arXiv subjects

Hans Plesner Jakobsen

Publications and source records attributed to Hans Plesner Jakobsen.

13 recordsLinked to original sources

arXiv2021

Determinantal Ideals and the Canonical Commutation Relations. Classically or Quantized

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.

math-ph↗

arXiv2019

Algebras of Variable Coefficient Quantized Differential Operators

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients should be. It is an emediate point that even $0$th order operators, given as multiplications by polynomials, have to be specified as e.g. left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings which allows us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings. The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra ${\mathcal W}eyl_q(n,n)$ introduced by T. Hyashi (Comm. Math. Phys. 1990) plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over ${\mathcal W}eyl_q(n,n)$. We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.

math.QA↗

arXiv2019

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

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 $ϕ^1_{Λ,Λ_1}$.

math.QA↗

arXiv2018

The center of ${\mathcal U}_q({\mathfrak n}_ω)$

We determine the center of a localization of ${\mathcal U}_q({\mathfrak n}_ω)\subseteq {\mathcal U}^+_q({\mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster algebras. In our set-up, ${\mathfrak g}$ is any finite-dimensional complex Lie algebra and $ω$ is any element in the Weyl group $W$. The non-zero complex parameter $q$ is mostly assumed not to be a root of unity, but our method also gives many details in case $q$ is a primitive root of unity. We point to a new and very useful direction of approach to a general set of problems which we exemplify here by obtaining the result that the center is determined by the null space of $1+ω$. Further, we use this to give a generalization to double Schubert Cell algebras where the center is proved to be given by $ω^{\mathfrak a}+ω^{\mathfrak c}$. Another family of quadratic algebras is also considered and the centers determined.

math.QA↗

arXiv2012

Quantized Matrix Algebras and Quantum seeds

We determine explicit quantum seeds for classes of quantized matrix algebras. Furthermore, we obtain results on centers and block diagonal forms {of these algebras.} In the case where $q$ is {an arbitrary} root of unity, this further determines the degrees.

math.QA↗

arXiv2012

Double-partition Quantum Cluster Algebras

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines $L$. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another. Compatible pairs can be written down. The algebras are equal to their upper cluster algebras. The variables of the quantum seeds are given by elements of the dual canonical basis. This is the final version, where some arguments have been expanded and/or improved and several typos corrected. Full bibliographic details: Journal of Algebra (2012), pp. 172-203 DOI information: 10.1016/j.jalgebra.2012.09.015

math.QA↗

arXiv2012

Indecomposable finite-dimensional representations of a class of Lie algebras and Lie superalgebras

In the article at hand, we sketch how, by utilizing nilpotency to its fullest extent (Engel, Super Engel) while using methods from the theory of universal enveloping algebras, a complete description of the indecomposable representations may be reached. In practice, the combinatorics is still formidable, though. It turns out that the method applies to both a class of ordinary Lie algebras and to a similar class of Lie superalgebras. Besides some examples, due to the level of complexity we will only describe a few precise results. One of these is a complete classification of which ideals can occur in the enveloping algebra of the translation subgroup of the Poincaré group. Equivalently, this determines all indecomposable representations with a single, 1-dimensional source. Another result is the construction of an infinite-dimensional family of inequivalent representations already in dimension 12. This is much lower than the 24-dimensional representations which were thought to be the lowest possible. The complexity increases considerably, though yet in a manageable fashion, in the supersymmetric setting. Besides a few examples, only a subclass of ideals of the enveloping algebra of the super Poincaré algebra will be determined in the present article.

math.RT↗

arXiv2005

The exponential nature and positivity

In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an anti-automorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusztig's canonical basis of $U_q(n^-)$, the positivity properties of the basis as well as the positivity properties of the canonical basis of the modified quantum enveloping algebra of type $A$, which has been conjectured by Lusztig, are proved.

math.QA↗

arXiv2001

Quantized rank R matrices

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ matrices as well as quantized factor algebras of $M_q(n)$ are analyzed. The latter are the quantized function algebra of rank $r$ matrices obtained by working modulo the ideal generated by all $(r+1)\times (r+1)$ quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one. In all cases, the quantum parameter is a primitive $m$th roots of unity. The degrees and centers of the algebras are determined when $m$ is a prime and the general structure is obtained for arbitrary $m$.

math.QA↗

arXiv2000

Quantized Dirac Operators

We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.

math.QA↗

arXiv1999

Q-differential operators

We set up a framework for discussing `$q$-analogues' of the usual covariant differential operators for hermitian symmetric spaces. This turns out to be directly related to the deformation quantization associated to quadratic algebras satisfying certain conditions introduced by Procesi and De Concini.

math.QA↗

arXiv1999

Quantized Heisenberg Space

We investigate the algebra $F_q(N)$ introduced by Faddeev, Reshetikhin and Takhadjian. In case $q$ is a primitive root of unity the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini-Kac-Procesi Conjecture is proved for this case. In the case of $q$ generic, the primitive ideals are described. A related algebra studied by Oh is also treated.

math.QA↗

arXiv1999

A class of quadratic matrix algebras arising from the quantized enveloping algebra ${\s U}_q(A_{2n-1})$

A natural family of quantized matrix algebras is introduced. It includes the two best studied such. Located inside ${\s U}_q(A_{2n-1})$, it consists of quadratic algebras with the same Hilbert series as polynomials in $n^2$ variables. We discuss their general properties and investigate some members of the family in great detail with respect to associated varieties, degrees, centers, and symplectic leaves. Finally, the space of rank r matrices becomes a Poisson submanifold, and there is an associated tensor category of $\rank\leq r$ matrices.

math.QA↗