebra, a rather different algebraic structure, denoted here by . In a recent paper, the Fock spaces of order p for were determined. In the current paper, we summarize some of the main properties of and its Fock spaces. In particular, we concentrate on the Fock space for p = 1, and indicate how it reduces to an ordinary boson-fermion Fock space.

Z 2 × Z 2 generalizations of 𝒩=2 super Schrödinger algebras and their representations

We generalize the real and chiral N=2 super Schrödinger algebras to Z2×Z2-graded Lie superalgebras. This is done by D-module presentation, and as a consequence, the D-module presentations of Z2×Z2-graded superalgebras are identical to the ones of super Schrödinger algebras. We then generalize the calculus over the Grassmann number to Z2×Z2 setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of Z2×Z2-graded superalgebras. A vector field realization of the Z2×Z2 generalization of N=1 super Schrödinger algebra is also presented.

Minimal graded Lie algebras and representations of quadratic algebras

Let (g0,B0) be a quadratic Lie algebra (i.e. a Lie algebra g0 with a non-degenerate symmetric invariant bilinear form B0) and let (ρ,V) be a finite dimensional representation of g0. We define on Γ(g0,B0,V)=V⁎⊕g0⊕V a structure of local Lie algebra in the sense of Kac ([4]), where the bracket between g0 and V (resp. V⁎) is given by the representation ρ (resp. ρ⁎), and where the bracket between V and V⁎ depends on B0 and ρ. This implies the existence of two Z-graded Lie algebras gmax(Γ(g0,B0,V)) and gmin(Γ(g0,B0,V)) whose local part is Γ(g0,B0,V). We investigate these graded Lie algebras, more specifically in the case where g0 is reductive. Roughly speaking, the map (g0,B0,V)⟼gmin(Γ(g0,B0,V)) is a bijection between triplets and a class of graded Lie algebras. We show that the existence of “associated sl2-triples” is equivalent to the existence of non-trivial relative invariants on some orbit, and we define the “graded Lie algebras of polynomial type” which give rise to some dual airs.

On integrable wave interactions and Lax pairs on symmetric spaces

Multi-component generalizations of derivative nonlinear Schrödinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann–Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup–Newell (KN) and Gerdjikov–Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end.

Once more on parastatistics

Equivalence between algebraic structures generated by parastatistics triple relations of Green (1953) and Greenberg-Messiah (1965), and certain orthosymplectic ℤ2 × ℤ2-graded Lie superalgebras is found explicitly. Moreover, it is shown that such superalgebras give more complex para-Fermi and para-Bose systems then ones of Green-Greenberg-Messiah.

Classification of maximal transitive prolongations of super-Poincaré algebras

Let V be a complex vector space with a non-degenerate symmetric bilinear form and S an irreducible module over the Clifford algebra Cℓ(V) determined by this form. A supertranslation algebra is a Z-graded Lie superalgebra m=m−2⊕m−1, where m−2=V and m−1=S⊕⋯⊕S is the direct sum of an arbitrary number N≥1 of copies of S, whose bracket [⋅,⋅]|m−1⊗m−1:m−1⊗m−1→m−2 is symmetric, so(V)-equivariant and non-degenerate (that is the condition “s∈m−1,[s,m−1]=0” implies s=0). We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for dim⁡V≥3 and classify them in terms of super-Poincaré algebras and appropriate Z-gradings of simple Lie superalgebras.

The universal enveloping algebra of the Witt algebra is not noetherian

This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces.As a consequence of our main result, we also show that the enveloping algebras of many other (infinite dimensional) Lie algebras are not noetherian. These Lie algebras include the Virasoro algebra and all infinite dimensional Z-graded simple Lie algebras of polynomial growth.

Задача факторизации с пересечением

We propose a generalization of the factorization method to the case when G is a finite-dimensional Lie algebra G = G 0 ⊕ �� ⊕ �� (direct sum of vector spaces), where G 0 is a subalgebra in G , ��, �� are G 0-modules, and G 0 + �� , G 0 + �� are subalgebras in G. In particular, our construction involves the case when G is a Z-graded Lie algebra. Using this generalization, we construct certain top-like systems related to algebra so (3, 1). According to the general scheme, these systems can be reduced to solving systems of linear equations with variable coefficients. For these systems we find polynomial first integrals and infinitesimal symmetries.