The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie}{G}_2\stackrel{i}{\hookrightarrow }so\left(7\right)$, and generalized conformal $so\left(7\right)$-Verma modules of scalar type. As a result, we classify the $i\left(\mathrm{Lie}{G}_2\right)\cap 𝔭$-singular vectors for this class of $so\left(7\right)$-modules.

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers–Tymoczko. In particular, when an ideal subarrangement is equal to the entireWeyl arrangement, our...

We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category $̃\left({ℳ}_k\right)$. More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke’s theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.

A list of known quantum spheres of dimension one, two and three is presented.

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants $A^G$ is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...

We characterize an important class of generalized projective geometries $(X,X^{\text{'}})$ by the following essentially equivalent properties: (1) $(X,X^{\text{'}})$ admits a central null-system; (2) $(X,X^{\text{'}})$ admits inner polarities: (3) $(X,X^{\text{'}})$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s...

Let $𝒩=N_n\left(R\right)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi$ on $𝒩$ is called preserving commutativity in both directions if $xy=yx\iff \varphi \left(x\right)\varphi \left(y\right)=\varphi \left(y\right)\varphi \left(x\right)$. In this paper, we prove that each invertible linear map on $𝒩$ preserving commutativity in both directions is exactly a quasi-automorphism of $𝒩$, and a quasi-automorphism of $𝒩$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.

Let $L_n=K[x_1^{\pm 1},...,x_n^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_n:={\mathrm{Der}}_K\left(L_n\right)$ the Lie algebra of $K$-derivations of the algebra $L_n$, the so-called Witt Lie algebra, and let $\mathrm{Vir}$ be the Virasoro Lie algebra which is a $1$-dimensional central extension of the Witt Lie algebra. The Lie algebras $W_n$ and $\mathrm{Vir}$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${\mathrm{Aut}}_{\mathrm{Lie}}\left(\mathrm{Vir}\right)\simeq {\mathrm{Aut}}_{\mathrm{Lie}}\left(W_1\right)\simeq \{\pm 1\}⋉K^{*}$, and give a short proof that ${\mathrm{Aut}}_{\mathrm{Lie}}\left(W_n\right)\simeq {\mathrm{Aut}}_{\mathrm{K}-\mathrm{alg}}\left(L_n\right)\simeq {\mathrm{GL}}_n\left(\mathbb{Z}\right)⋉K^{*n}$.

Let $G/K$ a noncompact symmetric space with Iwasawa decomposition $KAN$. The Harish-Chandra homomorphism is an explicit homomorphism between the algebra of invariant differential operators on $G/K$ and the algebra of polynomials on $A$ that are invariant under the Weyl group action of the pair $(G,A)$. The main result of this paper is a generalization to the quantum setting of the Harish-Chandra homomorphism in the case of $G/K$ being an hermitian (classical) symmetric space