Representations of Hamiltonian Lie algebras

Koszul Lie algebras and their subalgebras

This article proposes two types of unscented particle filters (UPFs) that leverage unscented transformation (UT) from a geometric perspective to compute the proposal distribution. An UPF on Lie groups is first developed. Specifically, both the propagation of the sigma points and the computation of the mean and covariance are performed on the Lie groups, while the weight update and resampling are conducted on the Lie algebra. Second, we introduce the log-linear property of group elements to streamline particle propagation by reducing redundant operations, thereby optimizing the proposed UPF framework. In the update process, intermittent measurements that are caused by factors such as packet dropouts and stochastic sensor scheduling are considered. While lowering computational demands, these measurements pose challenges to filter stability. To this end, the introduced property is used to prove that the estimation error remains bounded under certain assumptions. We further establish a critical threshold for the arrival rate of intermittent measurements and derive an upper bound for the expected state error covariance. Moreover, a detailed computational complexity analysis is conducted to evaluate the efficiency of the proposed method. Finally, with the original method serving as a benchmark, simulation and real-world GNSS/INS integrated navigation experiments confirm that the redesigned approach delivers comparable performance and significantly improved computational efficiency.

Irreducible weight modules with infinite-dimensional weight spaces for Lie algebra of type A2

On maximal solvable extensions of nilpotent Lie algebras

Abstract We show that there is an equivalence of $$\infty $$ ∞ -categories between Lie algebroids and certain kinds of curved Lie algebras. For this we develop a method to study the $$\infty $$ ∞ -category of curved Lie algebras using the homotopy theory of algebras over a complete operad.

Given a color Lie algebra [Formula: see text] and an [Formula: see text]-module [Formula: see text], we obtain a color Lie algebra [Formula: see text] which is a subalgebra of the derivation algebra of the semidirect sum [Formula: see text]. By using a representation of [Formula: see text] on the second cohomology group [Formula: see text] we obtain the Wells mapping, which is shown to be applicable to measure extensibility of pairs of derivations. We use [Formula: see text], [Formula: see text] and the Wells mapping to deduce the Wells sequence for any abelian extension of [Formula: see text] by [Formula: see text].

We investigate the Magnus expansion for a generic time-dependent two-level system under single-axis driving. By virtue of the su(2) Lie algebra, the expansion is decomposed into a commutator-free form. To illustrate the usefulness of the gained expression, we then revisit the Landau-Zener-Stückelberg-Majorana model, with a focus on non-adiabatic transitions as well as the Stokes phase. In addition, the semiclassical Rabi model is systematically treated by determining the Floquet quasienergy up to different orders. We demonstrate how to employ suitable picture transformations as well as how to enforce the symmetry of the underlying model to guarantee convergence of the expansion as well as to achieve satisfactory agreement with the exact results. For both models that we studied, it turns out that a third order approximation yields results that are in next to perfect agreement with exact analytical ones. Surprisingly, in the case of the semiclassical Rabi model, even the second order Magnus approximation in the adiabatic picture produces almost exact results for a large parameter range.

We develop the theory of ħ-vertex algebras, algebraic structures closely related to vertex algebras but with a deformed translation covariance axiom. We establish their structure theory, including analogues of Goddard's Uniqueness Theorem, the Reconstruction Theorem, Borcherds Identity, and the OPE Expansion Formula, and introduce the associated notions of ħ-Lie conformal and ħ-Poisson vertex algebras. The formalism provides a natural and simplified construction of the Zhu algebra. The main application is to the chiralization of classical star-products: we show that every star-product on the symmetric algebra of a Lie algebra (or its central extensions) admits a chiralization, and we derive explicit formulae for these chiral star-products, including the Moyal–Weyl and Gutt star-products. Setting ħ = 0 recovers explicit deformation quantizations of a broad class of Poisson vertex algebras, including the classical limits of free-boson, βγ-system, affine, and Virasoro vertex algebras.

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In this paper, we study the representation theory of affine Nappi-Witten Lie algebra H 4 ̂ corresponding to the Nappi-Witten Lie algebra H 4 . We completely classify Cartan-free modules of rank one for the Nappi-Witten Lie algebra H 4 . With the help of Cartan free H 4 modules we classify Cartan-free modules of rank one over affine Nappi Witten Lie algebras. We also give a necessary and sufficient condition for these modules to be irreducible. Finally as an application we classify Cartan free modules of rank one for affine-Virasoro Nappi-Witten Lie algebras.

Preserving spin symmetry in variational quantum algorithms is essential for producing physically meaningful electronic wave functions. Implementing spin-adapted transformations on quantum hardware, however, is challenging because the corresponding fermionic generators translate into noncommuting Pauli operators. In this study, we introduce an exact and computationally efficient factorization of spin-adapted unitaries derived from fermionic double excitation and deexcitation rotations. These unitaries are expressed as ordered products of exponentials of Pauli operators. Our method exploits the fact that the elementary operators in these generators form small Lie algebras. By working in the adjoint representation of these algebras, we reformulate the factorization problem as a low-dimensional nonlinear optimization over matrix exponentials. This approach enables precise numerical reparametrization of the unitaries without relying on symbolic manipulations. The proposed factorization provides a practical strategy for constructing symmetry-conserving quantum circuits within variational algorithms. It preserves spin symmetry by design, reduces implementation cost, and ensures the accurate representation of electronic states in quantum simulations of molecular systems.

Abstract We introduce color Heisenberg-Lie (super)algebras graded by the abelian groups Z 3 2 , Z 2 p × Z 3 2 for p = 1 , 2 , 3 , and investigate the properties of their associated multi-particle quantum paraoscillators. In the Rittenberg–Wyler’s color Lie (super)algebras framework the above abelian groups are the simplest ones which induce mixed brackets interpolating commutators and anticommutators. These mixed brackets allow to accommodate two types of parastatistics: one based on the permutation group (beyond bosons and fermions in any space dimension) and an anyonic parastatistics based on the braid group. In both such cases the two broad classes of paraparticles are given by parabosons and parafermions. Mixed-bracket parafermions are created by nilpotent operators; they satisfy a generalized Pauli exclusion principle leading to roots-of-unity truncations in their multi-particle energy spectrum (braided Majorana qubits and their Gentile-type parastatistics are recovered in this color Lie superalgebra setting). Mixed-bracket parabosons do not admit truncations of the spectrum; the minimal detectable signature of their parastatistics is encoded in the measurable probability density of two indistinguishable parabosonic oscillators in a given energy eigenstate.

L∞-algebraic extensions of non-Lorentzian kinematical Lie algebras, gravities, and brane couplings

We determine the irreducible representations of the simple Lie algebras with maximum weight multiplicity 2.

To address the challenges of high-precision pose alignment, dynamic path smoothness, and efficient multi-objective optimization in three-dimensional UAV path planning under complex environments, this study proposes the Lie Group-based Griffon Vulture Grey Wolf Hybrid Optimizer (L-VGWO) framework. First, the framework employs the rotation-vector representation of the Lie group [Formula: see text] to parameterize the path. Through the use of exponential and logarithmic mappings, the proposed model inherently preserves geometric continuity and avoids singularities in rotational motion. Second, to overcome the difficulty of efficient optimization in the high-dimensional Lie algebra space, we design the core solver of L-VGWO, which integrates the local refinement capability of the Grey Wolf Optimizer (GWO) with the global exploration ability of the Griffon Vulture Optimizer (GVOA). This cooperative mechanism substantially improves the balance between exploration and exploitation when optimizing the multi-objective cost function. Under a comprehensive objective formulation that includes obstacle avoidance, pose accuracy terms, and angular and linear velocity smoothness constraints, L-VGWO is validated through multi-scenario simulations and statistical analyses.

ABSTRACT The classification of all possible Poisson structures associated with a given Lie (super)algebra constitutes a fundamental problem in Poisson algebra theory. In this paper, we investigate transposed Poisson algebra structures on the N = 1 Heisenberg-Virasoro superalgebra. Our main results establish that all 1 2 -derivations on this superalgebra are trivial, which consequently implies the nonexistence of non-trivial transposed Poisson algebra structures on the N = 1 Heisenberg-Virasoro superalgebra.

We present a symmetry-based framework for equilibrium statistical mechanics that formulates a single Lie group combining conventional spacetime symmetries with a recently identified phase-space gauge-shifting invariance [Müller et al., Phys. Rev. Lett. 133, 217101 (2024)0031-900710.1103/PhysRevLett.133.217101]. Using Noether's theorem, we obtain a set of general Ward identities together with previously unexplored cross-relations arising from the noncommutation of different symmetry generators. The approach extends standard many-body symmetries-such as translations, rotations, Galilean boosts, dilations, and particle exchange-by incorporating an internal gauge-shift symmetry within a unified group structure. The resulting Lie algebra suggests a hierarchy of exact identities that encompass established sum rules and indicate possible cross-coupling relations between distinct response and correlation functions. We also identify a Wigner-Eckart-Ward reduction that simplifies tensor-hyperforce correlators to two scalar radial spectra in isotropic fluids, and we outline an equivariant gauge-constrained-DFT formulation whose Euler-Lagrange equationsare constructed to satisfy the corresponding Ward and cross-Ward constraints. This framework provides a consistent organizational basis for phenomena in liquids, mixtures, and interfaces, and may offer a symmetry-based perspective connecting structure, mechanics, and dynamics in many-body systems.

Matrix integrable hierarchies connected with the symplectic Lie algebras sp(2m) and their bi-Hamiltonian structures and Darboux transformations

Exact unitary transformations play a central role in the analysis and simulation of many-body quantum systems, yet the conditions under which they can be carried out exactly and efficiently remain incompletely understood. We show that exact transformations arise whenever the adjoint action of a unitary's generator defines a linear map within a finite-dimensional operator space. In this regime, there exists a finite-degree polynomial that annihilates the adjoint map, rendering the Baker-Campbell-Hausdorff (BCH) expansion finite. We identify the role of Lie algebras and their modules in producing finite BCH expansions in all known cases. This perspective brings together previously disparate examples of exact transformations under a single unifying principle and clarifies how algebraic relations between generators and transformed operators determine the polynomial degree of the transformation. We illustrate this framework for previously known cases of efficient unitary transformations including unitary coupled-cluster and Pauli product generators. Using this framework, we propose a new class of Fermionic generators that can be used for efficient transformations. The result establishes sufficient algebraic conditions for when exact unitary transformations are possible and provides new strategies for reducing their computational cost in quantum simulation and constructing feasible unitary transformations.