Non-weight modules over the super-BMS3_3 algebra

In the present paper, a class of non-weight modules over the super-BMS$_3$ algebras $\S^{\epsilon}$ ($\epsilon=0$ or $\frac{1}{2}$) are constructed. These modules when regarded as $\S^{0}$-modules and further restricted as modules over the Cartan subalgebra $\mathfrak{h}$ are free of rank $1$, while when regarded as $\S^{\frac{1}{2}}$-modules and further restricted as modules over the Cartan subalgebra $\mathfrak{H}$ are free of rank $2$. We determine the necessary and sufficient conditions for these modules being simple, as well as determining the necessary and sufficient conditions for two $\S^{\epsilon}$-modules being isomorphic. At last, we present that these modules constitute a complete classification of free $U(\mathfrak{h})$-modules of rank $1$ over $\S^{0}$, and also constitute a complete classification of free $U(\mathfrak{H})$-modules of rank $2$ over $\S^{\frac{1}{2}}$.Comment: arXiv admin note: text overlap with arXiv:1906.07129 by other author

Transposed Poisson structures on Block Lie algebras and superalgebras

We describe transposed Poisson algebra structures on Block Lie algebras $\mathcal B(q)$ and Block Lie superalgebras $\mathcal S(q)$, where $q$ is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on $\mathcal B(q)$ are trivial whenever $q\not\in\mathbb Z$, and for each $q\in\mathbb Z$ there is only one (up to an isomorphism) non-trivial transposed Poisson structure on $\mathcal B(q)$. The superalgebra $\mathcal S(q)$ admits only trivial transposed Poisson superalgebra structures for $q\ne 0$ and two non-isomorphic non-trivial transposed Poisson superalgebra structures for $q=0$. As a consequence, new Lie algebras and superalgebras that admit non-trivial ${\rm Hom}$-Lie algebra structures are found

Non-Associative Algebraic Structures: Classification and Structure

These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agrega\c{c}\~ao em Matem\'atica e Applica\c{c}\~oes (University of Beira Interior, Covilh\~a, Portugal, 13-14/03/2023)