Lie Algebra Research Articles

Model-theoretic properties of nilpotent groups and Lie algebras

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP1. We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for 2<c, the generic c-nilpotent Lie algebra over Fp is strictly NSOP4 and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime p>c.

Hierarchical attitude stabilization controller design and analysis for underactuated spacecraft on SO(3)

In this paper, the complete attitude stabilization of underactuated spacecraft with two parallel single gimbal control moment gyroscopes (SGCMGs) is explored on 3-dimensional special orthogonal group (SO(3)) and its corresponding Lie algebra (so(3)). Instead of assumption on zero total angular momentum in existing article, a less conservative criterion is adopted to ensure system controllability, which does not simplify system dynamics. On the kinematic level, an ideal controller is proposed to globally stabilize attitude without singular initial condition and stagnation behavior. Besides, the ideal kinematic controller is developed on so(3), which avoids singularity or unwinding phenomenon caused by other attitude parameters. On the dynamic level, a hierarchical controller consisting of two parts is proposed. The high-level sliding mode part enforces finite time stabilization of angular velocity about underactuated axis while the low-level part tracks ideal angular velocity commands about actuated axes. A rigorous proof of the whole dynamic-kinematic system that has not been explored before is given. Analysis of hierarchical control from the perspective of linear algebra provides a novel insight into controller design for underactuated systems. Furthermore, the paradigm of controller design which is applicable to other underactuated systems is derived. Simulation results validate the effectiveness of the proposed control logic.

Toward Jordan decompositions for tensors

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

(α+2)-Dimensional fractional evolution equation: group classification, symmetries, reduction and conservation laws

ABSTRACT A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional ( α + 2 ) -dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem.

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

Twisted tensor products of quantum affine vertex algebras and coproducts

Let g be a symmetrizable Kac-Moody Lie algebra, and let Vgˆ,ħℓ, Lgˆ,ħℓ be the quantum affine vertex algebras constructed in [11]. For any complex numbers ℓ and ℓ′, we present an ħ-adic quantum vertex algebra homomorphism Δ from Vgˆ,ħℓ+ℓ′ to the twisted tensor product ħ-adic quantum vertex algebra Vgˆ,ħℓ⊗ˆVgˆ,ħℓ′. In addition, if both ℓ and ℓ′ are positive integers, we show that Δ induces an ħ-adic quantum vertex algebra homomorphism from Lgˆ,ħℓ+ℓ′ to the twisted tensor product ħ-adic quantum vertex algebra Lgˆ,ħℓ⊗ˆLgˆ,ħℓ′. Moreover, we prove the coassociativity of Δ.

Bigraded modified Toda hierarchy and its extensions

Modified Toda hierarchy is just the two-component first modified KP hierarchy, which is related to 2D Toda hierarchy through Miura transformation and also has been widely used in discussing the B-Toda and C-Toda hierarchies. In this paper, we firstly construct (N,M)-bigraded modified Toda hierarchy (BMTH) as a reduction of modified Toda hierarchy, and give corresponding Lie algebra interpretation. After that, we propose two kinds of extensions of (N,M)-BMTH. One is extended by using logarithmic flows, while the other is (2+1)D extension, which is corresponding to the toroidal Lie algebra slntor. At last, the relation of these two kinds of extensions also is discussed.

Lie algebras arising from two-periodic projective complex and derived categories

Let A be a finite-dimensional C-algebra of finite global dimension and A be the category of finitely generated right A-modules. By using of the category of two-periodic projective complexes C2(P), we construct the motivic Bridgeland's Hall algebra for A, where structure constants are given by Poincaré polynomials in t, then construct a C-Lie subalgebra g=n⊕h at t=−1, where n is constructed by stack functions about indecomposable radical complexes, and h is by contractible complexes. For the stable category K2(P) of C2(P), we construct its moduli spaces and a C-Lie algebra g˜=n˜⊕h˜, where n˜ is constructed by support-indecomposable constructible functions, and h˜ is by the Grothendieck group of K2(P). We prove that the natural functor C2(P)→K2(P) together with the natural isomorphism between Grothendieck groups of A and K2(P) induces a Lie algebra isomorphism g≅g˜. This makes clear that the structure constants at t=−1 provided by Bridgeland in [5] in terms of exact structure of C2(P) precisely equal to that given in [30] in terms of triangulated category structure of K2(P).

Quantum Sugawara operators in type A

The quantum Sugawara operators associated with a simple Lie algebra g are elements of the center of a completion of the quantum affine algebra Uq(gˆ) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ℓV(z) associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over Uq(gˆ) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed W-algebras via the Harish-Chandra homomorphism.We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.

Non-invertible symmetries and higher representation theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra \mathfrak{so}(N)𝔰𝔬(N), and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed ’t Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

Translation functors for locally analytic representations

Let [Formula: see text] be a [Formula: see text]-adic Lie group with reductive Lie algebra [Formula: see text]. In analogy to the translation functors introduced by Bernstein and Gelfand on the categories of [Formula: see text]-modules, we consider similarly defined functors on the category of modules over the locally analytic distribution algebra [Formula: see text] on which the center of [Formula: see text] acts locally finitely. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category [Formula: see text] via the functors from category [Formula: see text] to the category of coadmissible modules. We also investigate the effect of the translation functors on locally analytic representations [Formula: see text]la associated by the [Formula: see text]-adic Langlands correspondence for [Formula: see text] to two-dimensional Galois representations [Formula: see text].

Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra K is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of K⊗K¯ and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in C3 this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of C3. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy. Published by the American Physical Society 2024

Inverse Reduction for Hook-Type W-Algebras

Originating in the work of A.M. Semikhatov and D. Adamović, inverse reductions are embeddings involving W-algebras corresponding to the same Lie algebra but different nilpotent orbits. Here, we show that an inverse reduction embedding between the affine sln+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_{n+1}$$\\end{document} vertex operator algebra and the minimal sln+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_{n+1}$$\\end{document} W-algebra exists. This generalises the realisations for n=1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=1,2$$\\end{document} in Adamović (Commun Math Phys 366:1025–1067, 2019), Adamović (Math Ann 1–44, 2023). A similar argument is then used to show that inverse reduction embeddings exists between all hook-type sln+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_{n+1}$$\\end{document} W-algebras, which includes the principal/regular, subregular, minimal sln+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_{n+1}$$\\end{document} W-algebras, and the affine sln+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_{n+1}$$\\end{document} vertex operator algebra. This generalises the regular-to-subregular inverse reduction of Fehily (Commun Contemp Math 2250049, 2022), and similarly uses free-field realisations and their associated screening operators.

Bestiary of 6D (1, 0) SCFTs: Nilpotent orbits and anomalies

Many six-dimensional (1, 0) superconformal field theories (SCFTs) are known to fall into families labeled by nilpotent orbits of certain simple Lie algebras. For each of the three infinite series of such families, we show that the anomalies for the continuous zero-form global symmetries of a theory labelled by a nilpotent orbit O of g can be determined from the anomalies of the theory associated to the trivial nilpotent orbit (the parent theory), together with the data of O. In particular, knowledge of the tensor branch field theory is bypassed completely. We show that the known anomalies, previously determined from the geometric/atomic construction, are reproduced by analyzing the Nambu-Goldstone modes inside of the moment map associated to the g flavor symmetry of the parent SCFT. This provides further evidence for the physics underlying the labeling of the SCFTs by nilpotent orbits. We remark on some consequences, such as the reinterpretation of the 6D a-theorem for such SCFTs in terms of group theory. Published by the American Physical Society 2024

Transposed Poisson structures on Virasoro-type (super)algebras

We explore transposed Poisson structures on the Lie algebra: the deformed twisted Schrödinger-Virasoro algebra D(λ), as well as two Lie superalgebras: the super-BMS3 algebra and the twisted N=1 Schrödinger-Neveu-Schwarz algebra. Initially, we demonstrate the absence of non-trivial transposed Poisson structures on the Lie algebra D(λ) for λ≠1 and provide an example of a transposed Poisson algebra with associative and Lie parts isomorphic to the algebra of triadic extended Laurent polynomials and D(1). Subsequently, we establish that the super-BMS3 algebra possesses non-trivial 12-superderivations but lacks a non-trivial transposed Poisson structure. Finally, we prove that the twisted N=1 Schrödinger-Neveu-Schwarz algebra does not have non-trivial 12-superderivations and thus lacks non-trivial transposed Poisson structures.

A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits

Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit’s generators.

Characterizing barren plateaus in quantum ansätze with the adjoint representation

Variational quantum algorithms, a popular heuristic for near-term quantum computers, utilize parameterized quantum circuits which naturally express Lie groups. It has been postulated that many properties of variational quantum algorithms can be understood by studying their corresponding groups, chief among them the presence of vanishing gradients or barren plateaus, but a theoretical derivation has been lacking. Using tools from the representation theory of compact Lie groups, we formulate a theory of barren plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra, covering a large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks. Our theory provides, for the first time, the ability to compute the exact variance of the gradient of the cost function of the quantum compound ansatz, under mixing conditions that we prove are commonplace.

What can abelian gauge theories teach us about kinematic algebras?

The phenomenon of BCJ duality implies that gauge theories possess an abstract kinematic algebra, mirroring the non-abelian Lie algebra underlying the colour information. Although the nature of the kinematic algebra is known in certain cases, a full understanding is missing for arbitrary non-abelian gauge theories, such that one typically works outwards from well-known examples. In this paper, we pursue an orthogonal approach, and argue that simpler abelian gauge theories can be used as a testing ground for clarifying our understanding of kinematic algebras. We first describe how classes of abelian gauge fields are associated with well-defined subalgebras of the diffeomorphism algebra. By considering certain special subalgebras, we show that one may construct interacting theories, whose kinematic algebras are inherited from those already appearing in a related abelian theory. Known properties of (anti-)self-dual Yang-Mills theory arise in this way, but so do new generalisations, including self-dual electromagnetism coupled to scalar matter. Furthermore, a recently obtained non-abelian generalisation of the Navier-Stokes equation fits into a similar scheme, as does Chern-Simons theory. Our results provide useful input to further conceptual studies of kinematic algebras.

Generalized Hom–Lie–Rinehart algebras

The first-order Hom-differential operators on a commutative Hom-associative algebra with unit are introduced. We describe Hom–Lie algebra associated with a module of first-order Hom-differential operators. We give the notion of a generalized Hom–Lie–Rinehart algebra and we show the characterization of generalized Hom–Lie–Rinehart algebras in terms of Hom–Gerstenhaber–Jacobi algebras. We prove the necessary and sufficient condition to obtain a generalized Hom–Lie–Rinehart algebra structure on a Hom–Lie algebroid.

Canonical torsor bundles of prescribed rational functions on complex curves

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve [Formula: see text], we construct explicitly a bundle [Formula: see text] with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.