Related papers: Double Bruhat cells and total positivity

We prove a conjecture of A. S. Buch concerning the structure constants of the Grothendieck ring of a flag variety with respect to its basis of Schubert structure sheaves. For this, we show that the coefficients in this basis of the…

It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras.

We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties.

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $B_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb…

We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures…

We describe the structure of finite Boolean inverse monoids and apply our results to the representation theory of finite inverse semigroups. We then generalize to semisimple Boolean inverse semigroups.

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a…

Let $G$ be a connected semisimple Lie group. There are two natural duality constructions that assign to it the Langlands dual group $G^\vee$ and the Poisson-Lie dual group $G^*$. The main result of this paper is the following relation…

In this paper, we study non-abelian extensions of strict Lie 2-algebras via the cohomology theory. A non-abelian extension of a strict Lie 2-algebra $\g$ by $\frkh$ gives rise to a strict homomorphism from $\g$ to $\SOut(\frkh)$.…

Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two…

We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three…

Structures of commuting semigroups of isometries under certain additional assumptions like double commutativity or dual double commutativity are found.

By explicitly describing a cellular decomposition we determine the Borel invariant cycles that generate the Chow groups of the quotient of a reductive group by a Levi subgroup. For illustrations we consider the variety of polarizations…

In this work we study a large class of exact Lie bialgebras arising from noncommutative deformations of Poisson-Lie groups endowed with a left invariant Riemannian metric. We call these structures \emph{exact metaflat Lie bialgebras}. We…

This paper investigates cohomology and support varieties for Lie superalgebras and restricted Lie superalgebras over a field of characteristic 2. The existence of an underlying ordinary Lie algebra allows us to obtain results that are still…

We investigate a special class of Poisson--Lie T-plurality transformations of Bianchi cosmologies invariant with respect to non-semisimple Bianchi groups. For six-dimensional semi-Abelian Manin triples $\mathfrak{b}\bowtie \mathfrak{a}$…

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field…

We present a construction of noncommutative double mirrors to complete intersections in toric varieties. This construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their…

We define the totally nonnegative matroid Schubert variety $\mathcal Y_V$ of a linear subspace $V \subset \mathbb R^n$. We show that $\mathcal Y_V$ is a regular CW complex homeomorphic to a closed ball, with strata indexed by pairs of…

We study the SUSY/non-SUSY duality proposed by Aganagic et al. from Type IIA string and M-theory perspectives. We find that our brane configuration generalizes the so-called $extended$ Seiberg-Witten theory on the one hand, and provides a…