Formal deformations and extensions of "twisted" Lie algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite-dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.

Déformation par quantification et rigidité des algèbres enveloppantes

In these notes, we introduce formal hom-associative deformations of the quantum planes and the universal enveloping algebras of the two-dimensional non-abelian Lie algebras. We then show that these deformations induce formal hom-Lie deformations of the corresponding Lie algebras constructed by using the commutator as bracket.

The relationship between * - products (formal deformations) and quantization deformations (non formal deformations) on dual of nilpotent Lie algebras are studied. An explicit, tangential quantization deformation is given on algebras of polynomial functions and C°°, rapidly decreasing function on the dual of any nilpotent special Lie algebras.

Deforming the orthosymplectic Lie superalgebra inside the Lie superalgebra of superpseudodifferential operators

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general ``twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.

CONTENTS Introduction § 1. Main theorems Chapter I. Algebra § 2. Moyal deformations of the Poisson bracket and *-product on § 3. Algebraic construction § 4. Central extensions § 5. Examples Chapter II. Deformations of the Poisson bracket and *-product on an arbitrary symplectic manifold § 6. Formal deformations: definitions § 7. Graded Lie algebras as a means of describing deformations § 8. Cohomology computations and their consequences § 9. Existence of a *-product Chapter III. Extensions of the Lie algebra of contact vector fields on an arbitrary contact manifold §10. Lagrange bracket §11. Extensions and modules of tensor fields Appendix 1. Extensions of the Lie algebra of differential operators Appendix 2. Examples of equations of Korteweg-de Vries type References

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions.

$${\mathcal{O}}$$ -operators are important in broad areas of mathematics and physics, such as integrable systems, the classical Yang–Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $${\mathcal{O}}$$ -operators is established in consistence with the general principles of deformation theories. On the one hand, $${\mathcal{O}}$$ -operators are shown to be characterized as the Maurer–Cartan elements in a suitable graded Lie algebra. A given $${\mathcal{O}}$$ -operator gives rise to a differential graded Lie algebra whose Maurer–Cartan elements characterize deformations of the given $${\mathcal{O}}$$ -operator. On the other hand, a Lie algebra with a representation is identified from an $${\mathcal{O}}$$ -operator T such that the corresponding Chevalley–Eilenberg cohomology controls deformations of T, thus can be regarded as an analogue of the Andre–Quillen cohomology for the $${\mathcal{O}}$$ -operator. Thereafter, linear and formal deformations of $${\mathcal{O}}$$ -operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of an $${\mathcal{O}}$$ -operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota–Baxter operators of weight 0 and skew-symmetric r-matrices for the classical Yang–Baxter equation. For skew-symmetric r-matrices, there is an independent Maurer–Cartan characterization of the deformations as well as an analogue of the Andre–Quillen cohomology, which turn out to have an explicit relationship with the ones obtained as $${\mathcal{O}}$$ -operators associated to the coadjoint representations. Finally, linear deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.

On the hom-associative Weyl algebras

We consider the action of the Lie algebra \(\mathfrak {aff}(1)\), by the Lie derivative on the space of symbols \(\mathcal S_{\delta }=\bigoplus _{k=0}^{+\infty }\mathcal {F}_{\delta -k}\) and of pseudo-differential operators \(\Psi \mathcal {D}\mathcal {O}\). We investigate the first and the second cohomology space associated with the embedding of the affine Lie algebra \(\mathfrak {aff}(1)\) on the line \(\mathbb {R}\) in the modules \(\mathrm {M}\) where \(\mathrm {M}=\mathcal {F}_{\lambda }\), \(\mathrm {D}_{\lambda ,\mu }\), \(\mathcal {S}_\delta \), \(\mathcal {P}\), \(\Psi \mathcal {D}\mathcal {O}\). We study the deformations of the structure of the \(\mathfrak {aff}(1)\)-modules \(\mathcal {S}_\delta \), \(\mathcal {P}\) and \(\Psi \mathcal {D}\mathcal {O}\). We prove that any formal deformation is equivalent to its infinitesimal part. We complete our study by giving a example of deformation. Following the work of Guha we study some applications in mathematical physics.

We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

The aim of this paper is to extend to Hom-algebra structures the theory of 1-parameter formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis and Richardson. In this paper, formal deformations of Hom-associative and Hom-Lie algebras are studied. The first groups of a deformation cohomology are constructed and several examples of deformations are given. We also provide families of Hom-Lie algebras deforming Lie algebra sl(2)(K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl(2)(K).

We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

For infinite dimensional Lie algebras G, the theory of formal deformations does not generally give decisive results about analytic deformations. In particular, if Gt is a family of Lie algebras depending holomorphically on t with G 0=G, and if (Gt) is formally equivalent to G[[t]], it does not necessarily follow that Gt is isomorphic toG for small ∣t∣. However, one may ask if Ĝt is isomorphic to Ĝ for small ∣t∣,Ĝ t and Ĝ being suitable completions of Gt and G respectively. In this Letter, this question is discussed when G=W, the Witt algebra.