Denote m_0 the infinite dimensional N-graded Lie algebra defined by basis e_i, i>= 1 and relations [e_1,e_i] = e_(i+1) for all i>=2. We compute in this article the bracket structure on H1(m_0,m_0), H2(m_0,m_0) and in relation to this, we establish that there are only finitely many true deformations of m_0 in each nonpositive weight, by constructing them explicitely. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.
Denote $\fm_2$ the infinite dimensional $\N$-graded Lie algebra defined by the basis $e_i$ for $i\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\geq 3$. We compute in this article the bracket structure on $H^1(\fm_2,\fm_2)$, $H^2(\fm_2,\fm_2)$ and in relation to this, we establish that there are only finitely many true deformations of $\fm_2$ in each weight by constructing them explicitely. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation.
In this work it is considered the vector space composed by the infinitesimal deformations of the model $\mathbb{Z}_3$-filiform Lie algebra $L^{n,m,p}$. By using these deformations all the $\mathbb{Z}_3$-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work together to those obtained in [Integrable deformations of nilpotent color Lie superalgebras, J. Geom. Phys. 61(2011)1797-1808] and [Corrigendum to Integrable deformations of nilpotent color Lie superalgebras, J. Geom. Phys. 62(2012)1571], leads to compute the total dimension of t…
For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$, we show that there is a family ${\mathfrak m}_2^λ(p)$ of restricted Lie algebra structures parameterized by elements $λ\in {\mathbb F}^p$. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and $[p]$-operations in the corresponding restricted one-dimensional central extensions…
We show, for a field ${\mathbb F}$ of prime characteristic $p>0$, that the truncated filiform Lie algebra ${\frak m}_0(p)$ admits a family ${\frak m}_0^λ(p)$ of restricted Lie algebra structures parameterized by elements $λ\in {\mathbb F}^p$. We compute the ordinary cohomology groups $H^q({\frak m}_0^λ(p))$ and restricted cohomology groups $H^q_*({\frak m}_0^λ(p))$ for $q=1, 2$, and we give explicit descriptions of bases for these cohomology spaces. We apply our results to restricted one-dimensional central Extensions of the algebras ${\frak m}_0^λ(p)$.
In this work infinitesimal deformations of the model filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebra have been studied. All the filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebras can be obtained by means of infinitesimal deformations, hence the importance of these. Thus, in particular, we give a family of filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebras via linearly integrable deformations.
We continue the study of the filiform Z2xZ2-color Lie superalgebras. All of them can be obtained by using infinitesimal deformations, i.e. cocycles. In this work we give the total dimension of such cocycles (for any dimensions n, m, p and t of the Z2xZ2-color Lie superalgebras). Also, we give a basis of such cocycles in some generic and concrete cases.
We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is "almost" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2.
In this paper, we define and develop a cohomology and deformation theories of Jacobi-Jordan algebras. We construct a cohomology based on two operators, called zigzag cohomology, and detail the low degree cohomology spaces. We describe the relationships between first and second cohomology groups with extensions and deformations. Moreover, we consider cohomology properties of pseudo-euclidean Jacobi-Jordan algebras and provide a deformation theory that fits with our zigzag cohomology of Jacobi-Jordan algebras. Furthermore, the paper includes several examples and applications.
In this paper, we establish the cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we use this type of cohomology to characterize deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota-Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order $n$ deformation of a relative Rota-Baxter operator can be extended to an order $n+1$ deformation if and only if the obstruction class …
The purpose of this paper is to study global deformations of Hom-Leibniz algebras. We introduce a cohomology for Hom-Leibniz algebras with values in a Hom-module, characterize versal deformations and provide examples.
Dendriform coalgebras are the dual notion of dendriform algebras and are splitting of associative coalgebras. In this paper, we define a cohomology theory for dendriform coalgebras based on some combinatorial maps. We show that the cohomology with self coefficients governs the formal deformation of the structure. We also relate this cohomology with the cohomology of dendriform algebras, coHochschild (Cartier) cohomology of associative coalgebras and cohomology of Rota-Baxter coalgebras which we introduce in this paper. Finally, using those combinatorial maps, we introduce homotopy analogue of …
Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $λ\in {\bf k}$, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by $λ$-weighted relative Rota-Baxter operators. Using such characterization, we define the cohomology of a $λ$-weighted relative Rota-Baxter operator $T$, and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule.…
Given a finite dimensional Lie algebra $\mathfrak{g}$, let $Γ_\circ(\mathfrak{g})$ be the set of irreducible $\mathfrak{g}$-modules with non-vanishing cohomology. We prove that a $\mathfrak{g}$-module $V$ belongs to $Γ_\circ(\mathfrak{g})$ only if $V$ is contained in the exterior algebra of the solvable radical $\mathfrak{s}$ of $\mathfrak{g}$, showing in particular that $Γ_\circ(\mathfrak{g})$ is a finite set and we deduce that $H^*(\mathfrak{g},V)$ is an $L$-module, where $L$ is a fixed subgroup of the connected component of $\operatorname{Aut}(\mathfrak{g})$ which contains a Levi factor. We…
In this paper, we define representations and cohomology of weighted Rota-Baxter Lie algebras. As applications of cohomology, we study abelian extensions and formal $1$-parameter deformations weighted Rota-Baxter Lie algebras. Finally, we consider weighted Rota-Baxter paired operators that induces a weighted Rota-Baxter Lie algebra and a representation of it. We define suitable cohomology for such paired operators that govern deformation.
Higher derivations on an associative algebra generalizes higher order derivatives. We call a tuple consisting of an algebra and a higher derivation on it by an AssHDer pair. We define a cohomology for AssHDer pairs with coefficients in a representation. Next, we study central extensions, abelian extensions of AssHDer pairs and relate them to the second cohomology group with appropriate coefficients. Deformations of AssHDer pairs are also considered which are governed by the cohomology with self coefficient.
Rota-Baxter operators and more generally $\mathcal{O}$-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct a cohomology for an $\mathcal{O}$-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable representation. Next, we study linear and formal deformations of an $\mathcal{O}$-…
A cohomology theory of weighted Rota-Baxter $3$-Lie algebras is introduced. Formal deformations, abelian extensions, skeletal weighted Rota-Baxter $3$-Lie 2-algebras and crossed modules of weighted Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups.
The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Al…
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.
The aim of this paper is to provide a cohomology of $n$-Hom-Lie color algebras governing one parameter formal deformations. Then, we study formal deformations of a $n$-Hom-Lie color algebra and introduce the notion of Nijenhuis operator on an $n$-Hom-Lie color algebra, which could give rise to infinitesimally trivial $(n-1)$-order deformations. Furthermore, in connection with Nijenhuis operators we introduce and discuss the notion of a product structure on $n$-Hom-Lie color algebras.
We give a complete study of the Clifford-Weyl algebra ${\mathcal C}(n,2k)$ from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself). We show that ${\mathcal C}(n,2k)$ is rigid when $n$ is even or when $k \neq 1$. We find all non-trivial deformations of ${\mathcal C}(2n+1,2)$ and study their representations.
In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can induce all deformations and is unique on the infinitesimal level. We construct formal 1-parameter deformations of Jacobi-Jordan algebras up to dimension 5 and versal deformations for 3-dimensional Jacobi-Jordan algebras.
We consider the socle deformations arising from formal deformations of a class of Koszul self-injective special biserial algebras which occur in the study of the Drinfeld double of the generalized Taft algebras. We show, for these deformations, that the Hochschild cohomology ring modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.
Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler-Poincare characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.
Let X be a hyperkahler manifold deformation equivalent to a Hilbert scheme of n points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X, and derive a generating series for deducing the number of canonical Hodge classes on X. The formula indicates the number of Hodge classes on X that remain Hodge under any deformation.
Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that an…
We describe the infinitesimal deformations of the standard embedding of the Lie superalgebra $D(2, 1 ; α)$ into the Poisson superalgebra of pseudodifferential symbols on $S^{1|2}$. We show that for the standard embedding of $D(2, 1 ; α)$ into the Poisson superalgebra of differential operators on $S^{1|2}$, the infinitesimal deformations correspond to formal deformations. For the embedding of $D(2, 1 ; α)$ into the derived contact superconformal algebra ${K}'(4)$, the infinitesimal deformations are formal deformations.
We study the geometry and cohomology of the (generic fibres) of formal deformation schemes of one-dimensional formal modules of finite height. By the work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic etale cohomology of these spaces realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence for supercuspidal representations is realized by…
For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an André-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs. Cohen-Macaulay approximation induces maps between versal base spaces for pairs and cohomology conditions imply properties like smoothness, isomorphism and linear se…
