arXiv subjects

V. K. Dobrev

Publications and source records attributed to V. K. Dobrev.

At least 19 recordsLinked to original sources

arXiv2026

Langlands Duality and Invariant Differential Operators: the Case SL(2n+1)

Recently we started building a bridge between two cases of Langlands duality. The latter is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. We started with the case of the group ~$SL(2n)$. In the present paper we deal with the group $SL(2n+1)$. The two mentioned groups are similar, but their representation theories is rather different.

math.RT↗

arXiv2025

Invariant Differential Operators for the Real Exceptional Lie Algebra $F'_4$

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional Lie algebra $F'_4=F_{4(4)}$ which is split real form of the exceptional Lie algebra $F_4$. We consider induction from a maximal parabolic algebra. We classify the reducible Verma modules over $F_4$ which are compatible with this induction. Thus, we obtain the classification of the corresponding invariant differential operators.

math-ph↗

arXiv2025

Langlands Duality and Invariant Differential Operators

Langlands duality is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. In this paper we start building the bridge between the two programs.

math.RT↗

arXiv2024

Invariant Differential Operators for Non-Compact Lie Groups: the $Sp(n,1)$ Case

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $sp(n,1)$. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which are of split rank one, of which class the other cases were studied, some long time ago. We concentrate on the case $n=2$. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations for, including the necessary data for all relevant invariant differential operators. We also present explicit expressions for the singular vectors and for the intertwining differential operators.

math.RT↗

arXiv2024

Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras ~$U_{q{\bf q}}(gl(n))$ and ~$U_{q{\bf q}}(sl(n))$~ which are in duality with the multiparameter quantum groups ~$GL_{q{\bf q}}(n)$, ~$SL_{q{\bf q}}(n)$,~ respectively. These objects depend on ~$n(n-1)/2+1$~ deformation parameters ~$q,q_{ij}$ ($1\leq i <j\leq n$) which is the maximal possible number in the case of $GL(n)$. The representations are labelled by $n-1$ complex numbers ~$r_i$~ and are acting in the space of formal power series of ~$n(n-1)/2$~ non-commuting variables. These variables generate quantum flag manifolds of ~$GL_{q{\bf q}}(n)$, ~$SL_{q{\bf q}}(n)$. The case $n=3$ is treated in more detail.

math-ph↗

arXiv2023

Exceptional Lie Algebra $E_{7(-25)}$ (Multiplets and Invariant Differential Operators)

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra $E_{7(-25)}$. Our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of $n$-dimensional Minkowski space-time. This class of algebras is identified and summarized in a table. Another motivation is related to the AdS/CFT correspondence. We give the multiplets of indecomposable elementary representations, including the necessary data for all relevant invariant differential operators.

hep-th↗

arXiv2023

Invariant Differential Operators for Non-Compact Lie Groups: Parabolic Subalgebras

In the present paper we start the systematic explicit construction of invariant differential operators by giving explicit description of one of the main ingredients - the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalised to the supersymmetric and quantum group settings and is necessary for applications to string theory and integrable models.

hep-th↗

arXiv2022

Invariant differential operators for the Jacobi algebra $ {\cal G}_2$

In the present paper we construct explicitly the intertwining differential operators for the Jacobi algebra ${\cal G}_2.$ For the construction we use the singular vectors of the Verma modules over ${\cal G}_2$ which we have constructed earlier. We construct the function spaces on which the operators act. We display two versions of the left (representation) action and the right action. The latter is inserted in the singular vectors to provide the intertwining differential operators.

math.RT↗

arXiv2022

Heisenberg Parabolic Subgroups of Exceptional Noncompact $G_{2(2)}$ and Invariant Differential Operators

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$. We use both the minimal and the maximal Heisenberg parabolic subalgebras. We give the main multiplets of indecomposable elementary representations. This includes the explicit parametrization of the invariant differential operators between the ERs. These are new results applicable in all cases when one would like to use $G_{2(2)}$ invariant differential operators.

math.RT↗

arXiv2021

Invariant Differential Operators for the Real Exceptional Lie Algebra $F"_4$

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional Lie algebra $F"_4$ which is the split rank one form of the exceptional Lie algebra $F_4$. We classify the reducible Verma modules over $F_4$ which are compatible with this induction. Thus, we obtain the classification of the corresponding invariant differential operators.

math.RT↗

arXiv2021

Positive Energy Representations, Holomorphic Discrete Series and Finite-Dimensional Irreps

Let G be a semi-simple non-compact Lie group with unitary lowest/highest weight representations. We consider explicitly the relation between three types of representations of G: positive energy (unitary lowest weight)representations, (holomorphic) discrete series representations and non-unitary finite-dimensional irreps. We consider mainly the conformal groups SO(n,2) treating in full detail the cases n=1,3,4.

hep-th↗

arXiv2021

Classification of the Reducible Verma Modules over the Jacobi Algebra $ {\cal G}_2$

In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram.

math.RT↗

arXiv2021

Tenfold Way for Holography : AdS/CFT and Beyond

The main purpose of the present paper is to lay the foundations of generalizing the AdS/CFT (holography) idea beyond the conformal setting. The main tool is to find suitable realizations of the bulk and boundary via group theory. We use all ten families of classical real semisimple Lie groups $G$ and Lie algebras $\cal G$. For this are used several group and algebra decompositions: the global Iwasawa decomposition and the local Bruhat and Sekiguchi-like decomposititions. The same analysis is applied to the exceptional real semisimple Lie algebras.

hep-th↗

arXiv2020

On Reducible Verma Modules over Jacobi Algebra

With this paper we start the study of reducible representations of the Jacobi algebra with the ultimate goal of constructing differential operators invariant w.r.t. the Jacobi algebra. In this first paper we show examples of the low level singular vectors of Verma modules over the Jacobi algebra. According to our methodology these will produce the invariant differential operators.

math.RT↗

arXiv2019

Parabolic Verma Modules and Invariant Differential Operators

In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebras.

math.RT↗

arXiv2019

Multiplet Classification of Reducible Verma Modules over the $G_2$ Algebra

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$ which is split real form of $G_2$. We give the classification of reducible Verma modules $G_2$. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over $G_{2(2)}$.

math.RT↗

arXiv2018

Characters of the Positive Energy UIRs of D=4 Conformal Supersymmetry

We give character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). Using these we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of su(2,2/N) developed already in the 80s.

hep-th↗

arXiv2016

Multiparameter Quantum Minkowski Space-Time and Quantum Maxwell Equations Hierarchy

Earlier we have proposed new $q$ - Maxwell equations which are the first members of an infinite new hierarchy of $q$ - difference equations. We have used an indexless formulation in which all indices are traded for two conjugate variables, $z,\bar z$. Weproposed also new $q$ - Minkowski coordinates which together with $z,\bar z$ can be interpreted as the six local coordinates of a $SU_q(2,2)$ flag manifold. In the present paper we generalise the main ingredients of this construction to the multiparameter case using the seven-parameter quantum group deformation of $GL(4)$ and $U(gl(4))$ and the four-parameter quantum group deformation of $SL(4)$ and $U(sl(4))$. The main result is the explicit presentation of the multiparameter quantum Minkowski space-time within the corresponding deformed flag manifold.

math.QA↗