Orthosymplectic Lie Superalgebra Osp 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.

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.

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.

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).

Characters for projective modules in the BGG category [formula omitted] for the orthosymplectic Lie superalgebra [formula omitted

We determine the Verma multiplicities of standard filtrations of projective modules in integral atypical blocks in the BGG category O for the orthosymplectic Lie superalgebras osp(3|4) by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

The minimal representation of a semisimple Lie group is a 'small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which intertwines the Schr\"odinger model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$ with this new Fock model.

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

For the orthosymplectic Lie superalgebra , we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi‐Hamiltonian structures.

On the second fundamental theorem of invariant theory for the orthosymplectic supergroup

Let OSp(V) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension (m|2n). Lehrer and Zhang showed that there is a surjective algebra homomorphism Frr:Br(m−2n)→EndOSp(V)(V⊗r), where Br(m−2n) is the Brauer algebra of degree r with parameter m−2n. The second fundamental theorem of invariant theory in this setting seeks to describe the kernel KerFrr of Frr as a 2-sided ideal of Br(m−2n). In this paper, we show that KerFrr≠0 if and only if r≥rc:=(m+1)(n+1), and give a basis and a dimension formula for KerFrr. We show that KerFrr as a 2-sided ideal of Br(m−2n) is generated by KerFrcrc for any r≥rc, and we provide an explicit set of generators for KerFrcrc. These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra Endosp(V)(V⊗r) over the orthosymplectic Lie superalgebra osp(V) to be isomorphic to Br(m−2n).

Kazhdan-Lusztig theory of super type D and quantum symmetric pairs

We reformulate the Kazhdan-Lusztig theory for the BGG category O \mathcal {O} of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category O \mathcal {O} of the ortho-symplectic Lie superalgebra o s p ( 2 m | 2 n ) \mathfrak {osp}(2m|2n) .

Representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras

We construct all the irreducible representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras osp(1|2n), and show that their highest weights are given by the dominant words. We use the dominant Lyndon words to construct the cuspidal modules and show that the irreducible representations are the simple heads of standard representations constructed by induction from the cuspidal modules.

Haar expectations of ratios of random characteristic polynomials

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl group invariance, certain weight constraints and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N \ge 1. This completes earlier work by Conrey, Farmer, and Zirnbauer for the case of U(N).

Complexity of modules over classical Lie superalgebras

The complexity of the simple and the Kac modules over the general linear Lie superalgebra gl(m|n) of type A was computed by Boe, Kujawa, and Nakano in [2]. A natural continuation to their work is computing the complexity of the same family of modules over the ortho-symplectic Lie superalgebra osp(2|2n) of type C. The two Lie superalgebras are both of Type I which will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. We also compute a categorical invariant, z-complexity, introduced in [2], and we interpret this invariant geometrically in terms of a specific detecting subsuperalgebra. In addition, we compute the complexity and the z-complexity of the simple modules over the Type II Lie superalgebras osp(3|2), D(2,1;α), G(3), and F(4).

Joseph Ideals and Harmonic Analysis for

The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^{m-2}. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in the tensor algebra of g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n)=spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R^{m-2|2n} and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n).

Trivial module for ortho-symplectic Lie superalgebras and Littlewood’s formula

We show that the denominator identity for ortho-symplectic Lie superalgebras osp(k|2n) is equivalent to the Littlewood’s formula. Such an equivalence also implies the relation between the trivial module and generalized Verma modules for osp(k|2n). Furthermore, we discuss the harmonic representative elements of the Kostant’s u-cohomology with trivial coefficients.

Canonical Basis for Quantum $${\mathfrak{osp}(1|2)}$$

We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra osp(1|2). Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for sl(2). This provides a first example of canonical basis for quantum superalgebras.

Differential Representations of the Lie Superalgebra C(n+1)

In this paper, we realize Verma modules and the vector-coherent-state (VCS) representations of the orthosymplectic Lie superalgebra osp(2|2n) as differential operators with vector coefficients. We also characterize simple sub-representations of VCS representations as kernels of some finite number of differential operators. The singular vectors of the atypical representation of osp(2|2n) are explicitly given.

Cohomology of osp(2|2) acting on spaces of linear differential operators on the superspace ℝ1|2

We compute the first differential cohomology of the orthosymplectic Lie superalgebra osp(2|2) with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, 2)-dimensional real superspace. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomologies. This work is the simplest generalization of a result from [1].

Paraboson coherent states

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b i ± . In particular, the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Recently we constructed these representations V(p) giving the explicit actions of the osp(1|2n) generators. We apply these results for the n = 2 case in order to obtain “coherent state” representations of the paraboson operators.

The Paraboson Fock Space and Unitary Irreducible Representations of the Lie Superalgebra $${{\mathfrak{osp}}}$$ (1|2n)

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this implies that the ``parabosons of order p'' correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching $ osp(1|2n) \supset sp(2n) \supset u(n)$, we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters.

A class of unitary irreducible representations of the Lie superalgebra osp(1|2n) 1

Using the equivalence of the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) to the defining triple relations of n pairs of parabose operators b± i we construct a class of unitary irreducible (infinite-dimensional) lowest weight representations V (p) of osp(1|2n). We introduce an orthogonal basis of V (p) in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role and we present explicit actions of the osp(1|2n) generators.