23 problems

Fusion products are constructed from tensor products of highest-weight cyclic modules for a current Lie algebra, using evaluation maps associated with a finite subset of…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ be its dominant integral weights, and let $a_1,ldots,a_sincmathbb C$ be pairwise distinct with…

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra, let $\lambda\in P^+$, and let $N\in\mathbb Z_{>0}$. Write $W_N(\lambda)$ for the graded truncated Weyl module for…

Let $F_{\lambda_1,\lambda_2}$ be the modules constructed in the paper from a pair of dominant integral weights $\lambda_1,\lambda_2$, and let $V(\lambda_i)_{c_i}$ denote the corres…

Fusion-product independence conjecture. The corresponding fusion product does not depend on $Z$ as a representation of $\g\otimes\mathbb{C}[u]$.

Let $\widehat{\mathfrak{g}}$ be an affine Kac–Moody algebra, let $\Lambda=m\Lambda_0+\lambda$ be a dominant integral weight, and let $V(\Lambda)$ and $V(\lambda)$ denote the corres…

Let ${\mathfrak{g}}$ be a simple Lie algebra, let $\lambda=\lambda_1+\cdots+\lambda_n$ be a sum of dominant integral weights, and let $c_1,\ldots,c_n$ be pairwise distinct complex…

Let $\lambda^1,\dots,\lambda^n$ be weights, let $W^1(\lambda^i)$ be the level-one Weyl modules, and let $W^1(\lambda^i)^{[k]}$ denote their level-$k$ constructions. Let $*$ denote…

Let ${\mathfrak g}$ be a simple Lie algebra, let $\lambda^1,\dots,\lambda^n$ be weights, let $V(\lambda^i)$ be the corresponding finite-dimensional ${\mathfrak g}$-modules, and let…

Let $A_{(m)}(n)$ be the subalgebra generated by $\xi_0,\ldots,\xi_{n-1}$ in $A_{(m)}$, and let $I_{(m)}(n)$ be its defining ideal. For a family…

Fusion-product conjecture. In general, the structure of the cohomology groups as $\slt\otimes\mathbb C[t]$-modules can be described in terms of fusion products.

Evaluation-point independence conjecture. As a $\mathfrak g[t]$-module, the fusion product $V(\lambda_1)^{a_1}\ast\cdots\ast V(\lambda_k)^{a_k}$ is independent of the points of eva…

Let $W_1,\dots,W_r$ be the modules considered in the preceding construction, let $\mathcal{A}(\lambda_1,\dots,\lambda_r)$ be the associated parameter algebra, and let…

Let $\mathfrak{g}$ be a simple Lie algebra, let $P_{\ell}$ be the set of level-$\ell$ dominant weights, let $W_{\ell}$ be the affine Weyl group, and let $V(\lambda)$ denote the irr…

Let $M_1,\dots,M_k$ be the modules considered in the construction of the global fusion product, let $R(M_1,\dots,M_k)$ denote that global fusion product, let…

Let $P^+$ be the dominant integral weights of $cmathfrak{sl}_n$, let $cmathcal{P}_{clambda}$ be the set of decompositions of $clambdain P^+$ into two dominant weights, and let…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ denote its dominant integral weights, and fix $clambda_1,clambda_2in P^+$. Put $clambda=clambda_1+clambda_2$, and…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $cmathfrak{g} cotimes cmathbb C[t]$ be its current algebra, and let $V(clambda_1)_{a_1} cast V(clambda_2)_{a_2}$ denote…

Local-product and fusion-product conjecture. The natural surjection from the local product to the fusion product is an isomorphism:

Asymptotic normality conjecture. The central string functions and the basic specialization of $\mathcal{F}_{\mu^{(N)}}$ behave asymptotically normally as $N\to\infty$.

For an SLE variant, let the space of its local martingales be regarded as a Virasoro module, and let the corresponding fusion product of boundary condition changing fields be its a…

By analogy with quantum affine algebras, consider maximal $l$-highest weight integrable representations, called Weyl modules, and fundamental representations of the quantum toroida…

Let $V_i$ be Kirillov–Reshetikhin–Chari-type modules for a simple Lie algebra current algebra, let $M_{\lambda;\{V_i\}}(q)$ be the graded multiplicity polynomial of $V(\lambda)$ in…