Related papers: Nilpotency indices for quantum Schubert cell algeb…
n this paper we study ad-nilpotent ideals of a complex simple Lie algebra $\ccg$ and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer and…
Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a $\mathbb{Z}$-graded algebra isomorphism…
In this paper we study ad-nilpotent ideals of a complex simple Lie algebra $\ccg$ and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer…
For the Drinfeld-Jimbo quantum enveloping algebra $U_q(\frak{sl}_{n+1})$, we show that the span of Lusztig's positive root vectors, with respect to Littlemann's nice reduced decompositions of the longest element of the Weyl group, form…
Automorphisms of the quantum Schubert cell algebras ${\mathcal U}_q^\pm[w]$ of De Concini, Kac, Procesi and Lusztig and their restrictions to some key invariant subalgebras are studied. We develop some general rigidity results and apply…
Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of…
In this note we study simple modules for a reduced enveloping algebra U_chi(g) in the critical case when chi element of g^* is ``nilpotent''. Some dimension formulas computed by Jantzen suggest modified versions of Weyl's dimension formula,…
A regular nilpotent Hessenberg Schubert cell is the intersection of a regular nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe a set of minimal generators of the defining ideal of a regular nilpotent Hessenberg…
A well-developed classification program for 4d $\mathcal{N}=2$ super conformal field theories (SCFTs) leverages Seiberg-Witten geometry on the Coulomb branch of vacua; theories are arranged by increasing $\mathfrak{rank}$, the complex…
Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that a left Leibniz algebra, all of whose maximal subalgebras are right ideals, is nilpotent. A…
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link…
Let $U_S$ be the localization of $U(\mathfrak{sp}_{2n})$ with respect to the Ore subset $S$ generated by the root vectors $X_{\epsilon_1-\epsilon_2},\dots,X_{\epsilon_1-\epsilon_n}, X_{2\epsilon_1}$. We show that the minimal nilpotent…
Let \H_r be the generic type A Hecke algebra defined over \ZZ[u, u^{-1}]. The Kazhdan-Lusztig bases \{C_w\}_{w \in \S_r} and \{C'_w\}_{w \in \S_r} of \H_r give rise to two different bases of the Specht module M_\lambda, \lambda \vdash r, of…
De Concini, Kac, and Procesi defined a family of subalgebras Uq[w] of the quantized enveloping algebra Uq(g) associated to elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We…
In the present paper, we introduce and study counterparts of Rickart involutive algebras, i.e., almost inner Rickart algebras. We prove that a nilpotent associative algebra, which has no nilpotent elements with nonzero square roots, is an…
Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive…
Let $P_J$ be the standard parabolic subgroup of $SL_n$ obtained by deleting a subset $J$ of negative simple roots, and let $P_J = L_JU_J$ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue…
In this paper, a nilpotency criterion is given for finite dimensional alternative superalgebras in the spirit of Engel's Theorem for Jordan superalgebras over infinite fields provided by Shestakov and Okunev. For alternative superalgebras,…
Let $A$ be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to generalize the results proven in CGS. Precisely, we determine which vertices of $Q_A$ are sufficient to be…
Let A be an associative algebra of arbitrary dimension over a field F and G a finite soluble group of automorphisms of A oforder n, prime to the characteristic of F. We prove that if the fixed-point subalgebra of A under the action of G…