Related papers: Description of $\mathsf{B}(\infty)$ through Kashiw…

In this paper, we consider polyhedral realizations for crystal bases $B(\lambda)$ of irreducible integrable highest weight modules of a quantized enveloping algebra $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a classical affine Lie algebra…

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible…

Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $\Gamma(\Lambda^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdot\Gamma(A)}$, where $R=C^\infty(M)$,…

We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum…

We generalize some results concerning affine algebras at the critical level to the corresponding quantum algebras. In particular, we show that the Wakimoto realization provides a homomorphism of Poisson algebras from the center of a quantum…

We give several generalizations of the Kodaira vanishing and embedding theorems for K\"ahler manifolds to the case where the relevent line bundle has a small region of negative curvature. To prove the vanishing theorems we adapt techniques…

Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in \cite{LZ:2009} and one can use them to define some ideals of the modified quantized…

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four dimensional base space. We focus on the base geometry which consists of intersecting F0=CP1xCP1…

We show that if a countably generated Lie algebra $H$ does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras $A$ and $B$ (satisfying some mild conditions), then $H$ embeds into a quotient of $A \ast B$ that…

Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We reduce the conjectured triviality of the reduced Whitehead group SK_1(QG) of the algebra…

Let $G$ be a connected Lie group, with Lie algebra $g$. In 1977, Duflo constructed a homomorphism of $g$-modules $Duf: S(g) -> U(g)$, which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture…

We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell-Hausdorff series in the form…

We give (conjectural) analogs of Berenstein-Zelevinsky data for affine type $A$. Moreover, by using these affine analogs of Berenstein-Zelevinsky data, we realize the crystal basis of the negative part of the quantized universal enveloping…

We develop a theory of bicrystalline ideals, synthesizing Gr\"obner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and…

We provide the unique affine crystal structure for type E_6^{(1)} Kirillov-Reshetikhin crystals corresponding to the multiples of fundamental weights s Lambda_1, s Lambda_2, and s Lambda_6 for all s \geq 1 (in Bourbaki's labeling of the…

Let $\ell\in\mathbb{N}$ with $\ell>2$ and $I:=\mathbb{Z}/2\ell\mathbb{Z}$. In this paper we give a new realization of the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ using the modular representation theory of the affine Hecke…

In the preceding paper, we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\gl_\infty$. In the present paper, we prove the…

Let Q be an affine quiver of type A. Let C be the associated generalized Cartan matrix. Let U^- be the negative part of the quantized enveloping algebra attached to C. In terms of perverse sheaves on the moduli space of representations of a…

The presentation of two-parameter quantum groups of type E-series in the sense of Benkart-Witherspoon [BW1] is given, which has a Drinfel'd quantum double structure. The universal $R$-matrix and a convex PBW-type basis are described for…

Based on empirical evidence obtained using the {\sf CHEVIE} computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan--Lusztig cells for type $B_n$ with unequal parameters. These…