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.

Classification of Simple Lie Superalgebras in Characteristic 2

All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.

Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

Cohomology of Lie Conformal Algebra Vir⋉Cur g

In this article, we compute cohomology groups of the semisimple Lie conformal algebra [Formula: see text] with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra [Formula: see text].

Centers and Azumaya loci for finite W-algebras in positive characteristic

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS

We prove that all classical affine W-algebras 𝒲(𝔤; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.

Uprolling unrolled quantum groups

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

Levi Factors and Admissible Automorphisms

Let $\mathfrak {g}$ g be a complex simple Lie algebra. We consider subalgebras $\mathfrak {m}$ m which are Levi factors of parabolic subalgebras of $\mathfrak {g}$ g , or equivalently $\mathfrak {m}$ m is the centralizer of its center. We introduced the notion of admissible systems on finite order $\mathfrak {g}$ g -automorphisms 𝜃, and show that 𝜃 has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize such automorphisms, and look for automorphisms of minimal order.

Yang-Yang functions, monodromy and knot polynomials

We derive a structure of ℤ[t, t−1]-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the ℤ[t, t−1]-module bundle is equivalent to the braid group representation induced by the universal R-matrices of Uh(g). We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.