Equivariant multiplicities of simply-laced type flag minors

Let g \mathfrak {g} be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D ¯ \overline {D} on the coordinate ring C [ N ] \mathbb {C}[N] related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram’s strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g \mathfrak {g} is of type A n A_n or D 4 D_4 , the map D ¯ \overline {D} takes similar distinguished values on the set of all flag minors of C [ N ] \mathbb {C}[N] , raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D ¯ \overline {D} on flag minors belonging to the same standard seed, and we show that in any A D E ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh’s monoidal categorification of the cluster structure of C [ N ] \mathbb {C}[N] via representations of quiver Hecke algebras.

Algebras with representable representations

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ of derivations on $X$ . In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field ${\mathbb {K}}$ , in such a way that these generalized derivations characterize the ${\mathbb {K}}$ -algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$ .

On Well-Founded and Recursive Coalgebras

This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor’s General Recursion Theorem that every well-founded coalgebra is recursive, and we study conditions which imply the converse. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.

Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in \mathbb{Q}[X] . We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results, where polynomials are replaced by packed words. We obtain, in particular, a non-commutative duality principle.

More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

For two Banach algebrasAandB, theT-Lau productA×TB, was recently introduced and studied for some bounded homomorphismT:B→Awith ∥T∥ ≤ 1. Here, we give general nessesary and sufficent conditions forA×TBto be (approximately) cyclic amenable. In particular, we extend some recent results on (approximate) cyclic amenability of direct productA⊕BandT-Lau productA×TBand answer a question on cyclic amenability ofA×TB.

A NOTE ON CYCLIC AMENABILITY OF THE LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

Let $T$ be a Banach algebra homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$ with $\Vert T\Vert \leq 1$. Recently, Bhatt and Dabhi [‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc.87 (2013), 195–206] showed that cyclic amenability of ${\mathcal{A}}\times _{T}{\mathcal{B}}$ is stable with respect to $T$, for the case where ${\mathcal{A}}$ is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra ${\mathcal{A}}$.

BIPROJECTIVITY AND BIFLATNESS OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

Let ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

On the conjectural Leibniz cohomology for groups

This article presents results which are consistent with conjectures about Leibniz (co)homology for discrete groups, due to J. L. Loday in 2003. We prove that rack cohomology has properties very close to the properties expected for the conjectural Leibniz cohomology. In particular, we prove the existence of a graded dendriform algebra structure on rack cohomology, and we construct a graded associative algebra morphism H•(−) → HR•(−) from group cohomology to rack cohomology which is injective for ● = 1.

ARENS REGULARITY AND AMENABILITY OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

Given a morphism T from a Banach algebra ℬ to a commutative Banach algebra 𝒜, a multiplication is defined on the Cartesian product space 𝒜×ℬ perturbing the coordinatewise product resulting in a new Banach algebra 𝒜×Tℬ. The Arens regularity as well as amenability (together with its various avatars) of 𝒜×Tℬ are shown to be stable with respect to T.