Orthosymplectic Lie Superalgebra

• Orthosymplectic Lie superalgebra is a complex simple superalgebra that unifies orthogonal and symplectic symmetries through a supergeometric framework.
• Its explicit construction via Killing vector fields and differential operators offers a clear realization essential for supersymmetric harmonic analysis.
• The structure underpins Howe duality, quantum symmetries, and parastatistics, revealing deep connections with invariant theory and integrable models.

The orthosymplectic Lie superalgebra, denoted $\mathfrak{osp}(m|2n)$, is a complex simple Lie superalgebra that occupies a pivotal role in the theory of supergeometry, superinvariant theory, and supersymmetric harmonic analysis. Its structure unifies the orthogonal and symplectic symmetries within a superalgebraic framework, leading to rich representation theory, dualities, and connections to mathematical physics, parastatistics, and quantum symmetries. The following sections provide a technical survey of its construction, properties, and applications as developed and synthesized in recent literature.

1. Algebraic Structure and Realization

$\mathfrak{osp}(m|2n)$ is defined as the algebra of Killing vector fields on the Riemannian superspace $\mathbb{R}^{m|2n}$ that stabilize the origin. Explicitly, it consists of all infinitesimal isometries (linear transformations preserving the supermetric) fixing the origin.

Differential Operator Realization

Let $X = (x_1,\dots,x_m,\, \xi_1,\dots,\xi_{2n})$ be coordinates on $\mathbb{R}^{m|2n}$, where $x_i$ are bosonic (even) and $\xi_j$ fermionic (odd) coordinates. The generators are given by

$$L_{ij} = X_i \frac{\partial}{\partial X_j} - (-1)^{[i][j]} X_j \frac{\partial}{\partial X_i}$$

where $[i]=0$ for $x_i$, $[i]=1$ for $\xi_i$.

The defining relations are encoded by a metric

$$g = \begin{pmatrix} I_m & 0 \ 0 & J_{2n} \end{pmatrix}$$

where $I_m$ is the identity (orthogonal structure) and $J_{2n}$ is a standard $2n\times 2n$ symplectic matrix (skew-symmetric). Invariance under $\mathfrak{osp}(m|2n)$ means $L_{ij}$ preserve the canonical quadratic form

$$R^2 = \sum_{i=1}^m x_i^2 - \sum (\text{fermionic quadratic terms}).$$

These $L_{ij}$ close under the supercommutator to form a Lie superalgebra.

2. Representation Theory and Harmonic Analysis

Fischer Decomposition and Spherical Harmonics

Let $\mathcal{P}$ denote the super-polynomial algebra on $\mathbb{R}^{m|2n}$. The super-Laplace operator $\Delta$ and $R^2$ (norm squared), together with the Euler operator $E$, satisfy

$$\left[-\frac{1}{2} \Delta,\, \frac{1}{2}R^2 \right] = E + \frac{M}{2},\quad M = m-2n.$$

They generate an $\mathfrak{sl}(2)$ triple that commutes with the $\mathfrak{osp}(m|2n)$ action.

The decomposition

$$\mathcal{P} = \bigoplus_{k} \bigoplus_{j} R^{2j} \mathcal{H}_{k-2j}$$

holds with $\mathcal{H}_k =$ homogeneous superharmonic polynomials of degree $k$. Each $\mathcal{H}_k$ is either an irreducible or indecomposable module for $\mathfrak{osp}(m|2n)$. For $m-2n \not\in -2\mathbb{N}$, complete reducibility holds and $\mathcal{H}_k$ realizes the irreducible module $L_{(k,0,\dots,0)}$.

Branching and Tensor Product Structure

The tensor and restriction (branching) structure of irreducible modules satisfy explicit rules:

• The restriction

$$L_{(k,0,\dots,0)} \downarrow \mathfrak{osp}(m-1|2n) = \bigoplus_\ell L_{(1,0,\dots,0)}^{\otimes ?}$$

is governed by precise combinatorial conditions on $k$ and the "superdimension" $m-2n$.

• The highest weight (traceless symmetric) part of $S^k(V)$ coincides with $L_{(k,0,\dots,0)}$ in completely reducible cases, i.e., the Cartan product inside the tensor power of the natural module captures exactly the irreducible with highest weight $(k,0,\dots,0)$.

Indecomposability and non-semisimplicity appear when $m-2n\in -2\mathbb{N}$, although the Cartan product and highest-weight logic extend via projective covers and unique indecomposable summands.

3. Howe Duality: $\mathfrak{osp}(m|2n)\times\mathfrak{sl}(2)$

A central result is the mutual commutant relationship:

• The triple $\{X_+, X_-, H\}$, with $X_- = -\frac{1}{2}\Delta$, $X_+ = \frac{1}{2}R^2$, $H = E+\frac{m-2n}{2}$, generates an $\mathfrak{sl}(2)$ algebra commuting with $\mathfrak{osp}(m|2n)$.
• Each irreducible $\mathfrak{osp}(m|2n)$-module in the polynomial representation appears with multiplicity one alongside an $\mathfrak{sl}_2$ irreducible. Thus, the decomposition is multiplicity-free and the full polynomial algebra realizes a (Howe) dual pair,

$$\mathcal{P} = \bigoplus_{k} V_k^{\mathfrak{sl}_2} \otimes L_{(k,0,\dots,0)}^{\mathfrak{osp}(m|2n)}.$$

This structure underpins explicit harmonic analysis, projection operators, and the analysis of symmetries in both geometric and physical models.

4. Invariants, Integration, and Uniqueness on the Supersphere

Integration over the "supersphere", i.e., the locus $R^2=1$, is uniquely determined by orthosymplectic invariance and the constraint $T(R^2 f) = T(f)$. The unique invariant distribution $T$ is constructed so that for any polynomial $f$,

$T[f] = \sum_\ell c_\ell (\Delta^\ell f)(0),$

where $c_\ell$ are explicit constants.

This property generalizes the classical Pizzetti formula and is crucial for supersymmetric field-theoretic applications where invariance under $\mathfrak{osp}(m|2n)$ determines all integration functionals.

5. Parastatistics, Graded Generalizations, and Physical Models

$\mathfrak{osp}(2m+1|2n)$ governs the algebraic structure of models incorporating $m$ parafermions and $n$ parabosons with specific "relative parafermion relations". Altering the relative commutation relations produces color (i.e., $\mathbb{Z}_2\times\mathbb{Z}_2$-graded) generalizations such as $pso(2m+1|2n)$ and more generally $osp(2m_1+1, 2m_2|2n_1, 2n_2)$ (Stoilova et al., 19 Feb 2024). In these settings, the block matrix realization must accommodate multi-grading, and the resulting algebras model systems of several types of parastatistics (including their mutual triple relations).

In the ordinary (super) case, creation and annihilation operators generating the Fock space representations of $\mathfrak{osp}(2m+1|2n)$ satisfy triple relations modeled directly on the structure constants of the superalgebra. As the number of parafermions and/or parabosons diverges ($n\to\infty$), the infinite-rank limit $B(\infty,\infty)$ provides a well-defined framework for studying parastatistics quantum fields (Stoilova et al., 2019).

6. Advanced Topics: Quantum Symmetries, Yangians, and Invariant Theory

Quantum Deformations and Howe Duality

The $q$-analog of Howe duality introduces commuting actions of quantum superalgebras and $i$-quantum groups on $q$-deformed supersymmetric spaces. This allows the development of semisimple categories analogous to the classical $(\mathfrak{g},G)$-duality—specifically for orthosymplectic $\mathfrak{g}$ and $G$ one of $O_\ell$ or $Sp_{2\ell}$—with a flat degeneration to the undeformed case (Bae et al., 6 Jun 2025).

Yangian and Drinfeld-Type Presentations

The Yangian $Y(\mathfrak{osp}_{N|2m})$ is constructed via an $RLL$-presentation using the super $R$-matrix and its Gauss decomposition (Fuksa et al., 2016, Molev, 2021). There exist recursive embeddings demonstrating that $Y(\mathfrak{osp}_{N|2m})$ contains as a subalgebra the Yangian $Y(\mathfrak{osp}_{N|2m-2})$, analogous to known results for non-superalgebraic cases.

Invariant Theory

The First Fundamental Theorem (FFT) for the orthosymplectic supergroup asserts that all polynomial invariants in tensor powers are generated by "Brauer contractions"—quadratic in the supermetric—with the Brauer algebra parameter corresponding to the superdimension $m-2n$ (Lehrer et al., 2014). When considering the Lie superalgebra (as opposed to the supergroup), the algebra of invariants is generated by the quadratic invariants and, at higher degrees, the super Pfaffian (Lehrer et al., 2015, Lehrer et al., 2014). This generator is an explicit highest weight vector related to the determinant in the even sector and the Berezinian in the odd sector, and its square is an invariant of the supergroup.

7. Segal–Sugawara Vectors, Vertex Algebras, and Central Elements

The Segal–Sugawara vectors of $\mathfrak{osp}(M|2n)$ generate the center of the affine vertex algebra at the critical level. These vectors are constructed using symmetrizers from the Brauer algebra, and their explicit formulas generalize classical results for orthogonal and symplectic Lie algebras. The development of an extended Brauer-type algebra resolves singularities in the symmetrizer, allowing for robust construction and evaluation at the critical value of parameters (Molev et al., 30 Jul 2025).

These central elements underpin the description of higher quantum Hamiltonians in integrable systems such as the Gaudin model, and the explicit techniques developed there provide substantial tools for both representation theory and the theory of integrable systems.

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The orthosymplectic Lie superalgebra thus serves as a foundation for a unified theory blending classical harmonic analysis, supergeometry, representation theory, and modern quantum symmetry, with far-reaching algebraic and physical implications in both mathematics and theoretical physics.