# Special classes of homomorphisms between generalized Verma modules for ${\mathcal U}_q(su(n,n))$

## TL;DR

This paper investigates special homomorphisms between quantized generalized Verma modules for ${
m U}_q(su(n,n))$, characterizing when compositions produce the quantum determinant and exploring conditions on parameters for such maps.

## Contribution

It provides a classification of homomorphisms between quantum Verma modules, especially those composing to the quantum determinant, and analyzes parameter conditions for these maps.

## Key findings

- Homomorphisms of degree 1 can compose to form the quantum determinant.
- Parameter conditions depend on whether maps are ${
m U}_q({
m g}^{f C})$-homomorphisms or only ${
m U}_q^-({
m g}^{f C})$-homomorphisms.
- Explicit characterization of the full set of such homomorphisms is provided.

## Abstract

We study homomorphisms between quantized generalized Verma modules $M(V_{\Lambda})\stackrel{\phi_{\Lambda,\Lambda_1}}{\rightarrow}M(V_{\Lambda_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $\phi^k_{\Lambda,\Lambda_1}$. We examine when one can have a series of such homomorphisms $\phi^1_{\Lambda_{n-1},\Lambda_{n}} \circ \phi^1_{\Lambda_{n-2}, \Lambda_{n-1}} \circ\cdots\circ \phi^1_{\Lambda,\Lambda_1} = \textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_{\Lambda})\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{\Lambda_n})$. If, classically, $su(n,n)^{\mathbb C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus {\mathbb C})\oplus {\mathfrak p}^+$, then $\Lambda = (\Lambda_L,\Lambda_R,\lambda)$ and $\Lambda_n =(\Lambda_L,\Lambda_R,\lambda+2)$. The answer is then that $\Lambda$ must be one-sided in the sense that either $\Lambda_L=0$ or $\Lambda_R=0$ (non-exclusively). There are further demands on $\lambda$ if we insist on ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${\mathcal U}^-_q({\mathfrak g}^{\mathbb C})$ homomorphisms, in which case the conditions on $\lambda$ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms $\phi^1_{\Lambda,\Lambda_1}$.