Related papers: On Borel subalgebras of quantum groups
Let $U_q(\hat{sl}_2)^{\geq 0}$ denote the Borel subalgebra of the quantum affine algebra $U_q(\hat{sl}_2)$. We show that the following hold for any choice of scalars $\epsilon_0, \epsilon_1$ from the set ${1,-1}$. (i) Let $V$ be a…
We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements…
We define two subalgebras which can be seen as the quantization of the coordinate rings of the unipotent radical of the standard positive (respectively negative) Borel subgroup of $SL_{n+1}$. We give a presentation for these algebras and…
Given an algebra $A$ and an $A-A$-bimodule $U$ with co-algebra structure, a bocs, the algebras of endomorphisms of $A$ as left or right module of the bocs are known as Burt-Butler algebras (up to an appropriate opposite). Here we give a…
Given a state on an algebra of bounded quantum-mechanical observables (the self-adjoint part of a C*-algebra), we investigate those subalgebras that are maximal with respect to the property that the given state's restriction to the…
We study the structure of an algebraic supergroup $\mathbb{G}$ and establish the Borel-Weil theorem for $\mathbb{G}$ to give a systematic construction of all simple supermodules over an arbitrary field. Especially when $\mathbb{G}$ has a…
We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.
This paper generalizes the classification in a paper of Dimitrov and Penkov of Borel subalgebras of gl_infty. Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras along inclusions preserving the root…
Let $H$ and $K$ be the bosonizations of the Jordan and super Jordan plane by the group algebra of a cyclic group; the algebra $K$ projects onto an algebra $L$ that can be thought of as the quantum Borel of $\mathfrak{sl}(2)$ at $-1$. The…
We formulate several basic properties of Verma supermodules over regular symmetrizable Kac--Moody Lie superalgebras, exhibiting $\mathfrak{gl}(1|1)$-nature as revealed through changing Borel subalgebras. We investigate variants of Verma…
Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half…
We study a class of algebras B(n,l) associated with integrable models with boundaries. These algebras can be identified with coideal subalgebras in the Yangian for gl(n). We construct an analog of the quantum determinant and show that its…
The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.
The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove…
Green and Schroll give an easy criterion for a monomial algebra $A$ to be quasi-hereditary with respect to some partial order $\leq_A$. A natural follow-up question is under which conditions a monomial quasi-hereditary algebra $(A, \leq_A)$…
This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundation for this study. For any vector space $W$,…
We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the…
In the present paper we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive chararacteristic called quasi-Verma modules. We provide a purely algebraic construction of…
The Verma modules over the quantum groups $\mathrm U_q(\mathfrak{gl}_{l + 1})$ for arbitrary values of $l$ are analysed. The explicit expressions for the action of the generators on the elements of the natural basis are obtained. The…
In [I. Arzhantsev and M. Zaidenberg, Borel subgroups of the automorphism groups of affine toric surfaces, arXiv:2507.09679 (2025)] we described the Borel subgroups and maximal solvable subgroups of the automorphism groups of affine toric…