scholarly journals FREE-FIELD REPRESENTATION OF GROUP ELEMENT FOR SIMPLE QUANTUM GROUPS

1998 ◽  
Vol 13 (10) ◽  
pp. 1651-1707 ◽  
Author(s):  
ALEXEI MOROZOV ◽  
LUC VINET
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Free Field ◽  
Group Element ◽  
Field Representation ◽  
Universal Formula ◽  
Simple Quantum ◽  
Free Fields ◽  
Usual Formula

A representation of the group element (also known as "universal [Formula: see text]-matrix") which satisfies Δ(g)=g⊗g, is given in the form [Formula: see text]where [Formula: see text], qi= q‖αi‖2/2 and Hi=2Hαi/ ‖αi‖2 and T±i are the generators of quantum group associated respectively with Cartan algebra and the simple roots. The "free fields" χ, ϕ, ψ form a Heisenberg-like algebra: [Formula: see text] We argue that the d G -parametric "manifold" which g spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication g→ g′ · g′′. The universal ℛ-matrix with the property that ℛ(g⊗ I)(I⊗g)= (I⊗ g)(g⊗ I)ℛ is given by the usual formula [Formula: see text]

FUSION CONSTRUCTION OF THE VERTEX OPERATORS IN HIGHER LEVEL REPRESENTATION OF THE ELLIPTIC QUANTUM GROUP

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1995-2001
Author(s):  
HITOSHI KONNO
Keyword(s):  
Quantum Group ◽  
Free Field ◽  
Vertex Operators ◽  
Type I ◽  
Field Representation ◽  
Short Summary ◽  
Higher Spin ◽  
Elliptic Algebra ◽  
Elliptic Quantum ◽  
Weight Coefficients

After a short summary on the elliptic quantum group [Formula: see text] and the elliptic algebra [Formula: see text], we present a free field representation of the Drinfeld currents and the vertex operators (VO's) in the level k. We especially demonstrate a construction of the higher spin type I VO's by fusing the spin 1/2 type I VO's and fix a rule of attaching the screening current S(z) associated with the q-deformed ℤk-parafermion theory. As a result we get a free field representation of the higher spin type I VO's which commutation relation by the fused Boltzmann weight coefficients is manifest.

WESS-ZUMINO-WITTEN MODEL AS A THEORY OF FREE FIELDS

1990 ◽  
Vol 05 (13) ◽  
pp. 2495-2589 ◽  
Author(s):  
A. GERASIMOV ◽  
A. MOROZOV ◽  
M. OLSHANETSKY ◽  
A. MARSHAKOV ◽  
S. SHATASHVILI
Keyword(s):  
Riemann Surfaces ◽  
Central Charge ◽  
Free Field ◽  
Heisenberg Algebra ◽  
Momentum Tensor ◽  
Point Of View ◽  
Special Role ◽  
Field Representation ◽  
Conformal Blocks ◽  
Free Fields

The free field representation or "bosonization" rule1 for Wess-Zumino-Witten model (WZWM) with arbitrary Kac-Moody algebra and arbitrary central charge is discussed. Energy-momentum tensor, arising from Sugawara construction, is quadratic in the fields. In this way, all known formulae for conformal blocks and correlators may be easily reproduced as certain linear combinations of correlators of these free fields. Generalization to conformal blocks on arbitrary Riemann surfaces is straightforward. However, projection rules in the spirit of Ref. 2 are not specified. The special role of βγ systems is emphasized. From the mathematical point of view, the construction involved represents generators of Kac-Moody (KM) algebra in terms of generators of a Heisenberg one. If WZW Lagrangian is considered as d−1 of Kirillov form on an orbit of KM algebra,3 then the free fields of interest (i.e. generators of the Heisenberg algebra) diagonalize Kirillov form and the action. Reduction of KM algebra within the same construction should naturally lead to arbitrary coset models.

A STREAMLINED HIGHEST WEIGHT DERIVATION OF THE BILINEAR VIRASORO CENTRE

1989 ◽  
Vol 04 (18) ◽  
pp. 1789-1796 ◽  
Author(s):  
N. GORMAN ◽  
L. O’RAIFEARTAIGH ◽  
W. McGLINN
Keyword(s):  
Virasoro Algebra ◽  
Simple Derivation ◽  
Field Representation ◽  
Universal Formula ◽  
Highest Weight ◽  
Free Fields

We derive the universal formula c=±(D+6J), where [Formula: see text] for the centre c of any Virasoro algebra constructed as a bilinear in D free fields of conformal spins ja. The derivation is extremely simple and relies only on the highest weight nature of the field representation. An analogous simple derivation of the corresponding result c=2K dim G/(2K+f2) for KM fields is also given.

scholarly journals THE FREE FIELD REPRESENTATION OF SU(3) CONFORMAL FIELD THEORY

1993 ◽  
Vol 08 (23) ◽  
pp. 4031-4053
Author(s):  
HOVIK D. TOOMASSIAN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Free Field ◽  
Field Representation ◽  
Conformal Field ◽  
Point Correlation

The structure of the free field representation and some four-point correlation functions of the SU(3) conformal field theory are considered.

scholarly journals GENERALIZED W3-STRINGS FROM FREE FIELDS

1995 ◽  
Vol 10 (06) ◽  
pp. 515-524 ◽  
Author(s):  
J. M. FIGUEROA-O'FARRILL ◽  
C. M. HULL ◽  
L. PALACIOS ◽  
E. RAMOS
Keyword(s):  
Bosonic Strings ◽  
Free Field ◽  
Space Time ◽  
General Construction ◽  
Special Cases ◽  
Free Fields ◽  
Rule Out ◽  
General Conditions ◽  
String Theories

The conventional quantization of w3-strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W3-string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W3-algebra. We study W3-like strings with a direct space-time interpretation — that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w3-algebra. We give the general conditions for the existence of W3-like strings, and comment on how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

scholarly journals From Quantum Groups to Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1073-1094 ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthias Neufang
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Large Classes ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Recent Developments ◽  
Compact Quantum Groups ◽  
Quantum Version ◽  
Quantum Point

AbstractIn this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group 𝔾 a locally compact group 𝔾˜ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group 𝔾. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of 𝔾 are encoded by 𝔾˜; the latter, despite being a simpler object, can carry very important information about 𝔾.

scholarly journals Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

2014 ◽  
Vol 57 (4) ◽  
pp. 708-720 ◽  
Author(s):  
Michael Brannan
Keyword(s):  
Strong Convergence ◽  
Quantum Group ◽  
Quantum Groups ◽  
Convergence Theorem ◽  
Convergence Result ◽  
Operator Norm ◽  
Matrix Coefficient ◽  
Strong Convergence Theorem ◽  
Convergence Results ◽  
Noncommutative Polynomials

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

BRST COHOMOLOGY IN QUANTUM AFFINE ALGEBRA $U_q \left( {\widehat{sl_2 }} \right)$

1994 ◽  
Vol 09 (14) ◽  
pp. 1253-1265 ◽  
Author(s):  
HITOSHI KONNO
Keyword(s):  
Free Field ◽  
Classical Case ◽  
Explicit Construction ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Field Representation ◽  
Highest Weight ◽  
Brst Charge ◽  
Brst Cohomology ◽  
Highest Weight Representation

Using free field representation of quantum affine algebra [Formula: see text], we investigate the structure of the Fock modules over [Formula: see text]. The analysis is based on a q-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra [Formula: see text]. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case (q=1), we obtain the irreducible highest weight representation space as a non-trivial cohomology group. This enables us to calculate a trace of the q-vertex operators over this space.

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