Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUniqueness of exact Borel subalgebras and bocses
65
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Together with Koenig and Ovsienko, the first author showed that every quasi-hereditary algebra can be obtained as the (left or right) dual of a directed bocs. In this monograph, we prove that if one additionally assumes that the bocs is basic, a notion we define, then this bocs is unique up to isomorphism. This should be seen as a generalisation of the statement that the basic algebra of an arbitrary associative algebra is unique up to isomorphism. The proof associates to a given presentation of the bocs an $A_infty$-structure on the $operatorname{Ext}$-algebra of the standard modules of the corresponding quasi-hereditary algebra. Uniqueness then follows from an application of Kadeishvilis theorem.
rate research
Read More
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $mathbbm k$ of characteristic zero. We consider the commuting variety $mathcal C(mathfrak u)$ of the nilradical $mathfrak u$ of the Lie algebra $mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $mathfrak u$ with only a finite number of orbits, we verify that $mathcal C(mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {em distinguished} $B$-orbits in $mathfrak u$. We observe that in general $mathcal C(mathfrak u)$ is not equidimensional, and determine the irreducible components of $mathcal C(mathfrak u)$ in the minimal cases where there are infinitely many $B$-orbits in $mathfrak u$.
A well-known Petersons theorem says that the number of abelian ideals in a Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly $2^r$. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which is a refinement of the Petersons theorem capturing more Lie algebra invariants.
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $mathcal C(mathfrak p) = {(X,Y) in mathfrak p times mathfrak p mid [X,Y] = 0}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $mathcal C(mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $mathfrak p$. As a consequence, for the case $P = B$ a Borel subgroup of $G$, we give a classification of when $mathcal C(mathfrak b)$ is irreducible; this builds on a partial classification given by Keeton. Further, in cases where $mathcal C(mathfrak p)$ is irreducible, we consider whether $mathcal C(mathfrak p)$ is a normal variety. In particular, this leads to a classification of when $mathcal C(mathfrak b)$ is normal.
We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.
The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $mathcal{J}$ of a fiat $2$-category $cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $cC$ associated with a diagonal $mathcal{H}$-cell in $mathcal{J}$. As an application, we classify simple transitive $2$-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_3$ and $B_4$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
