Differential Lie Algebra Research Articles

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.

Lie algebras with differential operators of any weights

<abstract><p>In this paper, we define a cohomology theory for differential Lie algebras of any weight. As applications of the cohomology, we study abelian extensions and formal deformations of differential Lie algebras of any weight. Finally, we consider homotopy differential operators on $ \mathrm{L}_{\infty} $ algebras and 2-differential operators of any weight on Lie 2-algebras, and we prove that the category of 2-term $ \mathrm{L}_{\infty} $ algebras with homotopy differential operators of any weight is same as the category of Lie 2-algebras with 2-differential operators of any weight.</p></abstract>

Deformations, cohomologies and integrations of relative difference Lie algebras

In this paper, first using the higher derived brackets, we give the controlling algebra of relative difference Lie algebras, which are also called crossed homomorphisms or differential Lie algebras of weight 1 when the action is the adjoint action. Then using Getzler's twisted L∞-algebra, we define the cohomology of relative difference Lie algebras. In particular, we define the regular cohomology of difference Lie algebras by which infinitesimal deformations of difference Lie algebras are classified. We also define the cohomology of difference Lie algebras with coefficients in arbitrary representations, and using the second cohomology group to classify abelian extensions of difference Lie algebras. Finally, we show that any relative difference Lie algebra can be integrated to a relative difference Lie group globally in a functorial way.

On (t2, t3)-Zakharov–Shabat equations of generalized Kadomtsev–Petviashvili hierarchies

We review the integration of the Kadomtsev–Petviashvili (KP) hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation, an algebra of functions equipped with a derivation that generalizes the gradient operator, an algebra of quaternion-valued functions, a differential Lie algebra, an algebra of polynomials equipped with the Pincherle differential, and a Moyal algebra. In all these cases, we can formulate and solve the Cauchy problem of the KP hierarchy. In addition, in each of these cases, we derive different Zakharov–Shabat (t2, t3)-equations—that is, different Kadomtsev–Petviashvili equations—and we prove the existence of solutions arising from solutions to the corresponding KP hierarchy.

Twisted Forms of Differential Lie Algebras over $${\mathbb {C}}(t)$$ Associated with Complex Simple Lie Algebras

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over $${\mathbb {C}}(t)$$ associated with complex simple Lie algebras. Hopf–Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors.

The solution to the embedding problem of a (differential) Lie algebra into its Wronskian envelope

ABSTRACTAny commutative algebra equipped with a derivation may be turned into a Lie algebra under the Wronskian bracket. This provides an entirely new sort of a universal envelope for a Lie algebra, the Wronskian envelope. The main result of this paper is the characterization of those Lie algebras which embed into their Wronskian envelope as Lie algebras of vector fields on a line. As a consequence we show that, in contrast to the classical situation, free Lie algebras almost never embed into their Wronskian envelope.

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D K → D L , defining the tensor product L ⊗ K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].

Cartan Connections as a Tool for Kinematic Chain Calculation

In this paper, the application of the concepts of Cartan connection and covariant derivative to robotics and mechanisms is presented. In particular, how Cartan connection can be used in kinematic calculation for rigid serial manipulators. The ideas were inspired in books of differential geometry, Lie groups and Lie algebras and its applications. The important concept of extended Newton’s law (or covariant formulations of dynamics) is also presented. It is presented first for the case of pure rotations and then for general motion of the robot’s links. Some calculations are presented for the case of a two-link planar robot, which is an example of serial mechanism, and for a planar n-bar mechanism, which is a closed-loop system. Finally, conclusions and future work directions are presented.

Lie symmetries of the Ricci and energy-momentum tensors

Though the Ricci and energy-momentum tensors have the same algebraic symmetries and appear in a similar way in the Einstein equations, in that interchanging them and inverting the gravitational coupling leaves the equations invariant, their differential symmetry Lie algebras are not identical in general. For cylindrically symmetric static spacetimes their inter-relationship is studied in this paper. Neither set of symmetries is a subset of the other in general.

String bracket and flat connections

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\Mc$ is the Maurer-Cartan moduli space of the graded differential Lie algebra $\Omega^\ast (M, \adp)$, the differential forms with values in the associated adjoint bundle of $P$. For a 2-dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman in \cite{G2}. We treat different Lie algebra structures on $\H_{2\ast}(LM)$ depending on the choice of the linear reductive Lie group $G$ in our discussion.

On maximally supersymmetric Yang–Mills theories

We consider ten-dimensional supersymmetric Yang–Mills theory (10D SUSY YM theory) and its dimensional reductions, in particular, BFSS and IKKT models. We formulate these theories using algebraic techniques based on application of differential graded Lie algebras and associative algebras as well as of more general objects, L ∞- and A ∞-algebras. We show that using pure spinor formulation of 10D SUSY YM theory equations of motion and isotwistor formalism one can interpret these equations as Maurer–Cartan equations for some differential Lie algebra. This statement can be used to write BV action functional of 10D SUSY YM theory in Chern–Simons form. The differential Lie algebra we constructed is closely related to differential associative algebra (Ω, ∂ ̄ ) of (0, k)-forms on some supermanifold; the Lie algebra is tensor product of (Ω, ∂ ̄ ) and matrix algebra. We construct several other algebras that are quasiisomorphic to (Ω, ∂ ̄ ) and, therefore, also can be used to give BV formulation of 10D SUSY YM theory and its reductions. In particular, (Ω, ∂ ̄ ) is quasiisomorphic to the algebra ( B, d), constructed by Berkovits. The algebras (Ω 0, ∂ ̄ ) and ( B 0, d) obtained from (Ω, ∂ ̄ ) and ( B, d) by means of reduction to a point can be used to give a BV-formulation of IKKT model. We introduce associative algebra SYM as algebra where relations are defined as equations of motion of IKKT model and show that Koszul dual to the algebra ( B 0, d) is quasiisomorphic to SYM.

Theory of Finite Pseudoalgebras

Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of conformal algebras. They are defined as Lie algebras in a certain ``pseudotensor'' category instead of the category of vector spaces. A pseudotensor category (as introduced by Lambek, and by Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a way to compose them. This allows for the definition of Lie algebras, representations, cohomology, etc. An instance of such a category can be constructed starting from any cocommutative (or more generally, quasitriangular) Hopf algebra $H$. The Lie algebras in this category are called Lie $H$-pseudoalgebras. The main result of this paper is the classification of all simple and all semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-modules. We also start developing the representation theory of Lie pseudoalgebras; in particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems. We show that the cohomology theory of Lie pseudoalgebras describes extensions and deformations and is closely related to Gelfand-Fuchs cohomology. Lie pseudoalgebras are closely related to solutions of the classical Yang-Baxter equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian formalism in the theory of nonlinear evolution equations. As an application of our results, we derive a classification of simple and semisimple linear Poisson brackets in any finite number of indeterminates.

Gauge theories with graded differential Lie algebras

We present a mathematical framework of gauge theories that is based upon a skew-adjoint Lie algebra and a generalized Dirac operator, both acting on a Hilbert space.

What a Classical r-Matrix Really Is

To my friend and colleague K.C. Reddy on occasion of his retirement. The notion of classical r-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, – where the standard definitions are shown to be deficient, – is proposed, the notion of an O-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differential generality relation End (V ) V ∗ ⊗ V for a vector space V . Examples considered include a quadratic Poisson bracket on the dual space to a Lie algebra; generalized symplectic-quadratic models of such brackets (aka Clebsch representations); and Drinfel’d’s 2-cocycle interpretation of nondegenate classical r-matrices.

Operational calculus in differential algebras*

Outlines of operational calculus i.e., techniques, reducing the solution of linear differential equations with constant coefficients to the solution of appropriate algebraic systems of linear equations, in differential algebras are described. It is shown that in the classical differential algebra such techniques cannot be realized, because of the lack of non-trivial exponentials. At the same time, the differential Lie algebra of vertical derivations, possesses a full spectrum of exponentials, and the corresponding operational calculus (secondary operational calculus )is developed.*

The cobar construction as a Hopf algebra

We show that the integral cobar construction ΩC * X of a 1-reduced simplicial set X has the structure of a homotopy Hopf algebra. This leads to a rational differential Lie algebra ℒ(X) given by a differential d ℒ on the free Lie algebra L(s −1 * X). The universal enveloping algebra Uℒ(X) is naturally isomorphic to ΩC * X⊗ℚ. Let X be a 1-reduced simplicial set and let s −1 * X be the desuspension of the reduced chain complex * X of X. The cobar construction is given by a differential d Ω on the tensor algebra T(s −1 * X) defined over ℤ. The homology of ΩC * X coincides with the homology of the loop space Ω|X|. A formula for d Ω was given by Adams [Ad]. We show that the cobar construction ΩC * X has over ℤ the additional natural structure of a “homotopy Hopf algebra” consisting of a coassociative diagonal and a derivation homotopy E:?ψ≃ψ. Explicit formulas for ψ and E are given in (2.9). The result is based on the geometric cobar construction in [B]. Using the fundamental theory of Anick [A] we derive from the data (ΩC * X,ψ,E) a differential Lie algebra which is defined by a differential d ℒ on the free rational Lie algebra L(s −1 * X). The homology of ℒ(X) coincides with the rational homotopy groups of Ω|X|. Moreover we describe a cocommutative diagonal together with a derivation homotopy H:ψ⊗ℚ≃φ and a natural iso- morphism of differential Hopf algebras Formulas for φ, H and β are obtained by an induction. A map f:X→Y between 1-reduced simplicial sets induces the map f *=L(s −1 * f): L(s −1 * X)→L(s −1 * Y) between free Lie algebras which is compatible with the differential d ℒ. Hence X↦ℒ(X) yields a canonical functor from 1-reduced simplicial sets to free differential Lie algebras. This functor induces an isomorphism of rational homotopy categories. In fact, ℒ is up to homotopy naturally equivalent to the functor λ=NPG of Quillen [Q] which is the composite of four intricate functors. We describe in [B”] inductively a sequence of elements β(2),β(3),… which determine the differential d ℒ and we compute the quadratic part of d ℒ. A complete formula for d ℒ, however, is highly intricate since it can be considered as a kind of “simplicial form” of the classical Hausdorff formula; see (4.9) (4). We cannot expect a simple form of the differential d ℒ since the homology of ℒ(X) yields the rational homotopy groups of X; that is This supplements the result of Kan [K] who gave a combinatorial description of π*(X) in terms of the free simplicial group GX. Hence π*(X) requires computations in free groups while the formula for π*(X)⊗ℚ requires only the computation of the homology of a chain complex of rational vector spaces. We point out that ℒ(X) is a chain complex of finite type if X has only finitely many non degenerate simplices. Hence in this case d ℒ yields a finite formula for π n (X)⊗ℚ. Such a formula was not yet achieved in the literature. For example using the Sullivan method one obtains π*(X)⊗ℚ only implicitly by computing a minimal model of the De Rham algebra A(X); see Bousfield-Gugenheim [BG]. Here A(X) is defined explicitly in terms of X but is not of finite type if X has only finitely many non degenerate simplices so that A(X) does not give a finite formula for π n (X)⊗ℚ. In this case the minimal model M X of A(X) is of finite type and hence also gives a finite formula for π n (X)⊗ℚ. There is, however, no explicit formula for M X in terms of X in the literature and M X is not natural in X. On the other hand also the model λ(X) of Quillen is considerably larger than ℒ(X) above and is only defined as the normalization of a simplicial object; see (4.9) (4) below.

Noncommutative geometry with graded differential Lie algebras

Starting with a Hilbert space endowed with a representation of a unitary Lie algebra and an action of a generalized Dirac operator, we develop a mathematical concept towards gauge field theories. This concept shares common features with the Connes–Lott prescription of noncommutative geometry, differs from that, however, by the implementation of unitary Lie algebras instead of associative * -algebras. The general scheme is presented in detail and is applied to functions ⊗ matrices.

Derivation of prime rings of positive characteristic

Let L be a finite-dimensional differential Lie algebra acting on a prime ring R and let the inner part {ie49-1} of L be quasi-Frobenius. Then a constant ring RL is prime iff {ie49-2} is a differentially simple ring. A ring of constants is semiprime iff {ie49-3} is a direct sum of differentially simple rings, and the prime dimension of a constant ring is equal to the number of differentially simple summands {ie49-4}. The Galois closure of L is obtained from L by adding all the inner derivations of a symmetric Martindale quotient ring which agree with elements from {ie49-5}.

The center of a graded connected Lie algebra is a nice ideal

Soit (𝕃(V),d) une algèbre de Lie différentielle graduée définie sur le corps des nombres rationnels. Un idéal I dans l’algèbre de Lie H(𝕃(V),d) est dit gentil si pour tout cycle α∈𝕃(V) dont la classe appartient à I,I contient le noyau de l’application H(𝕃(V),d)→H(𝕃(V⊕ℚx),d),d(x)=α. Nous montrons que le centre de H(𝕃(V),d) est un gentil idéal et nous donnons dans ce cas des informations sur la structure de H(𝕃(V⊕ℚx),d). Ceci est ensuite appliqué à l’étude de la structure de l’algèbre de Lie d’homotopie rationnelle L X =π * (ΩX)⊕ℚ d’un CW complexe simplement connexe X.

Nonlinear dynamical systems, first integrals, Bose operators, and Lie algebras

A connection between nonlinear autonomous systems of ordinary differential equations, first integrals, Bose operators and Lie algebras is described. An extension to nonlinear partial differential equations is given.