Activity (UTC+0)

Chairman: Shavkat Ayupov

08.00 Zerui Zhang (China)

Some results on Novikov algebras and Novikov-Poisson algebras [pdf, slides] [video, youtube]

Abstract: We first prove that a left Novikov algebra is right nilpotent if and only if it is solvable. And we show that the ideal generated by all the commutators of a Lie nilpotent Novikov algebra is nilpotent. Then a connection between Novikov algebras and differential commutative algebras will be discussed. Finally, we show that such a connection have an analogue between unital Novikov-Poisson algebras and special Novikov-Poisson admissible algebras.

09.00 Jiefeng Liu (China)

Cohomology and deformation quantization of Poisson conformal algebras [pdf. slides] [video, youtube]

Abstract: In this talk, we first recall the notion of (noncommutative) Poisson conformal algebras and give some constructions of them. Then we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.

10.00 Chengming Bai (China)

Parity duality of super r-matrices via O-operators and pre-Lie superalgebras [pdf, slides] [video, youtube]

Abstract: We interpret the homogeneous solutions of the super classical Yang-Baxter equation, also called super r-matrices, in terms of O-operators by a unified treatment. Furthermore, by a parity reversion of Lie superalgebra representations, a duality is established between the even and odd O-operators. This leads to a parity duality of the super r-matrices induced by the O-operators in semi-direct product Lie superalgebras. Therefore a pre-Lie superalgebra naturally defines an even O-operator, and hence an odd O-operator by the duality, thereby giving rise to a parity pair of super r-matrices. This is a joint work with Li Guo and Runxuan Zhang.

11.00 Break

Chairman: Yunhe Sheng

12.00 Xiangui Zhao (China)

Growth and Gelfand-Kirillov dimension of brace algebras [pdf, slides] [video, youtube]

Abstract: A brace algebra over a field is a vector space equipped with a family of linear operations satisfying certain identities. Brace algebras have strong connections with other important classes of algebras such as pre-Lie algebras, ε-bialgebras, and dendriform algebras. The Gelfand-Kirillov dimension of a (not necessarily associative) algebra is an important invariant for the study of the growth of the algebra. In this talk, we discuss the growth and possible values of the Gelfand-Kirillov dimension of brace algebras. In particular, we construct examples to show that the Bergman's gap theorem for the Gelfand-Kirillov dimension of associative algebras does not hold for brace algebras. This is joint work with Qiuhui Mo, Yu Li, and Wenchao Zhang.

13.00 Farkhod Eshmatov (Uzbekistan)

Necklace Lie algebra and derived Poisson structure [pdf, slides] [video, youtube]

Abstract: We introduce the notion of a derived Poisson structure on an associative (not necessarily commutative) algebra. Then we will discuss how Necklace Lie algebra structure can be used to construct derived Poisson bracket for some interesting classes of algebras.

14.00 Uzi Vishne (Israel)

Identities of the tensor square of the octonion algebra [video, youtube]

Abstract: We describe the nonassociative polynomial identities of minimal degree, which is 7, for the algebras O х O and M2(O), where O is the octonion algebra. After being discovered by a computer, the proofs are rather elegant. We also discuss some related open problems on varieties of nonassociative algebras.

15.00 Break

Chairman: Alberto Elduque

16.00 Daniel Fox (Spain)

Sectional nonassociativity of metrized algebras [pdf, slides] [video, youtube]

Abstract: The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to what is known as the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras.

17.00 Xingting Wang (USA)

Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation [pdf, slides] [video, youtube]

Abstract: Let H be a Hopf algebra over a field k such that H is Z-graded as an algebra. In this talk, we introduce the notion of a twisting pair for H and show that the Zhang twist of H by such a pair can be realized as a 2-cocycle twist. We use twisting pairs to describe twists of Manin’s universal quantum groups associated to quadratic algebras. Furthermore, we discuss a strategy to twist a solution to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction. If time permits, we illustrate this result for the quantized coordinate rings of GLn(k). This is joint work with Hongdi Huang, Van Nguyen, Charlotte Ure, Kent Vashaw and Padmini Veerapen.

18.00 Li Guo (USA)

Coherent categorical structures for Lie bialgebras, Manin triples, classical r-matrices and pre-Lie algebras [pdf, slides] [video, youtube]

Abstract: The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and O-operators of Lie algebras owe their importance to the close relationship among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, we introduce, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras. This is a joint work with Chengming Bai and Yunhe Sheng.

19.00 Break

Next talks will be given on

European Non-Associative Algebra Seminar

[Every Monday, 15:00 (+0 UTC) ]

Organizers

• Ivan Kaygorodov

• Jobir Adashev