In the last section, we deal with the Sridharan enveloping algebras of ﬁnite dimensional Lie al- gebras. In general, a Sridharan enveloping algebra is no longer a Hopf algebra. However, a Sridharan enveloping algebra is a cocycle deformation of a cocommutativeHopfalgebras, and the CY property of the Sridharan enveloping algebras is closely related to that of the universal enveloping algebras. Hence it is proper to include the discussion of Sridharan enveloping algebras in this paper. Sridharan enveloping algebras were introduced in  in order to discuss certain representations of Lie alge- bras. Let g be a ﬁnite dimensional Lie algebra. A Sridharan enveloping algebra is related to a 2-cocycle f ∈ Z 2 (g, k) of g , and is usually denoted by U f (g) (the deﬁnition is recalled in the ﬁnal section). The class of Sridharan enveloping algebras includes many interesting algebras, such as Weyl algebras. Ho- mological properties, especially the Hochschild (co)homology and cyclic homology, are studied by several authors [24,11,18,19]. Berger proposed at the end of his recent paper  a question: to ﬁnd some necessary and suﬃcient conditions for a Sridharan enveloping algebra to be CY. We get the following result (Theorem 5.3) which in part answers Berger’s question .
The structure of the contribution goes as follows. Section 2 is devoted to introducing the main aspects of LH systems and Poisson–Hopfalgebras. The general approach to construct Poisson–Hopf algebra deformations of LH systems  is summarized in Section 3. For our further purposes, the (non- standard) Poisson–Hopf algebra deformation of sl(2) is recalled in Section 4. The novel unifying approach to deformations of Poisson–Hopf Lie systems with a LH algebra isomorphic to a fixed Lie algebra g are treated in Section 5. Such a procedure is explicitly illustrated in Section 6 by applying it to the three non-diffeomorphic classes of sl(2)-LH systems on the plane, so obtain- ing in a straightforward way their corresponding deformation. Next, a new method to construct (non-deformed) LH systems is presented in Section 6. Finally, our results are summarised and the future work to be accomplished is briefly detailed in the last Section.
R#H is an ordinary Hopf algebra with a projection to H , see . This fact plays a crucial role in the classification of finite-dimensional pointed Hopfalgebras  and in the description of quantum groups arising from deformation of enveloping algebras of semisimple Lie algebras, in particular their quantum Borel subalgebras. Let A be a Hopf algebra with bijective antipode such that its coradical A 0 is a
are Frobenius algebrasand bialgebras whose equations witness an interaction between a commutative monoid andcocommutative comonoid.
Lack  showed that several such situations can be understood as arising from PROP composition where a distributive law —a notion closely related to standard distributive laws between monads —witnesses the interaction. The beauty of this approach is that one can consider distributive laws to be responsible for the newly introduced equations, resulting in a pleasantly mod- ular account of the composite algebraic theory. For example, the equations of (strongly separable) Frobenius algebra  can be obtained in this way. Another example is the theory of bialgebras: here monoids and comonoids interact through a different distributive law, thus yielding different equations. Our chief original contribution is the study of the interaction of the PROP HA R of Hopfalgebras, parametrised over a principal ideal domain R, and its opposite HA op R . As in the case of the PROP of commutative monoids and its opposite, two different distributive laws can be defined, yielding IH Sp
An important question to be addressed is whether this approach can provide an effective procedure to derive a deformed analogue of superposition principles for deformed LH systems. Also, it would be interesting to know whether such a description is simultaneously applicable to the various non-equivalent deformations, like an extrapolation of the notion of Lie algebra contraction to Lie systems. Another open problem worthy to be considered is the possibility of getting a unified description of such systems in terms of a certain amount of fixed ‘elementary’ systems, thus implying a first rough systematization of LH-related systems from a more general perspective than that of finite-dimensional Lie algebras. Work in these directions is currently in progress.
a group algebra by a 2-pseudo-cocycle twist but not by a 2-cocycle twist . Here we would like to initiate a project for the study of Hopfalgebras whose coradicals are low-dimensional neither commutative nor cocommutative semisim- ple Hopfalgebras by running procedures of the lifting method. One important step is to study the Nichols algebras over those low-dimensional semisimple Hopfalgebras. Nichols algebras were studied first by Nichols . These are connected graded braided Hopfalgebras  generated by primitive elements, and all primi- tive elements are of degree one. In the past decades, the study of Nichols algebras was mainly focused on categories of Yetter–Drinfel’d modules over group algebras. Under the assumption that the base field has characteristic 0, the classification of finite-dimensional Nichols algebras over abelian groups was completely solved in [30, 31] by using Lie theoretic structures, and the result of the classification played an important role later in the significant work . The problem of classifying finite-dimensional Nichols algebras over non-abelian groups is difficult in general for lack of systematic method; for related works please refer to , , , , , , , etc.
In the above settings, the algebras considered are both commutative andcocommutative. However more general Hopfalgebras, perhaps not even symmetric, are a ubiquitous structure in mathematical physics, finding applications in gauge theory , topological quantum field theory  and topological quantum computing . In this paper we take the first steps towards generalising the concept of Hopf-Frobenius algebra to the non-commutative case, and opening the door to applications of categorical quantum theory in other areas of physics.
To explain our other results, recall that an important class of Hopf actions is the class of Hopf-Galois actions, i.e., those defining a Hopf-Galois extension. Question 3.5 of [CEW] asks if an inner faithful semisimple Hopf action on a (central) division algebra must be Hopf-Galois. Our second result, given in Section 3, answers it in the affirmative for finite group actions. This is a generalization of the standard result in classical Galois theory. We also give an example showing that if the ground field is not algebraically closed, then the answer is negative, even for fields. Hence, the result of [EW1] fails: there is an inner faithful action of a non-cocommutative semisimple Hopf algebra on a field that is not Hopf-Galois.
One of the main motivations for studying Hopf-(bi)Galois objects is an important result by Schauenburg  stating that the comodule categories over two Hopfalgebras H and L are monoidally equivalent if and only if there exists an H -L- bi-Galois object. The knowledge of the full cogroupoid structure (rather than “only” the bi-Galois object) might be useful to exactly determine the image of an object by the monoidal equivalence. It is also the aim of the notes to present several applications of this: construction of new explicit resolutions from old ones in homological algebra, invariant theory, monoidal equivalences between categories of Yetter-Drinfeld modules with applications to bialgebra cohomology and Brauer groups. The use of the cogroupoid structure is probably not necessary everywhere, but we believe that it can help!
Abstract. Let k be an algebraically closed field of characteristic zero. We construct several families of finite-dimensional Hopfalgebras over k with- out the dual Chevalley property via the generalized lifting method. In par- ticular, we obtain 14 families of new Hopfalgebras of dimension 128 with non-pointed duals which cover the eight families obtained in our unpublished version, arXiv:1701.01991 [math.QA].
some of these facts in Section 2. We investigate, in Section 3, modules over these algebras whose G-isotypic components are 1-dimensional and classify indecompos- able modules of this kind. We find conditions on a given G-character under which it can be extended to a representation of the algebra. We apply these results to the representation theory of two families of pointed Hopfalgebras over S n . In Sec-
The paper is organized as follows. In Section 2, we recall the definitions and ba- sic properties of semisolvability, characters and Radford’s biproducts, respectively. Some useful lemmas are also obtained in this section. In particular, we give a partial answer to Kaplansky’s conjecture. We prove that if dimH is odd and H has a simple module of dimension 3 then 3 divides dimH . This result has already appeared in [1, Corollary 8] and [10, Theorem 4.4], respectively. In the first paper, Burciu does not assume that the characteristic of the base field is zero, but adds the assumption that H has no even-dimensional simple modules. Accordingly, his proof is rather different from ours. Our proof here is also different from that in the second paper. Under the assumption that H does not have simple modules of dimension 3 or 7, we also prove that if dimH is odd and H has a simple module of dimension 5 then 5 divides dimH.
of the space of conformal blocks with chiral insertion of type V i at the ith marked point of Σ g , n .
In the particular case that the category C is ﬁnitely semisimple, the structure of a modular functor is reasonably well understood. Speciﬁcally, precise conditions are known under which the representa- tion category of a vertex algebra V is a modular tensor category. In this case the Reshetikhin–Turaev construction allows one to obtain a modular functor just on the basis of C as an abstract category. In a remarkable development, Lyubashenko and others (see [KL] and references cited there) have ex- tended many aspects of this story to a larger class of monoidal categories that are not necessarily semisimple any longer. In particular, given an abstract monoidal category with adequate additional properties, one can still construct representations of mapping class groups.
[ACh] Artamonov V.A., Chubarov I.A., Properties of some semisimple Hopfalgebras, Contemp. Math. 483, Algebras, representations and applications, A conference in honour of Ivan Shestakov’s 60th birthday, August 26–September 1, 2007, Maresias, Brazil. Edited by: Vyacheslav Futorny, Victor Kac, Iryna Kashuba and E. Zelmanov, Amer. Math. Soc., 2009, 23–36. 2, 6.1, 6.2
mined by just two values: its lower and upper bound. The aim of this work is to develop a logic that has intervals as truth values. The intended semantics are residuated lattices on the set of closed subintervals of the unit interval. We call this set the triangularization of the unit interval. A particular subset of this triangularization is its so-called diagonal, consisting of those intervals for which the lower and upper bound coincide. These intervals are called exact intervals and rep- resent truth values of propositions about which the knowledge is complete. Intuitively, the truth values of formulas constructed with these propositions should be exact intervals as well (because in these cases, the situation is similar to working with formulas in fuzzy logics). The semantics of so-called interval-valued fuzzy logics have already been examined by different authors. Especially interval-valued triangular norms, triangular conorms and implicators have received ample atten- tion. Most of these authors [1, 7, 26, 37, 44] only consider interval-valued operations that map the diagonal on the diagonal, although the most general definitions of triangular norms, triangu- lar conorms and implicators allow other operations as well [15, 21, 19, 48]. Generally speaking, interval-valued operations satisfy not as many properties as operations on the unit interval. For example, standard interval-valued residuated lattices can never satisfy prelinearity . A lot of other properties can hold though. There are even interval-valued implicators that satisfy all the Smets-Magrez axioms [17, 70].
Por ejemplo, ´ este es el caso tambi´ en del an´ alisis sobre las ´ algebras finito dimensionales de , las de intercambio en  o las ´ algebras de caminos de Leavitt puramente infinitas simples que describimos en la Secci´ on 1.3. Es interesante observar que el cuerpo subyacente K no juega ning´ un papel en estos teoremas de caracterizaci´ on. Sin embargo, cabe mencionar que por ejemplo en el trabajo  realizado por los autores G. Aranda Pino, K.M. Rangaswamy y L. Vaˇs, apareci´ o el primer teorema de caracterizaci´ on (sobre la ∗ -regularidad o regularidad con involuci´ on propia de un ´ algebra de caminos de Leavitt) que involucra adem´ as una propiedad como anillo sobre el cuerpo K, es decir, esquem´ aticamente en forma:
r such factors, allowing us to switch the order of a(z), b(w) by their locality. The terms with 0 ≤ s ≤ r have (z − u) appearing to a power of at least r, which allows us to move c(u) through a(z) while also still having (w − u) to the rth power, so that we can move c(u) through b(w). Similarly, on the right hand side, the terms with r < s ≤ 2r will vanish, and the other terms give us the same expression as on the left hand side. This establishes (3.4) and the lemma.
(5) Behrend’s interpretation of Donaldson-Thomas invariants as weighted Euler characteristics .
The credit for the development of motivic Hall algebras as a tool for studying moduli spaces of sheaves on Calabi-Yau threefolds is due jointly to Joyce and to Kontsevich and Soibelman. Joyce introduced motivic Hall algebras in a long series of papers [17, 18, 19, 20, 21, 22]. He used this framework to define generalizations of the naive Donaldson-Thomas invariants considered above, which apply to moduli stacks containing strictly semistable sheaves. He also worked out the wall-crossing formula for these invariants and proved a very deep no-poles theorem. Kontsevich and Soibelman  constructed an alternative theory which incorporates motivic vanishing cycles, and therefore applies to genuine DT invaraints and motivic versions thereof. They also produced a more conceptual statement of the wall-crossing formula. Some of their work was conjectural and is still being developed today. Joyce and Song  later showed how to directly incorporate the Behrend function into Joyce’s framework, and so obtain rigorous results on DT invariants.
Originalmente, no campo da topologia algébrica, o termo 'Hopf alge- bróide' foi usado por Douglas C. Ravenel em para descrever objetos cogrupóides na categoria de álgebras comutativas. Estes são exemplos de Hopf algebróides com estrutura de álgebra subjacente comutativa. Também em , ainda na área da topologia algébrica, encontra-se uma aplicação-exemplo de Hopf algebróide. Em , Hopf algebróides não comutativos têm sido usados, mas ainda sobre álgebras base co- mutativas, como uma ferramenta de um estudo da geometria dos feixes de brados principais com simetrias de grupóides. Além da topolo- gia algébrica, podemos citar outras áreas em que foram aplicados Hopf algebróides, como geometria de Poisson e topologia.