行列式理想与正则对易关系。经典或量化
Determinantal Ideals and the Canonical Commutation Relations. Classically or Quantized
摘要：
构造了$su(n,n)^{\mathbb C}$在Weyl代数${\mathcal H}_{2nr}$中的同态象.更准确地说，并使用${\mathcal H}_{2nr}$的Bernstein过滤，将$su(n,n)^{\mathbb C}$映射到度$2$元素，其中负非紧根空间被映射到二阶创建算子。利用${\mathcal H}_{2nr}$的Fock表示，这些同态给出了$su(n,n)^{\mathbb C}$的所有酉最高权表示，从而重建了Segal-Shale-Weil表示的Kashiwara-Vergne列表。正如在推导它们的列表中一样，我们构造了$u(r)$在Fock空间中与$su(n,n)^{\mathbb C}$交换的表示，这给出了重数。该构造也给出了$(r+1)\times (r+1)$子式的理想是素的（$r\leq n-1)$ .所有的多项式由这样的理想的导数进行更奇异的表示。因此，这些表示可以在麦克斯韦型方程的解的空间中实现。我们实际上更进一步，确定我们列表中的哪些表示缺少某些${\mathfrak k}^{\mathbb C}$ -类型，从而准确地揭示哪些协变微分算子具有酉零空间。我们证明了${\mathcal U}_q(su(n,n)^{\mathbb C})$的类似结果。用Hayashi-Weyl代数${\mathcal H}{\mathcal W}_{2nr}$代替Weyl代数，用$q$-Fock空间代替Fock空间.进一步，用$q$ -行列式代替行列式，构造了${\mathcal U}_q(u(r))$在$q$ -Fock空间中的交换表示.为此，使用了Drinfeld Double。我们提到一个区别：量子化的负非紧根空间，虽然仍然是2度的，但不再完全由二阶生成算子给出。
英文摘要：
We construct homomorphic images of $su(n,n)^{\mathbb C}$ in Weyl Algebras ${\mathcal H}_{2nr}$. More precisely, and using the Bernstein filtration of ${\mathcal H}_{2nr}$, $su(n,n)^{\mathbb C}$ is mapped into degree $2$ elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of ${\mathcal H}_{2nr}$, these homomorphisms give all unitary highest weight representations of $su(n,n)^{\mathbb C}$ thus reconstructing the Kashiwara--Vergne List for the Segal--Shale--Weil representation. Just as in the derivation of the their list, we construct a representation of $u(r)$ in the Fock space commuting with $su(n,n)^{\mathbb C}$, and this gives the multiplicities. The construction also gives an easy proof that the ideals of $(r+1)\times (r+1)$ minors are prime ($r\leq n-1)$. The quotients of all polynomials by such ideals carry the more singular of the representations. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly which representations from our list are missing some ${\mathfrak k}^{\mathbb C}$-types, thereby revealing exactly which covariant differential operators have unitary null spaces. We prove the analogous results for ${\mathcal U}_q(su(n,n)^{\mathbb C})$. The Weyl Algebras are replaced by the Hayashi--Weyl Algebras ${\mathcal H}{\mathcal W}_{2nr}$ and the Fock space by a $q$-Fock space. Further, determinants are replaced by $q$-determinants, and a commuting representation of ${\mathcal U}_q(u(r))$ in the $q$-Fock space is constructed. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
