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In this paper, we classify 1-cocycles of the Witt algebra with coefficients in the tensor product of two arbitrary tensor density modules. In a special case, we recover a theorem originally established by Ng and Taft in [24]. Furthermore, by these 1-cocycles, we determine Lie bialgebra structures over certain infinite-dimensional Lie algebras containing the Witt algebra. [ABSTRACT FROM AUTHOR]

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX , where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q -cycle sets, q -braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting. [ABSTRACT FROM AUTHOR]

An exceptional cycle in a triangulated category with Serre functor is a generalization of a spherical object. Suppose that A and B are Gorenstein algebras, given a perfect exceptional n -cycle E ⁎ in K b (A - proj) and a perfect exceptional m -cycle F ⁎ in K b (B - proj) , we construct an A - B -bimodule N , and prove the product E ⁎ ⊠ F ⁎ is an exceptional (n + m − 1) -cycle in K b (Λ - proj) , where Λ = ( A N 0 B ). Using this construction, one gets many new exceptional cycles which is unknown before for certain class of algebras. [ABSTRACT FROM AUTHOR]

A close relation between categories of color Lie algebras and sets of pairs of homomorphisms of modules of Lie algebras is discussed, which is applicable to construct color Lie algebras from Lie algebras and their modules. It is also shown that such construction yields equivalent categories of color Lie algebras under an equivalent relation between symmetric bicharacters. As an application we construct and classify all simple color Lie algebras from the three dimensional simple complex Lie algebra and its finite-dimensional simple modules. [ABSTRACT FROM AUTHOR]

In this paper we introduce the Hom-analogue of the definition of 2-cocycle for Hopf algebras, called Hom-2-cocycle, and study its properties in order to give a theory of multiplication alteration by Hom-2-cocycles for Hom-Hopf algebras. We show that, just like in the classical setting, if H is a Hom-Hopf algebra with associated endomorphism α and σ is a convolution invertible Hom-2-cocycle, it is possible to define a new product in H to get a new Hom-Hopf algebra if α 4 = α. Moreover we introduce the notions of matched pair and skew pairing in the Hom case and, by the close connection between Hom-2-cocycles and Hom-skew pairings, we show that a special case of Hom-matched pair can be obtained as a deformation of a Hom-Hopf algebra by a Hom-2-cocycle built by a Hom-skew pairing. [ABSTRACT FROM AUTHOR]

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles. [ABSTRACT FROM AUTHOR]

Involving a symmetric Hochschild 1-cocycle condition, we equip the space of decorated planar rooted forests with a coproduct which turns the space into a dendriform-Nijenhuis bialgebra. We combine dendriform-Nijenhuis bialgebras with operated algebras and introduce the notation of an Ω-operated DN-bialgebra. Applying the universal property of the underlying operated algebras, we construct free objects in the category of Ω-cocycle DN-bialgebras. We introduce the notation of a DN-associative Yang-Baxter equation (AYBE) and show that the dendriform-Nijenhuis bialgebra offers an algebraic framework of the DN-associative Yang-Baxter equation. We construct a Leroux's TD operator from a solution of the DN-AYBE. We also give two different ways to derive Lie algebras from quasitriangular DN-bialgebras. Finally, we classify the solutions of the DN-AYBE in the unitary algebras of dimensions two and three over the field of complex numbers. [ABSTRACT FROM AUTHOR]

Recently E.R. Alvares, M.M. Alves and M.J. Redondo introduced a cohomology for a group G with values in a module over the partial group algebra K par (G) , which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action α of G on a (unital) algebra A we consider A as a K par (G) -module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in A. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that A is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group H p a r n (G , A) is isomorphic to the usual cohomology group H n (G , M (B)) , where M (B) is the multiplier algebra of B and B is the algebra under the enveloping action of α. [ABSTRACT FROM AUTHOR]

Abstract We give an efficient algorithm to compute equations of twists of hyperelliptic curves C of arbitrary genus over any perfect field k (of characteristic different from 2) starting with a cocycle in H 1 (Gal (k ‾ / k) , Aut (C)). We also discuss some interesting examples. [ABSTRACT FROM AUTHOR]

Abstract We describe Hom-Lie structures on affine Kac–Moody and related Lie algebras, and discuss the question when they form a Jordan algebra. [ABSTRACT FROM AUTHOR]

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature. [ABSTRACT FROM AUTHOR]

Let f : G × G ⟶ L be a weak 2-cocycle, where L / K is a finite Galois field extension with Galois group G , and A f = ( L / K , f ) be the associated weak crossed product K -algebra. We associate a partition of G to A f and construct idempotent 2-cocycles of A f . [ABSTRACT FROM AUTHOR]

In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-Lie Leibniz algebras are described up to isomorphism. It is shown that k -dimensional central extensions ( k ≥ 5 ) of these algebras are split. [ABSTRACT FROM AUTHOR]

The abstract arithmetic of non-commutative non-singular arithmetic curves (equivalently: the ideal theory of hereditary orders) is revisited in the framework of quantum B-algebras. It is shown that multiplication of ideals can be transformed into composition of functions. This yields a non-commutative “fundamental theorem of arithmetic” extending the classical one. Local hereditary arithmetics are presented by generators and relations and correlated with tubular quantum B-algebras. Main results are achieved by a divisor theory which furnishes the divisor group with a ring-like structure satisfying a 1-cocycle condition. [ABSTRACT FROM AUTHOR]

For J an integral domain and F its field of fractions, we construct a map from the 3-skeleton of the classifying space for Γ = S L 2 ( J [ t , t − 1 ] ) to a Euclidean building on which Γ acts. We then find an infinite family of independent cocycles in the building and lift them to the classifying space, thus proving that the cohomology group H 2 ( S L 2 ( J [ t , t − 1 ] ) ; F ) is infinite-dimensional. [ABSTRACT FROM AUTHOR]

We suggest a cohomological framework to describe groups of I -type and involutive Yang–Baxter groups. These groups are key in the study of involutive non-degenerate set-theoretic solutions of the quantum Yang–Baxter equation. Our main tool is a lifting criterion for 1-cocycles, established here in a general non-abelian setting. [ABSTRACT FROM AUTHOR]

We study cocycle twists of a 4-dimensional Sklyanin algebra A and a factor ring B which is a twisted homogeneous coordinate ring. Twisting such algebras by the Klein four-group G , we show that the twists A G , μ and B G , μ have very different geometric properties to their untwisted counterparts. For example, A G , μ has only 20 point modules and infinitely many fat point modules of multiplicity 2. The ring B G , μ falls under the purview of Artin and Stafford's classification of noncommutative curves, and we describe it using a sheaf of orders over an elliptic curve. [ABSTRACT FROM AUTHOR]

The theory of differential equations has an arithmetic analogue [8] in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear differential equations is the same as the study of the differential cocycle from GL n into its Lie algebra given by the logarithmic derivative [14] (equivalently by the Maurer–Cartan connection [17] ). However we prove here that there are no such cocycles in the context of arithmetic differential equations. In sequels of this paper [10,11] we will remedy the situation by introducing arithmetic analogues of Lie algebras and a skew version of differential cocycles; this will lead to a theory of linear arithmetic differential equations. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, the representation rings (or the Green rings) for a family of Hopf algebras of tame type, the 2-rank Taft algebra (at ) and its two relatives twisted by 2-cocycles are explicitly described via a representation theoretic analysis. It turns out that the Green rings can serve to detect effectively the twist-equivalent Hopf algebras here. [Copyright &y& Elsevier]

Abstract: Let G be a finite p-group, K a field of characteristic p, and the twisted group algebra of G over K with a 2-cocycle . Necessary and sufficient conditions for to be a uniserial algebra are presented in the paper. [Copyright &y& Elsevier]

Abstract: In this paper, we consider weak crossed product orders with coefficients in the integral closure S of a discrete valuation ring R in a tamely ramified Galois extension of the field of fractions of R. In the first section, we compute the Jacobson radical of when S is local, and we give a characterization of the hereditarity of the order in terms of the cocycle values. In the second section, we prove (again in the local case) that every σ in the inertia group for must belong to . In the final section, we compute the Jacobson radical in the general case (S is semilocal) and show how the hereditarity of can be determined locally under an additional hypothesis. [Copyright &y& Elsevier]

Abstract: In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the n-cycle , up to a power of q. This twisting is needed because c does not naturally induce an automorphism of the quantum Grassmannian, as it does classically and semi-classically. We extend this construction to give a quantum analogue of the action on the Grassmannian of the dihedral subgroup of generated by c and , the longest element, and this analogue takes the form of a groupoid. We show that there is an induced action of this subgroup on the torus-invariant prime ideals of the quantum Grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive Grassmannians. Then we see that this dihedral subgroup action exists classically, semi-classically (by Poisson automorphisms and anti-automorphisms, a result of Yakimov) and in the quantum and nonnegative settings. [Copyright &y& Elsevier]

Abstract: It is well known that Clifford algebras are group algebras deformed by a 2-cocycle. Furthermore, these algebras, which are not commutative in the usual sense, can be viewed as commutative algebras in certain symmetric monoidal categories of graded vector spaces. In this note we invent a Clifford process for coalgebras that will allow us to show that Clifford algebras have also cocommutative coalgebra structures, and consequently commutative and cocommutative weak braided Hopf algebras structures, within the same symmetric monoidal categories where they lie as commutative algebras. Also, we will show that they are selfdual weak braided Hopf algebras, monoidal Frobenius algebras and monoidal coFrobenius coalgebras. [Copyright &y& Elsevier]

Abstract: We study groups of bi-Galois objects over a Hopf algebra H in a braided monoidal category . We assume H to be cocommutative in a certain sense; this does not mean that H is a cocommutative coalgebra with respect to the braiding given in , but it is cocommutative with respect to a different braiding subject to specific axioms. The type of cocommutative Hopf algebras under consideration (investigated in previous papers) occurs naturally, for example in Majid''s transmutation construction. We show that for cocommutative H the suitably defined cocommutative bi-Galois objects form a subgroup in the group of H-H-bi-Galois objects. We also show that all cocycles on H are lazy, and that second (lazy) cohomology describes the subgroup of cleft bi-Galois extensions in the group of cocommutative ones. [Copyright &y& Elsevier]

Abstract: We give an explicit formula for the correspondence between simple Yetter–Drinfeld modules for certain finite-dimensional pointed Hopf algebras H and those for cocycle twists of H. This implies an equivalence between modules for their Drinfeld doubles. To illustrate our results, we consider the restricted two-parameter quantum groups under conditions on the parameters guaranteeing that is a Drinfeld double of its Borel subalgebra. We determine explicit correspondences between -modules for different values of r and s and provide examples where no such correspondence can exist. Our examples were obtained via the computer algebra system Singular::Plural. [Copyright &y& Elsevier]

Abstract: We consider a class of quasiHopf algebras which we call generalized twisted quantum doubles. They are abelian extensions (G is a finite group, a homomorphic image, and * denotes the dual algebra), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if G is a subgroup of then H exhibits an orbifold McKay Correspondence: certain fusion rules of H define a graph with connected components indexed by conjugacy classes of , each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of G stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when . [Copyright &y& Elsevier]

Abstract: On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac–Moody algebras. [Copyright &y& Elsevier]

Abstract: We construct a family of automorphisms of Mickelsson algebra, satisfying braid group relations. The construction uses ‘Zhelobenko cocycle’ and includes the dynamical Weyl group action as a particular case. [Copyright &y& Elsevier]

Abstract: We study some problems related to lazy 2-cocycles, such as: extension of (lazy) 2-cocycles to a Drinfeld double and to a Radford biproduct, Yetter–Drinfeld data obtained from lazy 2-cocycles, lifting of projective representations afforded by lazy 2-cocycles. [Copyright &y& Elsevier]

A new approach to the computation of the cohomology of groups acting on Buekenhout geometries is described. It is shown that in many cases, including BN-pairs and groups acting on dimensional linear spaces, to compute the kth cohomology group Hk(G,M) it is sufficient, roughly speaking, to consider only parabolic subgroups of rank ⩽k+1. As an application it is shown that Hk(An,Fp)=0 for k⩾1, p>k+2 (where An is the alternating group). [Copyright &y& Elsevier]