Differential geometry on linear quantum groups

Schupp, Peter; Watts, Paul; Zumino, Bruno

doi:10.1007/bf00398310

Cited by 80 publications

(97 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…That's why for generic q = 1 the four matrix elements of either (dT )T or (dT )T are independent, and make up alternative bases for both Ω * S and Ω * . Actually, one can check (we will give details in [18]) that (d, Ω * ) coincides with the bicovariant differential calculus on M q (2), GL q (2) [44,46], and (d, Ω * S ) coincides with the Woronowicz 4D-bicovariant one [53,43] on C(SU q (2)).…”

Section: Remarkmentioning

confidence: 99%

“…[12] are made to be isomorphic (as ⋆-algebras) if they are slightly extended so as to contain suitable rational functions of their respective central elements; therefore that noncommutative sphere can be regarded as a compactification of A. In section 3 we reformulate in q-quaternion language the SO q (4)-covariant differential calculus [this turns out to coincide with the bicovariant differential calculus on M q (2), GL q (2) [44,46], and after imposing the unit q-determinant condition with the Woronowicz 4D-bicovariant differential calculus [53,43] on C (SU q (2))], the SO q (4)-covariant q-epsilon tensor and Hodge map [21,19,37,20] on Ω * (R 4 q ). In section 4 we recall some basic notions about the standard framework [9] for gauge theories on noncommutative spaces, pointing out where it doesn't fit the present model, and we formulate (anti)selfduality equations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

q -quaternions and q-deformed su(2) instantons

Fiore

2007

Journal of Mathematical Physics

View full text Add to dashboard Cite

We construct (anti)instanton solutions of a would-be q-deformed su(2) Yang-Mills theory on the quantum Euclidean space R 4 q [the SO q (4)-covariant noncommutative space] by reinterpreting the function algebra on the latter as a q-quaternion bialgebra. Since the (anti)selfduality equations are covariant under the quantum group of deformed rotations, translations and scale change, by applying the latter we can generate new solutions from the one centered at the origin and with unit size. We also construct multiinstanton solutions. As they depend on noncommuting parameters playing the roles of 'sizes' and 'coordinates of the centers' of the instantons, this indicates that the moduli space of a complete theory should be a noncommutative manifold. Similarly, gauge transformations should be allowed to depend on additional noncommutative parameters.

show abstract

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

q -quaternions and q-deformed su(2) instantons

Fiore

2007

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…Besides, we shall suppose the R-matrix to be skew-invertible that is there exists an n 2 × n 2 matrix Ψ such that In the compact notations the above formula reads T r (2) R 12 Ψ 23 = P 13 = T r (2) Ψ 12 R 23 , (1.6) where the symbol T r (2) means the calculation of trace in the second space and P is the permutation matrix.…”

Section: Reflection Equation Algebramentioning

confidence: 99%

The Weyl approach to the representation theory of reflection equation algebra

Saponov¹

2004

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the Rmatrix is arbitrary (not necessary originated from quantum groups). We suggest a universal method of constructing finite dimensional irreducible non-commutative representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed which are parametrized by Young diagrams. The spectrum of central elements s k = T r q L k is calculated in the single-row and single-column representations. A rule for the decomposition of the tensor product of modules into the direct sum of irreducible components is also suggested. Reflection Equation AlgebraReflection equation and the corresponding algebra which will be called the reflection equation algebra (REA for short) play a significant role in the theory of integrable systems and non-commutative geometry. In application to integrable systems the reflection equation with a spectral parameter is mainly used. First it appears in the work by I. Cherednik [1]. Usually it comprises the information about the behaviour of a system at a boundary, for example, describes the reflection of particles on a boundary of the configuration space.The reflection equation without a spectral parameter is important for the non-commutative geometry. One of the first applications the corresponding REA found in the theory of differential calculus on quantum groups (see, e.g., [2]). In such a differential calculus REA with the Hecke type R-matrix is a non-commutative analog of the algebra of vector fields on the groups GL(N ) or SL(N ). Besides, REA serves as a base for a definition of quantum analogs of homogeneous spaces -orbits of the coadjoint representation of a Lie group, as well as quantum analogs of linear bundles over such orbits (see, e.g., [3, 4]).In this paper we turn to problems of the representation theory of REA without a spectral parameter. We are interested in the following main topics:i) a construction of finite dimensional non-commutative irreducible representations and the calculation of spectrum (characters) of central elements in these representations;ii) a rule for the decomposition of the tensor product of irreducible modules into irreducible components. * saponov@mx.ihep.su 1 Before reviewing the known results, we introduce some necessary definitions and notations. Consider an associative algebra L q with the unity e L over the complex field C generated by n 2 elementsl j i , 1 ≤ i, j ≤ n, n being a fixed positive integer. Let the generators satisfy the following quadratic commutation relationswhere the matrixL ∈ Mat n (L q ) is composed ofl j i :L = l j i . Here the lower index enumerates the rows while the upper one columns. In (1.1) and everywhere below the use is made of the compact matrix notations [5] when the index of an object indicates the vector space to which t...

show abstract

“…One can use differential structure of SL h (2), applying the same contraction proccess in the level of differentials too. To construct differential structure one should use the following relations [20,21,16]…”

Section: µ-Deformation Of the Two Dimensional Poincaré Groupmentioning

confidence: 99%

A Triangular Deformation of the Two-Dimensional Poincaré Algebra

et al. 1995

View full text Add to dashboard Cite

Contracting the h-deformation of SL(2, R), we construct a new deformation of two dimensional Poincaré algebra, the algebra of functions on its group and its differential structure. It is also shown that the Hopf algebra is triangular, and its universal R matrix is also constructed explicitly. Then, we find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation.

show abstract

Differential geometry on linear quantum groups

Abstract: An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group GL q (N ). SL q (N ) is then obtained by constructing matrices with determinant unity, and the induced calculus is found.

Cited by 80 publications

References 7 publications

q -quaternions and q-deformed su(2) instantons

q -quaternions and q-deformed su(2) instantons

The Weyl approach to the representation theory of reflection equation algebra

A Triangular Deformation of the Two-Dimensional Poincaré Algebra

Contact Info

Product

Resources

About