Lie Superalgebra Research Articles

From quantum loop superalgebras to super Yangians

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra glM|N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra Uq(LglM|N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of Uq(LglM|2n) with respect to the comultiplication.

Superspin chains solutions from 4D Chern-Simons theory

As a generalisation of the correspondence linking 2D integrable systems with 4D Chern-Simons (CS) gauge theory, superspin chains are realized by means of crossing electric and magnetic super line defects in the 4D CS with super gauge symmetry. The oscillator realization of Lax operators solving the RLL relations of integrability is obtained in the gauge theory by extending the notion of Levi decomposition to Lie superalgebras. Based on particular 3-gradings of Lie superalgebras, we obtain graded oscillator Lax matrices for superspin chains with internal symmetries given by A(m − 1 | n − 1), B(m | n), C(n) and D(m | n).

Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras

In this paper, we give the forms of local automorphisms (resp. superderivations) of model filiform Lie superalgebra Ln,m in the matrix version. Linear 2-local automorphisms (resp. superderivations) of Ln,m are also characterized. We prove that each linear 2-local automorphism of Ln,m is an automorphism.

Semi-abelian condition for color Hopf algebras

Recently, in [22], it was shown that the category of cocommutative Hopf algebras over an arbitrary field k is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the symmetric monoidal category of G-graded vector spaces with G an abelian group, given an arbitrary skew-symmetric bicharacter on G, when G is finitely generated and the characteristic of k is different from 2 (not needed if G is finite of odd cardinality). We also prove that this category is action representable and locally algebraically cartesian closed, then algebraically coherent. In particular, these results hold for the category of cocommutative super Hopf algebras by taking G=Z2. Furthermore, we prove that, under the same assumptions on G and k, the abelian category of abelian objects in the category of cocommutative color Hopf algebras is given by those cocommutative color Hopf algebras which are also commutative.

The Clifford Algebra Bundle on Loop Space

We construct a Clifford algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the first (fractional) Pontrjagin class of the manifold.

Twisted affine Lie superalgebras and integrability

A twisted affine Lie superalgebra is either a twisted affine Lie algebra or of one of the types X = A ( 2 m − 1 , 2 n − 1 ) ( 2 ) ( m , n ≠ 0 , ( m , n ) ≠ ( 1 , 1 ) ), A ( 2 m , 2 n ) ( 4 ) , A ( 2 m , 2 n − 1 ) ( 2 ) or D ( m + 1 , n ) ( 2 ) ( m ≥ 0 , n > 0 ). It is known that irreducible integrable highest weight modules over a twisted affine Lie superalgebra of type X do not exist if m ≠ 0. In this paper, we show that nonzero level irreducible integrable finite weight modules over a twisted affine Lie superalgebra of type X do not exist if m ≠ 0.

Representations of quantum toroidal superalgebras and plane s-partitions

We construct Fock and MacMahon modules for the quantum toroidal superalgebra [Formula: see text] associated with the Lie superalgebra [Formula: see text] and parity [Formula: see text]. The bases of the Fock and MacMahon modules are labeled by super-analogs of partitions and plane partitions with various boundary conditions, while the action of generators of [Formula: see text] is given by Pieri-type formulas. We study the corresponding characters.

T-duality/plurality of BTZ black hole metric coupled to two fermionic fields

We ask the question of classical super (non-)Abelian T-duality for BTZ black hole metric coupling to two fermionic fields. Our approach is based on super Poisson-Lie (PL) T-duality in the presence of spectator fields. In order to study the Abelian T-duality of the metric we dualize over the Abelian Lie supergroups of the types (1|2) and (2|2), in such a way that it is shown that both original and dual backgrounds of the models are conformally invariant up to one-loop order in the presence of field strength. Then, we study the non-Abelian T-duality of the BTZ vacuum metric coupling to two fermionic fields. The dualizing is performed on some non-Abelian Lie supergroups of the type (2|2), in such a way that we are dealing with semi-Abelian superdoubles which are non-isomorphic as Lie superalgebras in each of the models. In the non-Abelian T-duality case, it is interesting to mention that the models can be conformally invariant up to one-loop order in both cases of the absence and presence of field strength. In addition, starting from the decomposition of semi-Abelian Drinfeld superdoubles generated by some of the C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{C} $$\\end{document}3 ⨁ A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}1,1 Lie superbialgebras we study the super PL T-plurality of the BTZ vacuum metric coupled to two fermionic fields. However, our findings are interesting in themselves, but at a constructive level, can prompt many new insights into supergravity and manifestly have interesting mathematical relationships with double field theory.

Tensor Categories for Vertex Operator Superalgebra Extensions

Let V V be a vertex operator algebra with a category C \mathcal {C} of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A A be a vertex operator (super)algebra extension of V V . We employ tensor categories to study untwisted (also called local) A A -modules in C \mathcal {C} , using results of Huang-Kirillov-Lepowsky that show that A A is a (super)algebra object in C \mathcal {C} and that generalized A A -modules in C \mathcal {C} correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C \mathcal {C} -algebra and (under suitable conditions) of generalized A A -modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V V -modules to A A -modules is a vertex tensor functor. Two applications are given: First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z \frac {1}{2}\mathbb {Z} -graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z \mathbb {Z} -graded components, respectively. Second, we analyze parafermionic cosets C = C o m ( V L , V ) C=\mathrm {Com}(V_L,V) where L L is a positive definite even lattice and V V is regular. If the vertex tensor category of either V V -modules or C C -modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

Deformations and abelian extensions of compatible pre-Lie superalgebras

In this paper, we give cohomologies and deformations theory, as well as abelian extensions for compatible pre-Lie superalgebras. Explicitly, we first introduce the notation of a compatible pre-Lie superalgebra and its representation. We also construct a new bidfferential graded Lie superalgebra whose Maurer–Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie superalgebra which controls deformations of a compatible pre-Lie superalgebra. Then, we introduce a cohomology of a compatible pre-Lie superalgebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie superalgebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group [Formula: see text] is trivial, then the compatible pre-Lie superalgebra is rigid. Additionally, we give a cohomology of a compatible pre-Lie superalgebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie superalgebras using this cohomology. We show that abelian extensions are classified by the second cohomology group. Finally, we study the relation between compatible Lie superalgebras and compatible pre-Lie superalgebras and we give some examples of compatible pre-Lie superalgebras.

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.

Exceptionally simple super-PDE for F(4)

For the largest exceptional simple Lie superalgebra [Formula: see text], having dimension [Formula: see text], we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second- and third-order, respectively.

Ramond and Neveu–Schwarz Algebras and Narrow Lie Superalgebras

Ramond and Neveu–Schwarz Algebras and Narrow Lie Superalgebras

On some central operators for loop Lie superalgebras

Let g be either a basic classical Lie superalgebra or gl(m,n) over the field of complex numbers C. For any associative, commutative, and finitely generated algebra A with unity, we consider the loop Lie superalgebra g⊗A. In [11], Rao defined a class of central operators for g⊗A and conjectured that these central operators, which generalizes the classical Gelfand invariants, generate the algebra U(g⊗A)g for g=gl(m,n). In this article we prove this conjecture and we also give a spanning set for U(g⊗A)G where g is the orthosymplectic Lie superalgebra and G is the corresponding orthosymplectic supergroup.

Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

Contact and Frobenius Lie superalgebras

We introduce the concepts of complex finite-dimensional contact and Frobenius Lie superalgebras, including some relevant examples of this type of Lie superalgebras. We establish some basic properties satisfied by both contact and Frobenius Lie superalgebras, providing characterizations for both of them in terms of its associated structure supermatrix. Finally, if g = g 0 ⊕ g 1 is a homogeneous even Frobenius Lie superalgebra satisfying dim g 0 ≠ dim g 1 , we show that such g contains a contact superideal of codimension one.

Gaussian generators for the Yangian associated with the Lie superalgebra [formula omitted]

We give a new presentation of the Yangian for the orthosymplectic Lie superalgebra osp1|2m. It relies on the Gauss decomposition of the generator matrix in the R-matrix presentation. The defining relations between the Gaussian generators are derived from a new version of the Drinfeld-type presentation of the Yangian for osp1|2 and some additional relations in the Yangian for osp1|4 by an application of the embedding theorem for the super-Yangians.

Crystal bases of parabolic Verma modules over the quantum orthosymplectic superalgebras

We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a q-analogue of a polynomial representation of a general linear Lie superalgebra.

Bounded weight modules for basic classical Lie superalgebras at infinity

Bounded weight modules for basic classical Lie superalgebras at infinity

Non-weight representations of Lie superalgebras of Block type, II

In this paper, we study non-weight representations of another two classes of Lie superalgebras T(p,s) of Block type, where p∈C and s=0,1. We completely classify the free U(h)-modules of rank (1+1) over T(p,s). As byproducts, we also obtain the non-weight representation results for four classes of non-semisimple finitely graded super-extensions of the Virasoro algebra.