Lie Algebra Structure Research Articles

The role of positivity and causality in interactions involving higher spin

It is shown that the recently introduced positivity and causality preserving string-local quantum field theory (SLFT) resolves most No-Go situations in higher spin problems. This includes in particular the Velo–Zwanziger causality problem which turns out to be related in an interesting way to the solution of zero mass Weinberg–Witten issue. In contrast to the indefinite metric and ghosts of gauge theory, SLFT uses only positivity-respecting physical degrees of freedom. The result is a fully Lorentz-covariant and causal string field theory in which light- or space-like linear strings transform covariant under Lorentz transformation.The cooperation of causality and quantum positivity in the presence of interacting s≥1 particles leads to remarkable conceptual changes. It turns out that the presence of H-selfinteractions in the Higgs model is not the result of SSB on a postulated Mexican hat potential, but solely the consequence of the implementation of positivity and causality. These principles (and not the imposed gauge symmetry) account also for the Lie-algebra structure of the leading contributions of selfinteracting vector mesons.Second order consistency of selfinteracting vector mesons in SLFT requires the presence of H-particles; this, and not SSB, is the raison d'être for H.The basic conceptual and calculational tool of SLFT is the S-matrix. Its string-independence is a powerful restriction which determines the form of interaction densities in terms of the model-defining particle content and plays a fundamental role in the construction of pl observables and sl interpolating fields.

On characterizing nilpotent Lie algebras by their multiplier, $$s(L)=4$$

Let L be a non-abelian nilpotent Lie algebra of dimension n and $$s(L)=\frac{1}{2}(n-1)(n-2)+1- \dim {\mathcal {M}}(L)$$ , where $${\mathcal {M}}(L)$$ denotes the Schur multiplier of L. For a non-abelian nilpotent Lie algebra, we know $$ s(L)\ge 0 $$ and the structure of all nilpotent Lie algebras are well known for $$ s(L) \in \lbrace 0,1,2,3 \rbrace $$ in several papers. The current paper is devoted to obtain the structure of all nilpotent Lie algebras L, when $$ s(L)=4 $$ .

Gerstenhaber Structure on Hochschild Cohomology of Toupie Algebras

We study homological properties of a family of algebras called toupie algebras. Our main objective is to obtain the Gerstenhaber structure of their Hochschild cohomology, with the purpose of describing the Lie algebra structure of the first Hochschild cohomology space, together with the Lie module structure of the whole Hochschild cohomology. © 2019, Springer Nature B.V.

Algebraic Hopf invariants and rational models for mapping spaces

The main goal of this paper is to define an invariant $$mc_{\infty }(f)$$ of homotopy classes of maps $$f:X \rightarrow Y_{\mathbb {Q}}$$ , from a finite CW-complex X to a rational space $$Y_{\mathbb {Q}}$$ . We prove that this invariant is complete, i.e. $$mc_{\infty }(f)=mc_{\infty }(g)$$ if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure on certain convolution algebras. More precisely, given an operadic twisting morphism from a cooperad $$\mathcal {C}$$ to an operad $$\mathcal {P}$$ , a $$\mathcal {C}$$ -coalgebra C and a $$\mathcal {P}$$ -algebra A, then there exists a natural homotopy Lie algebra structure on $$Hom_\mathbb {K}(C,A)$$ , the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to model mapping spaces. More precisely, suppose that C is a $$C_\infty $$ -coalgebra model for a simply-connected finite CW-complex X and A an $$L_\infty $$ -algebra model for a simply-connected rational space $$Y_{\mathbb {Q}}$$ of finite $$\mathbb {Q}$$ -type, then $$Hom_\mathbb {K}(C,A)$$ , the space of linear maps from C to A, can be equipped with an $$L_\infty $$ -structure such that it becomes a rational model for the based mapping space $$Map_*(X,Y_\mathbb {Q})$$ .

Проективно-групповые свойства h-пространств типа {221}

The curvature of a 5-dimensional h-space H221 of the type {221} [3] is investigated, necessary and sufficient conditions are obtained for H221 to be a space of constant curvature K (Theorem 1). A general solution of the Eisenhart equation is found in an h-space H221 of non-constant curvature. The necessary and sufficient conditions for the existence of non-homothetical projective motion in an h-space H221 of non-constant curvature are established (Theorem 5) and, as a consequence, the structure of the non-homothetical projective Lie algebra in such a space is determined (Theorem 6).

Convolution algebras and the deformation theory of infinity-morphisms

Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.

Lie polynomials in q-deformed Heisenberg algebras

Let F be a field, and let q∈F. The q-deformed Heisenberg algebra is the unital associative F-algebra H(q) with generators A,B and relation AB−qBA=I, where I is the multiplicative identity in H(q). The set of all Lie polynomials in A,B is the Lie subalgebra L(q) of H(q) generated by A,B. If q≠1 or the characteristic of F is not 2, then the equation AB−qBA=I cannot be expressed in terms of Lie algebra operations only, yet this equation still has consequences on the Lie algebra structure of L(q), which we investigate. We show that if q is not a root of unity, then L(q) is a Lie ideal of H(q), and the resulting quotient Lie algebra is infinite-dimensional and one-step nilpotent.

Restricted one-dimensional central extensions of the restricted filiform Lie algebras [formula omitted

Consider the filiform Lie algebra m0 with nonzero Lie brackets [e1,ei]=ei+1 for 1<i<p, where the characteristic of the field F is p>0. We show that there is a family m0λ(p) of restricted Lie algebra structures parameterized by elements λ∈Fp. We explicitly describe both the ordinary and restricted 1-cohomology spaces and show that for p≥3 these spaces are equal. We also describe the ordinary and restricted 2-cohomology spaces and interpret our results in the context of one-dimensional central extensions.

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.

Derivations of Leavitt path algebras

In this paper, we describe the K-module HH1(LK(Γ)) of outer derivations of the Leavitt path algebra LK(Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of HH1(LK(Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C⁎-algebra.

Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.

Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning.

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology).

Non-Abelian gauge theory from the Poisson bracket

The Hamiltonian of a nonrelativistic particle coupled to non-Abelian gauge fields is defined to construct a non-Abelian gauge theory. The Hamiltonian which includes isospin as a dynamical variable dictates the dynamics of the particle and isospin according to the Poisson bracket that incorporates the Lie algebraic structure of isospin. The generalized Poisson bracket allows us to derive Wong’s equations, which describe the dynamics of isospin, and the homogeneous (sourceless) equations for non-Abelian gauge fields by following Feynman’s proof of the homogeneous Maxwell equations.It is shown that the derivation of the homogeneous equations for non-Abelian gauge fields using the generalized Poisson bracket does not require that Wong’s equations be defined in the time-axial gauge, which was used with the commutation relation. The homogeneous equations derived by using the commutation relation are not Galilean and Lorentz invariant. However, by using the generalized Poisson bracket, it can be shown that the homogeneous equations are not only Galilean and Lorentz invariant but also gauge independent. In addition, the quantum ordering ambiguity that arises from using the commutation relation can be avoided when using the Poisson bracket.From the homogeneous equations, which define the “electric field” and “magnetic field” in terms of non-Abelian gauge fields, we construct the gauge and Lorentz invariant Lagrangian density and derive the inhomogeneous equations that describe the interaction of non-Abelian gauge fields with a particle.

Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebanski and later analyzed in a series of articles. The AKS-algebraic and related $\mathcal{R}$-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie--Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equations under consideration. Moreover, all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a fascinating Lagrange--d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax--Sato equations is also discussed. We pay a special attention to a generalization of the devised Lie-algebraic scheme to a case of loop Lie superalgebras of superconformal diffeomorphisms of the $1|N$-dimensional supertorus. This scheme is applied to constructing the Lax--Sato integrable supersymmetric analogs of the Liouville and Mikhalev-Pavlov heavenly equation for every $N\in\mathbb{N}\backslash\lbrace 4;5\rbrace.$

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

Classical M. A. Buhl Problem, Its Pfeiffer–Sato Solutions, and the Classical Lagrange–D’Alembert Principle for the Integrable Heavenly-Type Nonlinear Equations

The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible linear vector field equations and their general Pfeiffer and modern Lax–Sato-type special solutions. In particular, we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlinear dynamical systems called dispersion-free heavenly-type equations, which were introduced by Plebanski and later analyzed in a series of papers. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding coadjoint actions intimately connected with the classical Lie–Poisson structures on them. It is shown that their compatibility condition coincides with the corresponding heavenly-type equations under consideration. It is also demonstrated that all these equations are originated in the indicated way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. In addition, we present typical examples of equations of this kind demonstrating in detail their integrability via the scheme proposed in the paper. The relationship between a very interesting Lagrange–d’Alembert-type mechanical interpretation of the devised integrability scheme and the Lax–Sato equations is also discussed.

Block algebras with HH1 a simple Lie algebra

The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP)⁠. In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules.

-Differential operators and -differential modules for the Virasoro algebra

ABSTRACTThe concept of -differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the -differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible . Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the -differential module structures on them.

The τ-symmetry algebra of the generalized Kaup–Newell hierarchy

Abstract In this paper, based on the generalized Kaup–Newell(KN) spectral problem, the corresponding isospectral and nonisospectral soliton hierarchies are derived. It is shown that its Lie algebraic structure of the corresponding zero curvature equation is worked out. Finally the τ − symmetry algebra of the isospectral and nonisospectral generalized KN hierarchies is deduced.

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.