Orthosymplectic 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.

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.

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.

Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs

Abstract We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over ${\mathbb {C}}$ with a triangular decomposition and a character $\zeta $ of the nilpotent radical, the associated Backelin functor $\Gamma _\zeta $ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category ${\mathcal {O}}$ . In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $\Gamma _\zeta $ and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type $AIII$ .

The Bargmann symmetry constraint of the new super Dirac hierarchy

For the orthosymplectic Lie superalgebra osp(2,2), we construct a new super Dirac integrable hierarchy, which can be written as the super Hamiltonian structure with the aid of supertrace identity. Upon choosing N distinct spectral parameters, a Bargmann symmetry constraint is proposed for the super Dirac integrable hierarchy. By substituting the symmetry constraint into N copies finite-dimensional super systems, we find the constrained super systems, defined over the supermanifold R4N|4N, are completely integrable in the Liouville sense.

Resolution of Singularities of the Odd Nilpotent Cone of Orthosymplectic Lie Superalgebras

Resolution of Singularities of the Odd Nilpotent Cone of Orthosymplectic Lie Superalgebras

A Drinfeld-Type Presentation of the Orthosymplectic Yangians

We use the Gauss decomposition of the generator matrix in the R-matrix presentation of the Yangian for the orthosymplectic Lie superalgebra ospN|2m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {osp}_{N|2m}$$\\end{document} to produce its Drinfeld-type presentation. The results rely on a super-version of the embedding theorem which allows one to identify a subalgebra in the R-matrix presentation which is isomorphic to the Yangian associated with ospN|2m-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {osp}_{N|2m-2}$$\\end{document}.

Ghost center and representations of the diagonal reduction algebra of osp(1|2)

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra A of the orthosymplectic Lie superalgebra osp(1|2). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of A is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for osp(1|2)). As another application, we classify all finite-dimensional irreducible representations of A. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

All basic quantizations of D=3, N=1 Lorentz supersymmetry

By the supersymmetrization of a simple algebraic technique proposed in Lukierski and Tolstoy (Eur Phys J C 77:226, 2017) we obtain the complete classification of all basic (nonisomorphic) quantum deformations for the orthosymplectic Lie superalgebra mathfrak {osp}(1|2;{mathbb {C}}) and its pseudoreal and real forms in terms of the classical r-matrices. In particular, we prove that pseudoreal compact form has only one quantum deformation (standart q-analog), and the D=3, N=1 Lorentz supersymmetry, which is the non-compact real form of mathfrak {osp}(1|2;{mathbb {C}}), has four different Hopf-algebraic quantum deformations: two standard q-analogs, and two (Jordanian and super-Jordanian) twist deformations. All basic Hopf-algebraic quantum deformations are presented in the explicit form.

Singular Conformal Oscillator Representations of Orthosymplectic Lie Superalgebras

Singular Conformal Oscillator Representations of Orthosymplectic Lie Superalgebras

Representations of the Yangians Associated with Lie Superalgebras mathfrak {osp}(1|2n)

We give a complete description of the finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras mathfrak {osp}_{1|2n}. The representations are classified in terms of their highest weights and are parameterized by n-tuples of monic polynomials in one variable. The arguments rely on explicit constructions of a family of elementary modules of the Yangian for mathfrak {osp}_{1|2}. We show that a wide class of irreducible representations of this Yangian can be produced by taking tensor products of the elementary modules.

First Cohomology Space of the Orthosymplectic Lie Superalgebra $$ \mathfrak{osp} $$(n|2) in the Lie Superalgebra of Superpseudodifferential Operators

First Cohomology Space of the Orthosymplectic Lie Superalgebra $$ \mathfrak{osp} $$(n|2) in the Lie Superalgebra of Superpseudodifferential Operators

First cohomology space of the orthosymplectic Lie superalgebra o s p ( n | 2 ) in the Lie superalgebra of superpseudodifferential operators

UDC 515.12 We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra o s p ( n | 2 ) on the ( 1 , n ) -dimensional superspace ℝ 1 | n in the Lie superalgebra 𝒮 Ψ D O ( n ) (for n ≥ 4 ) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles.This work is the simplest generalization of a result by Basdouri [<em>First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators</em>, Algebras and Representation Theory, <strong>16</strong>, 35-50 (2013)].

Representations of the Super Yangians of Types A and C

We classify the finite-dimensional irreducible representations of the super Yangian associated with the orthosymplectic Lie superalgebra mathfrak {osp}_{2|2n}. The classification is given in terms of the highest weights and Drinfeld polynomials. We also include an R-matrix construction of the polynomial evaluation modules over the Yangian associated with the Lie superalgebra mathfrak {gl}_{m|n}, as an appendix. This is a super-version of the well-known construction for the mathfrak {gl}_{n} Yangian and it relies on the Schur–Sergeev duality.

SOME HOMOLOGICAL PROPERTIES OF CATEGORY FOR LIE SUPERALGEBRAS

AbstractFor classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda )$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda )$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu ),\Delta (\lambda ))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$ . Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ .

Characters and quantum reduction for orthosymplectic Lie superalgebras

In this study, we study principal admissible representations for the affine Lie superalgebra [Formula: see text]. Using the character formula of irreducible admissible representations of [Formula: see text], we calculate a character formula of [Formula: see text]-modules which are obtained from the quantized Drinfeld–Sokolov reduction and principal admissible representations. As a by-product, we obtain the minimal series modules of the Neveu–Schwarz algebra through the [Formula: see text]-modules arising from the principal admissible modules over [Formula: see text].

Semisimplicity of the DS functor for the orthosymplectic Lie superalgebra

We prove that the Duflo-Serganova functor DSx attached to an odd nilpotent element x of osp(m|2n) is semisimple, i.e. sends a semisimple representation M of osp(m|2n) to a semisimple representation of osp(m−2k|2n−2k) where k is the rank of x. We prove a closed formula for DSx(L(λ)) in terms of the arc diagram attached to λ.

English

Let $${{\cal D}_{\lambda ,\mu }}$$ be the space of linear differential operators on weighted densities from $${{\cal F}_\lambda }$$ to $${{\cal F}_\mu }$$ as module over the orthosymplectic Lie superalgebra $$\mathfrak{osp}(3\left| 2 \right.)$$ , where $${{\cal F}_\lambda }$$ , $$\lambda \in \mathbb{C}$$ is the space of tensor densities of degree λ on the supercircle S1∣3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.

Blocks and characters of G(3)-modules of non-integral weights

We classify blocks in the BGG category O of modules of non-integral weights for the exceptional Lie superalgebra G(3). We compute the characters for tilting modules of non-integral weights in O. Reduction methods are established to connect non-integral blocks of G(3) with blocks of the special linear Lie algebra sl(2), the exceptional Lie algebra G2, the general linear Lie superalgebras gl(1|1), gl(2|1) and the ortho-symplectic Lie superalgebra osp(3|2).

Bethe Ansatz Equations for Orthosymplectic Lie Superalgebras and Self-dual Superspaces

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $\mathfrak{osp}_{2m+1|2n}$ and $\mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $\mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $\mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $\mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $\mathfrak{osp}_{2m|0}=\mathfrak{so}_{2m}$.