arXiv subjects
Publications and source records attributed to Hengyun Yang.
We study local derivations on a Block-type Lie algebra \(\mathcal B\). Using the structure of the derivation algebra of \(\mathcal B\) and a series of normalizations and coefficient comparisons, we prove that every local derivation on \(\mathcal B\) is a derivation.
We construct central elements of the degenerate quantum general linear group introduced by Cheng, Wang and Zhang. In particular, we give an explicit formula for the quantum Casimir element. Our method is based on the explicit $L$ operators. Moreover, we construct a universal $L$ operator, which is a spectral parameter-dependent solution of the quantum Yang-Baxter equation in the tensor product of the degenerate quantum general linear group and the endomorphism ring of its natural representation. This construction leads to the FRT approach to the degenerate quantum general linear group.
Let $\mathcal{G}$ be the planar Galilean conformal algebra and $\widetilde{\mathcal{G}}$ be its universal central extension. Then $\mathcal{G}$ (resp. $\widetilde{\mathcal{G}}$) admits a triangular decomposition: $\mathcal{G}=\mathcal{G}^{+}\oplus\mathcal{G}^{0}\oplus\mathcal{G}^{-}$ (resp. $\widetilde{\mathcal{G}}=\widetilde{\mathcal{G}}^{+}\oplus\widetilde{\mathcal{G}}^{0}\oplus\widetilde{\mathcal{G}}^{-}$). In this paper, we study universal and generic Whittaker $\mathcal{G}$-modules (resp. $\widetilde{\mathcal{G}}$-modules) of type $ϕ$, where $ϕ:\mathcal{G}^{+}=\widetilde{\mathcal{G}}^{+}\longrightarrow\mathbb{C}$ is a Lie algebra homomorphism. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker modules of type $ϕ$ is irreducible if and only if $ϕ$ is nonsingular. For the nonsingular case, we completely determine the Whittaker vectors in universal and generic Whittaker modules. For the singular case, we concretely construct some proper submodules of generic Whittaker modules.
In this paper, we construct a family of non-weight modules over the untwisted $N=2$ superconformal algebras. Those modules when regarded as modules over the Cartan subalgebra (modulo the center) are free of rank $2$. We give a classification of isomorphism classes of such modules. Moreover, all submodules of such modules are precisely determined. In particular, those modules are not simple. The corresponding simple quotient modules are classified. Furthermore, these simple modules when restricted as modules over $N=1$ superconformal algebras coincide with those modules constructed in [H. Yang, Y. Yao, L. Xia, A family of non-weight modules over the super-Virasoro algebras, J. Algebra 547 (2020), 538-555].
In this paper, we construct a family of non-weight modules over the super-Virasoro algebras. Those modules when regarded as modules of the Ramond algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{h}$ are free of rank $1$, while when regarded as modules of the Neveu-Schwarz algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{H}$ are free of rank $2$. We obtain a sufficient and necessary condition for such modules to be simple. Moreover, we determine the isomorphism classes of these modules. Finally, we show that these modules constitute a complete classification of free $U(\mathfrak{h})$-modules of rank $1$ over the super-Virasoro algebra of Ramond type, and also constitute a complete classification of free $U(\mathfrak{H})$-modules of rank $2$ over the super-Virasoro algebra of Neveu-Schwarz type.
The quantum supergroup ${\rm{U}}_q({\mathfrak {osp}}(1|2n))$ admits a finite dimensional spinor representation, which does not have a classical limit. We construct a realisation of this representation on the Fock space of $q$-fermions. We also generalise the construction to the infinite dimensional spinor representations of ${\rm{U}}_q({\mathfrak {osp}}(2m+1|2n))$ for $m, n\ge 1$ by using both quantum bosons and fermions. This leads to a new realisation different from those obtained before by other researchers.
In this paper, we describe nonstandard quantum deformation of the super-Virasoro algebra. Using the Drinfel'd twist quantization technique, we obtain the deformed coproduct and antipode. Hence we get a family of noncommutative and noncocommutative Hopf superalgebras.