A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions

Abstract

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U_q^{+}(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U_q^{+}(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for $C_2$. Our proof is based on a realization of $U_q^{+}(\mathfrak{g})$ in a quotient ring of $A_q(\mathfrak{g})$.

References in corpus (1)

• Tetrahedron equations and nilpotent subalgebras of U_q(sl_n)

Cited by in corpus (8)

• Tetrahedron equation and quantum R matrices for $q$-oscillator representations of $U_q(A_{2n}^{(2)}),U_q(C_n^{(1)})$ and $U_q(D_{n+1}^{(2)})$
• Integrable 3D lattice model in M-theory
• Tetrahedron equation and quantum R matrices for infinite dimensional modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2)
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps
• Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras
• I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type
• Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras
• Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases