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Denis V. Juriev

Publications and source records attributed to Denis V. Juriev.

At least 37 records · Page 2Linked to original sources
arXiv1998

Infinite dimensional geometry of $M_1=Diff_+(S^1)/PSL(2,R)$ and $q_R$--conformal symmetries

A geometric interpretation of approximate ($HS$-projective or $TC$-projective) representations of the Witt algebra $w^C$ by $q_R$-conformal symmetries in the Verma modules $V_h$ over the Lie algebra $sl(2,C)$ is established and some their characteristics are calculated. It is shown that the generators of representations coincide with the Nomizu operators of holomorphic $w^C$-invariant hermitean connections in the deformed holomorphic tangent bundles $T_h(M_1)$ over the infinite-dimensional Kähler manifold $M_1=\Diff_+(S^1)/PSL(2,R)$, whereas the deviations $A_{XY}$ of the approximate representations coincide with the curvature operators for these connections, which supply the determinant bundle $det T_h(M_1)$ by a structure of the prequantization bundle over $M_1$. At $h=2$ the geometric picture reduces to one considered by A.A.Kirillov and the author [Funct.Anal.Appl. 21(4) (1987) 284-293] (without any relation to the approximate representations) for ordinary tangent bundles.

math.RT↗
arXiv1998

Topics in hidden symmetries. V

This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the concrete ones as well as on the abstract objects themselves. The results of researches allow to state that the actual richness and attractiveness of the inverse problem of representation theory are based not only on a large scope of various interesting abstract algebraic structures, which may be concretely represented and somehow unravelled, but also on the diversity of the manners of representation, which may produce very intriguing unexpected and nontrivial effects under the intent look even in the simple and almost hackneyed situations.

funct-an↗
arXiv1998

Approximate representations and Virasoro algebra

In the paper there are investigated various approximate representations of the infinite dimensional $\Bbb Z$--graded Lie algebras: the Witt algebra of all Laurent polynomial vector fields on a circle and its one-dimensional nontrivial central extension, the Virasoro algebra, by the infinite dimensional hidden symmetries in the Verma modules over the Lie algebra $\sl(2,C)$. There are considered as asymptotic representations "$mod O(\hbar^n)$" and representations up to a class $\frak S$ of operators (compact operators, Hilbert-Schmidt operators and finite-rank operators) as cases, which combine both types of approximations and in these cases an effect of noncommutativity of the order of their perform is explicated, that perhaps is underlied by a more general fundamental fact of deviations between the asymptotical theory of pseudodifferential operators and the pseudodifferential calculus on the asymptotic manifolds.

math.RT↗
arXiv1998

Cross-projective representations of pairs of anticommutative algebras, alloys and finite-dimensional irreducible representations of some infinite-dimensional Lie algebras

The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their representations, quaternary algebras and their alloyability are discussed. Considered examples allow to conclude that new representations have some intriguing features (continuous moduli of finite-dimensional irreducible representations, sophisticated Clebsch-Gordan coefficient calculus, etc.).

math.RT↗
arXiv1997

On the nonhamiltonian interaction of two rotators. II. Unraveling the algebraic structure

This note being is devoted to the unraveling of the algebraic structure, which governs the quadratic nonhamiltonian interaction of two rotators described in the first part. It should be considered in the context of a general ideology of the inverse problem of representation theory (see mp_arc/96-477). The resulted objects are formalized as Lie algebras with operators (Lie algebras with additional unary operations).

q-alg↗
arXiv1997

Topics in hidden symmetries. VI

In this note devoted to some aspects of the inverse problem of representation theory the attention is concentrated on the interrelations between various algebraic structures (algebras with operators) unraveled by different solutions of the same inverse problem. The Lie algebras and Jordan triple systems with operators are considered.

q-alg↗
arXiv1997

On the infinite dimensional hidden symmetries. II. $q_R$-conformal modular functor

The article is devoted to the $q_R$-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category $Train(Diff_+(S^1))$, the train of the group $Diff_+(S^1)$ of all orientation preserving diffeomorphisms of a circle) in the class of all projective modular functors (the projective representations of the category $Train(PSL(2,R))$, the train of the projective group $PSL(2,R)$), may be regarded as its ``Berezin quantizations''.

funct-an↗
arXiv1997

On the infinite dimensional hidden symmetries. I. Infinite dimensional geometry of $q_R$-conformal symmetries

This paper opens the series of articles supplemental to the series (hep-th/9405050,q-alg/9610026,q-alg/9611003,q-alg/9611019,funct-an/9611003), which also lies in lines of general ideology exposed in the review (mp_arc/96-477). The main purpose of the activity, which has its origin and motivation presumably in the author's applied researches on the interactively controlled systems (i.e.the controlled systems, in which the control is coupled with unknown or uncompletely known feedbacks), is to explicate the essentially infinite-dimensional aspects of the hidden symmetries, which appear in the representation theory of the finite dimensional Lie algebras and related algebraic structures. The series is organized as a sequence of topics, which illustarate this basic idea on the simple and tame examples without superfluous difficulties and details as well as in the previous series. Many of objects, which will appear, are somehow related to ones discussed earlier. However, the matherial will be treated more geometrically, presumably, from the points of view of the infinite dimensional geometry, an infinite dimensional version of the nonlinear geometric algebra and the infinite dimensional noncommutative geometry. The intent reader will see some ideological and concrete similarities between the subject of the paper and the infinite dimensional geometric picture for a second quantized string.

funct-an↗
arXiv1996

Topics in hidden symmetries. IV

This note being devoted to some aspects of the inverse problem of representation theory explicates the links between researches on the Sklyanin algebras and the author's (based on the noncommutative geometry) approach to the setting free of hidden symmetries in terms of "the quantization of constants". Namely, the Racah-Wigner algebra for the Sklyanin algebra is constructed. It may be considered as a result of the quantization of constants in the Racah-Wigner algebra for the Lie algebra $sl(2,C)$. The Racah-Wigner algebra for the Sklyanin algebra is an example of the noncommutative weighted shift operator algebras (NWSO-algebras), which generalize the mho-algebras introduced by the author earlier. If the Sklyanin algebra is interpreted as an algebra of anomalous spins then the Racah-Wigner algebra for it may be regarded as an enlargement of the Sklyanin algebra by operators of the anomalous spin-spin interaction (of tensor type).

q-alg↗
arXiv1996

Topics in hidden symmetries. III

This short paper being devoted to some aspects of the inverse problem of the representation theory treats several themes, which have their origins in the researches of F.A.Berezin, D.P.Zhelobenko, V.P.Maslov and his group, in context of the author's approach to the setting free of hidden symmetries, which is based on the noncommutative geometry. Relations of the discussed topics to the general ideologies of the translator algebras in representation theory and of the dynamical symmetries or the spectrum--generating algebras in mathematical physics are pointed out. A possibility of quantum field generalizations of the constructions in terms of vertex (QFT--operator) and Zamolodchikov algebras is mentioned.

q-alg↗
arXiv1996

Topics in hidden symmetries. II

This short paper being devoted to some aspects of the inverse problem of the representation theory briefly treats the interrelations between the author's approach to the setting free of hidden symmetries and the researches of D.P.Zhelobenko in the generalized Mickelsson algebras and related topics.

q-alg↗
arXiv1996

On the dynamics of noncanonically coupled oscillators and its hidden superstructure

The classical and quantum dynamics of the noncanonically coupled oscillators is considered. It is shown that though the classical dynamics is well--defined for both harmonic and anharmonic oscillators, the quantum one is well--defined in the harmonic case, admits a hidden (super)Hamiltonian formulation, and thus, preserves the initial operator relations, whereas a na\"ıve quantization of the anharmonic case meets with principal difficulties.

solv-int↗
arXiv1995

Isotopic pairs and their representations. II. A general supercase

Isotopic pairs and their representations are considered in a general framework of the vector superalgebra. Numerous examples of finite-dimensional and infinite-dimensional isotopic pairs are discussed. Several types of their representations are analysed. Applications to physics and relations to other mathematical objects are specified. Some curious combinatorial aspects of representation theory of isotopic pairs (graph-representations) are explicated. Perspectives of the further researches are drawn.

q-alg↗