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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., Comment: 21 pages, 2 figures. arXiv admin note: text overlap in review part with arXiv:1311.7557, arXiv:1208.0409, arXiv:1412.8038; V2 : added references

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

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., Comment: 26 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1205.5521, arXiv:0812.2655, arXiv:0812.2690; text overlap with arXiv:1312.5998, arXiv:1402.0190, arXiv:1311.7557, arXiv:1208.0409

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., Comment: 19 pages, 4 postscript figures, Contribution to special issue of Symmetry "Manifest and Hidden Symmetries in Field and String Theories", V2: changes to conform with version accepted to the journal

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., Comment: 18 pages, Contribution to Peter Suranyi Festschrift; v2: small additions

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., Comment: 13 pages, no figure, version published in MPLA

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., Comment: 41pages, many figures

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., Comment: 24 pages, 5 tables, Latex2e; v2,3: corrected typos, v4 revised as accepted in IJMPA

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., Comment: 8 pages, Plenary talk at Bogolyubov Conference, Moscow-Dubna, 9-13.9.2019

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., Comment: 8 pages, V2 typos corrected, to appear in the Proceedings of QTS-11, Montreal, July 2019

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 $G_{2(2)}$., Comment: 9 pages, 7 figures, Invited talk at the XXXII International Colloquium on Group Theoretical Methods in Physics (Prague, July 2018); session dedicated to I.E. Segal (1918-1998)

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., Comment: 10 pages, Invited talk at the III National Congress of Bulgarian Physicists (Sofia, 2016), Bulg. J. Phys., to appear

In the present paper we continue the project of systematic construction of invariant differential operators on the example of representations of the conformal algebra induced from the maximal cuspidal parabolic., Comment: 13 pages, 5 figures; V2: small corrections

We continue the study of positive energy (lowest weight) unitary irreducible representations of the superalgebras $osp(1|2n,R)$. We update previous results and present the full list of these UIRs. We give also some character formulae for these representations., Comment: 27 pages, 2 figures, To appear in the Proceedings of the VIII Mathematical Physics Meeting, (Belgrade, 24-31 August 2014); v2: corrected sign error; arXiv admin note: text overlap in preliminaries with arXiv:hep-th/0402039, Proceedings of the VIII Mathematical Physics Meeting, (Belgrade, 24-31 August 2014) SFIN XXVIII (A1), eds. B. Dragovich et al, Belgrade Inst. Phys. 2013 [ISBN 978-86-82441-43-4], pp. 59-81

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $so^*(12)$. We give the main multiplets of indecomposable elementary representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebra $so(6,6)$ with suitably chosen maximal parabolic subalgebra., Comment: 7 pages, 1 figure, Latex2e, Invited talk at the XXX International Colloquium on Group Theoretical Methods in Physics (Ghent, July 2014), Text overlap in Preliminaries with arXiv:1402.0190, arXiv:1312.5998; V2: misprints corrected

In the present paper we review our project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n,n) for n=2,3,4. We give explicitly the main multiplets of indecomposable elementary representations and some reduced multiplets. We give explicitly the minimal representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras sl(2n,R) and when n=2k for the algebras su*(4k) with suitably chosen maximal parabolic subalgebras., Comment: 21 pages, 16 figures, Review paper based on Plenary talk at the International Conference on Integrable Systems and Quantum Symmetries, Prague, June 2014

Using our previous results on the systematic construction of invariant differential operators for non-compact semisimple Lie groups we classify the special reduced multiplets and minimal representations in the case of SO(p,q)., Comment: 26 pages, 11 figures, to appear in the Proceedings of the X International Workshop "Lie Theory and Its Applications in Physics}, (Varna, Bulgaria, June 2013), "Springer Proceedings in Mathematics and Statistics", Vol. 111

We give a group-theoretic interpretation of relativistic holography as equivalence between representations of the anti de Sitter algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators for arbitrary integer spin the framework of representation theory. Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the de Sitter case, we show that each bulk field has two boundary (shadow) fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary., Comment: 21 pages, Latex; V2,V3: improvements related to the publication process

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $su(n,n)$. Earlier were given the main multiplets of indecomposable elementary representations for $n\leq 4$, and the reduced ones for $n=2,3$. Here we give all reduced multiplets containing physically relevant representations including the minimal ones for $n=4$. Due to the recently established parabolic relations the results are valid also for the algebras $sl(8,\mathbb{R})$ and $su^*(8)$ with suitably chosen maximal parabolic subalgebras., Comment: 35 pages, 21 figures, Lectures at "Third International School on Symmetry and Integrable Systems" and at "Humboldt Kolleg on Symmetry and Integrable Systems", Tsakhkadzor, Armenia, July 2013; V2: small correction. V3: changed normalization of conformal weight. arXiv admin note: substantial text overlap with arXiv:1312.5998

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $su(n,n)$. Earlier were given the main multiplets of indecomposable elementary representations for $n\leq 4$, and the reduced ones for $n=2$. Here we give all reduced multiplets containing physically relevant representations including the minimal ones for the algebra $su(3,3)$. Due to the recently established parabolic relations the results are valid also for the algebra $sl(6,\mathbb{R})$ with suitably chosen maximal parabolic subalgebra., Comment: 13 pages, 9 figures, Plenary talk at the International Workshop 'Supersymmetries and Quantum Symmetries', Dubna, July 29 - August 3, 2013. V2: extended version with 5 more figures treating all reduced multiplets containing physically relevant representations. V3: systematic typos corrected. V4: changed normalization of conformal weight, results unchanged

We give a review of some group-theoretical results related to non-relativistic holography. Our main playgrounds are the Schr\"odinger equation and the Schr\"odinger algebra. We first recall the interpretation of non-relativistic holography as equivalence between representations of the Schr\"odinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant w.r.t. Schr\"odinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semi-simple Lie groups and adapted to the non-semisimple Schr\"odinger algebra. These requires development of the representation theory of the Schr\"odinger algebra which is reviewed in some detail. We also recall the $q$-deformation of the Schr\"odinger algebra. Finally, the realization of the Schr\"odinger algebra via difference operators is reviewed., Comment: 58 pages; V2: added 2 references, references rearranged to conform with journal requirements; V3: small changes

In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of {\it parabolic relation} between two non-compact semisimple Lie algebras $\cal G$ and $\cal G'$ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra $E_{7(7)}$ which is parabolically related to the CLA $E_{7(-25)}$. Other interesting examples are the orthogonal algebras $so(p,q)$ all of which are parabolically related to the conformal algebra $so(n,2)$ with $p+q=n+2$, the parabolic subalgebras including the Lorentz subalgebra $so(n-1,1)$ and its analogs $so(p-1,q-1)$. Further we consider the algebras $sl(2n,R)$ and for $n=2k$ the algebras $su^*(4k)$ which are parabolically related to the CLA $su(n,n)$. Further we consider the algebras $sp(r,r)$ which are parabolically related to the CLA $sp(2r,R)$. We consider also $E_{6(6)}$ and $E_{6(2)}$ which are parabolically related to the hermitian symmetric case $E_{6(-14)}$., Comment: 25 pages, 16 figures, Talk at the VIII International Symposium "Quantum Theory and Symmetries", Mexico City, August 5-9, 2013

In the present paper we continue the programme of systematic construction of invariant differential operators on the example of the non-compact groups Sp(n,R). Earlier in arXiv:1205.5521 we gave the main multiplets and the main reduced multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. Here we give the special reduced multiplets and the minimal representations of Sp(n,R)., Comment: 20 pages, 9 figures, LATEX, Invited talk at the VII Mathematical Physics Meeting, Belgrade, 9-19.9.2012

Using our previous results on the systematic construction of invariant differential operators for non-compact semisimple Lie groups we classify the conservation laws in the case of SO(p,q)., Comment: 6 pages, Invited talk at XXIX International Colloquium on Group-Theoretical Methods in Physics, Chern Institute of Mathematics, Nankai Univ., China, August 20-26, 2012, to appear in the Proceedings

This paper continues the project of constructing the character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). In the first paper we gave the bare characters which represent the defining odd entries of the characters. Now we give the full explicit character formulae for N=1 and for several important examples for N=2 and N=4., Comment: 48 pages, TeX with Harvmac, overlap in preliminaries with arXiv:hep-th/0406154; some comments and references added

In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of {\it parabolic relation} between two non-compact semisimple Lie algebras g and g' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E_{7(7)} which is parabolically related to the CLA E_{7(-25)}, the parabolic subalgebras including E_{6(6)} and E_{6(-6)} . Other interesting examples are the orthogonal algebras so(p,q) all of which are parabolically related to the conformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including the Lorentz subalgebra so(n-1,1) and its analogs so(p-1,q-1). We consider also E_{6(6)} and E_{6(2)} which are parabolically related to the hermitian symmetric case E_{6(-14)}, the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so(p,q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of {\it shadow fields}. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible {\it conservation laws} and {\it conserved currents}, unitary or not., Comment: LATEX, 42 pages, 10 figures, overlap in Preliminaries with arXiv:0812.2690, arXiv:1205.5521, arXiv:0812.2655, V3: text amended as accepted to JHEP, v4,v5: small additions

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,R), in detail for n=6. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of Minkowski space-time. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations for n=6, including the necessary data for all relevant invariant differential operators. In fact, this gives by reduction also the cases for n<6, since the main multiplet for fixed n coincides with one reduced case for n+1., Comment: Latex2e, 27 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:0812.2690, arXiv:0812.2655

In an earlier paper we gave the complete group-theoretical classification of BPS states of the N-extended D=4 conformal superalgebras su(2,2/N), but not all interesting cases were given in detail. In the present paper we spell out the interesting case of (1/N)-BPS and possibly protected states., Comment: LATEX2e, 8 pages, Plenary talk at the International Workshop 'Supersymmetries and Quantum Symmetries', Dubna, 18-23.7.2011, to appear in the Proceedings, editors Evgeny Ivanov et al

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrodinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov-Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others., Comment: 15 pages, LATEX; Published version

We point out that the nontrivial central extension of the superalgebra $su(2/2)$ is related to the some not so well-known Serre relations., Comment: 6 pages, LATEX, v2: added references; v3: small changes and added reference

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., Comment: 20 pages, 2 figures, TEX with input files harvmac.tex, amssym.def, amssym.tex; v2: added references; v3: change of normalization in f-lae (4.1) and (4.7); v4 corrected misprint. arXiv admin note: substantial text overlap with arXiv:0812.2655

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact exceptional group $E_{6(-14)}$ we give the multiplets of indecomposable elementary representations. This includes the data for all relevant invariant differential operators., Comment: 30 pages, 18 figures, TEX with input files harvmac.tex, amssym.def, amssym.tex, Proceedings, V School in Modern Mathematical Physics, Belgrade, Serbia, 2008 (Institute of Physics, Belgrade, 2009) pp. 95-124. v3: added acknowledgements

We present the exotic bialgebras that arise from a 9x9 unitary braid matrix., Comment: 14 pages, LATEX; v3: some rearrangement, added to appear

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., Comment: 28 pages, TEX with Harvmac using amssym.def, amssym.tex, epsf.tex; v2: new texts in Sections 1 & 3, new refs; v3: added 5 figures; v4,v5: small corrections

We present the list of irreducible (generalized) highest weight modules over the Virasoro algebra and N=1 super-Virasoro algebras obtained as factor-modules of (generalized) Verma modules. We present also the character formulae of all these modules and single out the unitary irreducible ones. Most formulas are valid for the three algebras under consideration, the different cases being distinguished by two parameters. This is a slightly extended version of an Encyclopedia entry., Comment: 16 pages, 2 figures, TEX with harvmac using amssym.def, amssym.tex, epsf.tex; Encyclopedia entry

The characters of the unitarizable highest weight modules over the N=2 superconformal algebras are presented. This is a slightly extended version of an Encyclopedia entry., Comment: 7 pages, 2 figures, TEX with Harvmac using amssym.def, amssym.tex, epsf.tex; Encyclopedia entry

A class of $(2n)^2\times(2n)^2$ multiparameter braid matrices are presented for all $n$ $(n\geq 1)$. Apart from the spectral parameter $\theta$, they depend on $2n^2$ free parameters $m_{ij}^{(\pm)}$, $i,j=1,...,n$. For real parameters the matrices $R(\theta)$ are nonunitary. For purely imaginary parameters they became unitary. Thus a unification is achieved with odd dimensional multiparameter solutions presented before., Comment: 07 pages

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., Comment: 44 pages; V2: important addition in Section 3 and misprints corrected; more corrections in Section 3; v3-v6: various corrections; v7: corrections in (11.7),(11.9),(11.11), and correspondingly in the Appendix; v8: added dimensions of N-factors where missing; v9: added missing case in 11.37; v10: corrected misprint in 11.17; v11: added missing case in 11.37; v12: typos corrected in (11.7),(11.9)

We construct $(2n)^2\times (2n)^2$ unitary braid matrices $\hat{R}$ for $n\geq 2$ generalizing the class known for $n=1$. A set of $(2n)\times (2n)$ matrices $(I,J,K,L)$ are defined. $\hat{R}$ is expressed in terms of their tensor products (such as $K\otimes J$), leading to a canonical formulation for all $n$. Complex projectors $P_{\pm}$ provide a basis for our real, unitary $\hat{R}$. Baxterization is obtained. Diagonalizations and block-diagonalizations are presented. The loss of braid property when $\hat{R}$ $(n>1)$ is block-diagonalized in terms of $\hat{R}$ $(n=1)$ is pointed out and explained. For odd dimension $(2n+1)^2\times (2n+1)^2$, a previously constructed braid matrix is complexified to obtain unitarity. $\hat{R}\mathrm{LL}$- and $\hat{R}\mathrm{TT}$-algebras, chain Hamiltonians, potentials for factorizable $S$-matrices, complex non-commutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements., Comment: 26 pages, 1 figure, Addendum and new references are added. To appear in Journal of Mathematical Physics

We give solutions of the q-deformed equations of quantum conformal Weyl gravity in terms of q-deformed plane waves., Comment: 13 pages, LATEX with revtex4; Proceedings of the XII International Conference on Symmetry Methods in Physics, (Yerevan, 2006) eds. G.S. Pogosyan et al; V2: added Summary and Outlook

We refute a recent claim in the literature of a "new" quantum deformation of GL(2)., Comment: 4 pages, LATEX