Lie Algebra Structure Research Articles

On the Lie algebra structure of integrable derivations

AbstractBuilding on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group.

Normal forms of double Hopf oscillators with radial nonlinearities

We are concerned with the simplest normal forms of non-resonant double Hopf oscillators with radial nonlinearities. Due to the required Lie algebraic structure for preserving the radial structure in the normal form process, we appeal to the Lie algebra of radial and rotational transformation generators. The structure constants of the Lie algebra are presented, where we notice a Lie ideal structure for the rotational vector fields. This gives rise to an effective reduction technique through the use of its corresponding quotient Lie algebra. Yet, computations of kernel spaces of the generalized homological maps seem a formidable project and thus, we employ the multiple Lie brackets method to skip explicit kernel space derivations. The simplest normal forms and orbital normal forms of the nonlinear family are presented. We also address their parametric normal forms as multiple parametric perturbations of the corresponding vector fields. Transformation symmetry groups associated with truncated normal forms are also derived. An efficient algorithm is provided for the normal form computations using computer algebra systems. Finally, we derive state-feedback controllers for a two-arm vibrator using normal forms of radial double Hopf singularity. This is to illustrate how our normal forms can be applied on classical control engineering problems as a new alternative nonlinear controller technique to the existing methods in nonlinear control theory.

Integrable Floquet systems related to logarithmic conformal field theory

We study an integrable Floquet quantum system related to lattice statistical systems in the universality class of dense polymers. These systems are described by a particular non-unitary representation of the Temperley-Lieb algebra. We find a simple Lie algebra structure for the elements of Temperley-Lieb algebra which are invariant under shift by two lattice sites, and show how the local Floquet conserved charges and the Floquet Hamiltonian are expressed in terms of this algebra. The system has a phase transition between local and non-local phases of the Floquet Hamiltonian. We provide a strong indication that in the scaling limit this non-equilibrium system is described by the logarithmic conformal field theory.

Certain lie algebraic structures on Riemannian manifolds with semi-symmetric non-metric connection

As a natural consequence of the Levi-Civita connection on a Riemannian manifold, there is a Lie algebra structure on a Riemannian manifold. Lie Algebras and Lie Groups are the mathematical structure of continuous symmetries in physics. In this paper, semi-symmetric non-metric connection is considered instead of Levi-Civita connection of Riemann manifold, and accordingly the existence of algebraic structures is investigated. First, it is shown that there is not always a Lie algebra structure on a Riemannian manifold with a semi-symmetric non-metric connection. Then, necessary and sufficient conditions for Lie admissible algebra, pre-Lie algebra and post Lie algebra on a Riemann manifold with semi-symmetric non-metric connection are obtained depending on geometric terms. In addition, the cases of the Riemannian manifold with such algebraic structures according to the semi-symmetric non-metric connection being Einstein manifold and being flat manifold have been also investigated.

CHARACTERIZATION OF FINITE DIMENSIONAL NILPOTENT LIE ALGEBRAS BY THE DIMENSION OF THEIR SCHUR MULTIPLIERS, s(L) = 5

It is known that the dimension of the Schur multiplier of a non-abelian nilpotent Lie algebra L of dimension n is equal to 1 2 (n − 1)(n − 2) + 1 − s(L) for some s(L) ≥ 0. The structure of all nilpotent Lie algebras has been given for s(L) ≤ 4 in several

Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of $\mathrm{HH}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the $B_\infty$-structure of the Hochschild cochain complex. In the appendix we explain how the $p$-power operation on Hochschild cohomology can be seen as an artifact of this $B_\infty$-structure. In particular, we establish well-definedness of the $p$-power operation, following some -- originally topological -- methods due to May, Cohen and Turchin, using the language of operads.

Автоморфные алгебры динамических систем и обобщенные контракции Иненю-Вигнера

Lie algebras a with a complex underlying vector space V are
 studied that are automorphic with respect to a given linear
 dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂
 Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a
 Lie algebraic structure to the vector space of trajectories of
 the group Gt. The basics of the general structure of automorphic
 algebras a are described in terms of the eigenspace
 decomposition of the operatorM ∈ der(a) that determines
 the dynamics. Symmetries encoded by the presence of nonabelian
 automorphic algebras are pointed out connected to
 conservation laws, spectral relations and root systems. It is
 shown that, for a given dynamics Gt, automorphic algebras
 can be found via a limit transition in the space of Lie algebras
 on V along the trajectories of the group Gt itself. This procedure
 generalises the well-known Inönü-Wigner contraction
 and links adjoint representations of automorphic algebras to
 isospectral Lax representations on gl(V ). These results can
 be applied to physically important symmetry groups and their
 representations, including classical and relativistic mechanics,
 open quantum dynamics and nonlinear evolution equations.
 Simple examples are given.

Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Martínez Alonso–Shabat equation

We study a Lagrangian extension of the 5d Martínez Alonso–Shabat equation \(\mathcal {E}\)$$\begin{aligned} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{aligned}$$that coincides with the cotangent equation \(\mathcal {T}^{*}\mathcal {E}\) to the latter. We describe the Lie algebra structure of its symmetries (which happens to be quite nontrivial and is described in terms of deformations) and construct two families of recursion operators for symmetries. Each family depends on two parameters. We prove that all the operators from the first family are hereditary, but not compatible in the sense of the Nijenhuis bracket. We also construct two new parametric Lax pairs that depend on higher-order derivatives of the unknown functions.

Quantization of restricted Lagrangian subvarieties in positive characteristic

Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization.

The primitive filtration of the Leibniz complex

Pirashvili exhibited a small subcomplex of the Leibniz complex $$(T(s {\mathfrak {g}}), d_{\textrm{Leib}})$$ of a Leibniz algebra $${\mathfrak {g}}$$ . The main result of this paper generalizes this result to show that the primitive filtration of $$T(s{\mathfrak {g}})$$ provides an increasing, exhaustive filtration of the Leibniz complex by subcomplexes, thus establishing a conjecture due to Loday. The associated spectral sequence is used to give a new proof of Pirashvili’s conjecture that, when $${\mathfrak {g}}$$ is a free Leibniz algebra, the homology of the Pirashvili complex is zero except in degree one. This result is then used to show that the desuspension of the Pirashvili complex carries a natural $$L_\infty $$ -structure that induces the natural Lie algebra structure on the homology of the complex in degree zero.

Systematic orbifold constructions of Schellekens' vertex operator algebras from Niemeier lattices

We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras $V$ of central charge 24 with non-zero weight-one space $V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras $V_N$ and certain 226 short automorphisms in $\operatorname{Aut}(V_N)$. We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in arXiv:1910.04947, of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in arXiv:1708.05990 and arXiv:1910.04947 this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in $\operatorname{Co}_0$. Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with non-zero weight-one space $V_1$ is uniquely determined by the Lie algebra structure of $V_1$.

The half-period addition formulae for genus two hyperelliptic ℘ functions and the Sp(4, ) Lie group structure

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4, )/ ≅ SO(3, 2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic ℘ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4, ) Lie group structure.

The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups

Abstract Let 𝐺 be one of the sporadic simple Mathieu groups M 11 M_{11} , M 12 M_{12} , M 22 M_{22} , M 23 M_{23} or M 24 M_{24} , and suppose 𝑘 is an algebraically closed field of prime characteristic 𝑝, dividing the order of 𝐺. In this paper, we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the 𝑝-blocks of k ⁢ G kG . In particular, we calculate the dimension of HH 1 ⁢ ( B ) \mathrm{HH}^{1}(B) for the 𝑝-blocks 𝐵 of k ⁢ G kG , and in almost all cases, we determine whether HH 1 ⁢ ( B ) \mathrm{HH}^{1}(B) is a solvable Lie algebra.

Conjugation matters. Bioctonionic Veronese vectors and Cayley–Rosenfeld planes

Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane [Formula: see text], we present a novel, explicit construction of the bioctonionic Cayley–Rosenfeld plane [Formula: see text], again by exploiting Veronese coordinates. We discuss the isometry groups of all generalized bioctonionic planes, recovering all complex and real forms of the exceptional groups [Formula: see text] and [Formula: see text], and characterizing such planes as symmetric and Hermitian symmetric spaces. We conclude by discussing some possible physical applications.

Bilinearization, Reachability, and Optimal Control of Control-Affine Nonlinear Systems: A Koopman Spectral Approach

This article considers the problem of bilinearization and optimal control of a control-affine nonlinear system by projecting the system dynamics onto the Koopman eigenspace. Although there are linearization techniques like Carleman linearization for embedding a finite-dimensional nonlinear system into an infinite-dimensional space, they depend on the analytic property of the vector fields and work only on polynomial space. The proposed method utilizes the Koopman canonical transform, specifically the Koopman eigenfunctions of the drift vector field, to transform the dynamics into a bilinear system under certain assumptions. While the bilinearization is exact, if there exists a Koopman-invariant finite-dimensional subspace for the drift vector field, sometimes this condition is too conservative. An approximate approach is to minimize an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}^2$</tex-math></inline-formula> norm on the truncated state-space. The approximation can be carried out from time-series data without explicit knowledge of the drift vector field. Controllability of the bilinear system is analyzed using the Myhill semigroup method and Lie algebraic structures. Pontryagin’s principle is applied to the bilinear system to yield a two-point boundary-value problem for the optimal control design. A single shooting method solves the boundary value problem in order to determine the control signal. Alternatively, a gradient-based method is also outlined to find the optimal control which exploits the bilinear structure. Several examples of control-affine nonlinear systems numerically illustrate the bilinearization and optimal control design, assuming a cost function quadratic in the states and control input.

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

The structure of Lie algebras with a derivation satisfying a polynomial identity

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity r(t) = 0. In the special case of the polynomial we obtain a uniform bound on the nilpotency class of Lie algebras admitting a periodic derivation of order n. We even find an optimal bound on the nilpotency class in characteristic p if p does not divide a certain invariant ρn .

Nilpotent Lie algebras of class 3 with 1-dimensional center and 3-dimensional derived subalgebra

In this paper, we obtain the structure of all non-isomorphic finite-dimensional nilpotent Lie algebras L of class 3 with derived subalgebra of dimension 3 when

Ward Identities in the $$\mathfrak {sl}_3$$ Toda Conformal Field Theory

Toda conformal field theories are natural generalizations of Liouville conformal field theory that enjoy an enhanced level of symmetry. In Toda conformal field theories this higher-spin symmetry can be made explicit, thanks to a path integral formulation of the model based on a Lie algebra structure. The purpose of the present document is to explain how this higher level of symmetry can manifest itself within the rigorous probabilistic framework introduced by R. Rhodes, V. Vargas and the first author. One of its features is the existence of holomorphic currents that are introduced via a rigorous derivation of the Miura transformation. More precisely, we prove that the spin-three Ward identities, that encode higher-spin symmetry, hold in the $\mathfrak{sl}_3$ Toda conformal field theory; as an original input we provide explicit expressions for the descendent fields which were left unidentified in the physics literature. This representation of the descendent fields provides a new systematic method to find the degenerate fields of the $\mathfrak{sl}_3$ Toda (and Liouville) conformal field theory, which in turn implies that certain four-point correlation functions are solutions of an hypergeometric differential equation of the third order.

Closed G2-Structures on Unimodular Lie Algebras with Non-trivial Center

We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G2-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G2-structures, up to isomorphism, and we show that six of them arise as the contactization of a symplectic Lie algebra. Finally, we prove that every semi-algebraic soliton on the contactization of a symplectic Lie algebra must be expanding, and we determine all unimodular Lie algebras with center of dimension at least two that admit semi-algebraic solitons, up to isomorphism.