An official website of the United States government
Abstract Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $${{\mathcal {V}}}$$, we have a surjective homomorphism of differential algebras $$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$$; equivalently, the singular support of $${{\mathcal {V}}}$$ is a closed subscheme of the arc space of the associated scheme $$X_{{{\mathcal {V}}}}$$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$$ for all positive integers $$n$$ and $$k$$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular $${{\mathcal {W}}}$$-algebra of $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.
more »
« less
This content will become publicly available on July 31, 2026
Two-Parameter Quantum Groups and $R$-Matrices: Classical Types
We construct finite $$R$$-matrices for the first fundamental representation $$V$$ of two-parameter quantum groups $$U_{r,s}(\mathfrak{g})$$ for classical $$\mathfrak{g}$$, both through the decomposition of $$V\otimes V$$ into irreducibles $$U_{r,s}(\mathfrak{g})$$-submodules as well as by evaluating the universal $$R$$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $$U^{\pm}_{r,s}(\mathfrak{g})$$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $$R$$-matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $$U_{r,s}(\widehat{\mathfrak{g}})$$ for classical $$\mathfrak{g}$$. The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras.
more »
« less
- Award ID(s):
- 2302661
- PAR ID:
- 10628583
- Publisher / Repository:
- SIGMA
- Date Published:
- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications
- ISSN:
- 1815-0659
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.more » « less
-
Webs give a diagrammatic calculus for spaces of $$U_{q}(\mathfrak{sl}_{r})$$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $$U_{q}(\mathfrak{sl}_{4})$$-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary $$U_{q}(\mathfrak{sl}_{r})$$. Here, we show that Fraser’s basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant $$U_{q}(\mathfrak{sl}_{r})$$-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general $$U_{q}(\mathfrak{sl}_{r})$$-web bases.more » « less
-
Abstract The 4 N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 N {4N} -carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ( u , v ) = ∫ F N ∇ u ⋅ ∇ v d x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ( u n , u n ) ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F .more » « less
-
The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $$E_{8}$$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $$U_{\zeta}({{\mathfrak g}})$$ where $${\mathfrak g}$$ is a complex simple Lie algebra and $$\zeta$$ ranges over roots of unity.more » « less
