${\mathcal U}_q(su(n,n))$的广义韦尔马模之间的特殊同态类
Special classes of homomorphisms between generalized Verma modules for ${\mathcal U}_q(su(n,n))$
摘要：
研究了${\mathcal U}_q(su(n,n))$的量化广义韦尔马模$M(V_Λ)\stackrel{ϕ_{Λ,Λ_1}}{\rightarrow}M(V_{Λ_1})$之间的同态.对于这样的映射，有一个自然的度概念，如果映射的度是$k$，我们写$ϕ^k_{Λ,Λ_1}$。我们检查当一个可以有一系列这样的同态$ϕ^1_{Λ_{n-1},Λ_{n}} \circ ϕ^1_{Λ_{n-2}, Λ_{n-1}} \circ\cdots\circ ϕ^1_{Λ,Λ_1} = \textrm{Det}_q$，其中$\textrm{Det}_q$表示的地图$M(V_Λ)\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{Λ_n})$。如果，经典地，$su(n,n)^{\mathbb C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus {\mathbb C})\oplus {\mathfrak p}^+$，那么$Λ= (Λ_L,Λ_R,λ)$和$Λ_n =(Λ_L,Λ_R,λ+2)$。答案是$Λ$必须是片面的，无论是$Λ_L=0$还是$Λ_R=0$（非排他性）。如果我们坚持${\mathcal U}_q({\mathfrak g}^{\mathbb C})$同态，则对$λ$有进一步的要求。然而，也有意思的是放松这一点，只考虑${\mathcal U}^-_q({\mathfrak g}^{\mathbb C})$同态，在这种情况下，$λ$上的条件消失了。利用对偶性，这些结果对协变量子化微分算子有一定的意义。最后，我们给出了一个明确的，虽然勾画，确定的全套${\mathcal U}_q({\mathfrak g}^{\mathbb C})$同态$ϕ^1_{Λ,Λ_1}$。
英文摘要：
We study homomorphisms between quantized generalized Verma modules $M(V_Λ)\stackrel{ϕ_{Λ,Λ_1}}{\rightarrow}M(V_{Λ_1})$ 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 $ϕ^k_{Λ,Λ_1}$. We examine when one can have a series of such homomorphisms $ϕ^1_{Λ_{n-1},Λ_{n}} \circ ϕ^1_{Λ_{n-2}, Λ_{n-1}} \circ\cdots\circ ϕ^1_{Λ,Λ_1} = \textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_Λ)\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{Λ_n})$. If, classically, $su(n,n)^{\mathbb C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus {\mathbb C})\oplus {\mathfrak p}^+$, then $Λ= (Λ_L,Λ_R,λ)$ and $Λ_n =(Λ_L,Λ_R,λ+2)$. The answer is then that $Λ$ must be one-sided in the sense that either $Λ_L=0$ or $Λ_R=0$ (non-exclusively). There are further demands on $λ$ 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 $λ$ 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 $ϕ^1_{Λ,Λ_1}$.
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