TY - JOUR

T1 - Determinantal Ideals and the Canonical Commutation Relations

T2 - Classically or Quantized

AU - Jakobsen, Hans Plesner

PY - 2023

Y1 - 2023

N2 - We construct homomorphic images of su(n,n)C in Weyl Algebras H2nr. More precisely, and using the Bernstein filtration of H2nr, su(n,n)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 H2nr, these homomorphisms give all unitary highest weight representations of su(n,n)C thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of u(r)C into H2nr whose image commutes with the image of su(n,n)C, and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of (r+1)×(r+1) minors is prime. Here, of course, r≤n−1 and for a fixed such r, the space of any irreducible representation of su(n,n)C is annihilated by this ideal. 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 for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for Uq(su(n,n)C). The Weyl Algebras are replaced by the Hayashi–Weyl Algebras HW2nr and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of Uq(u(r)C) into HW2nr is constructed with analogous properties. 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.

AB - We construct homomorphic images of su(n,n)C in Weyl Algebras H2nr. More precisely, and using the Bernstein filtration of H2nr, su(n,n)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 H2nr, these homomorphisms give all unitary highest weight representations of su(n,n)C thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of u(r)C into H2nr whose image commutes with the image of su(n,n)C, and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of (r+1)×(r+1) minors is prime. Here, of course, r≤n−1 and for a fixed such r, the space of any irreducible representation of su(n,n)C is annihilated by this ideal. 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 for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for Uq(su(n,n)C). The Weyl Algebras are replaced by the Hayashi–Weyl Algebras HW2nr and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of Uq(u(r)C) into HW2nr is constructed with analogous properties. 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.

U2 - 10.1007/s00220-022-04524-5

DO - 10.1007/s00220-022-04524-5

M3 - Journal article

VL - 398

SP - 239

EP - 259

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -