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Konrad Schmuedgen

Publications and source records attributed to Konrad Schmuedgen.

At least 19 recordsLinked to original sources
arXiv2009

Algebras of Fractions and Strict Positivstellensätze for *-Algebras

In this paper we investigate a *-algebra $\cX$ of fractions associated with a unital complex *-algebra $\cA$. The algebra $\cX$ and its Hilbert space representations are used to prove abstract noncommutative strict Positivstellensätze for $\cA$. Multi-grading of $\cA$ are studied as technical tools to verify the assumptions of this theorem. As applications we obtain new strict Positivstellensätze for the Weyl algebra and for the Lie algebra $\cg$ of the affine group of the real line. We characterize integrable representations of the Lie algebra $\cg$ in terms of resolvents of the generators and derive a new integrability criterion for representations of $\cg$.

math.OA↗
arXiv2007

Non-commutative Real Algebraic Geometry - Some Basic Concepts and First Ideas

We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A version of Stengle's Positivstellensatz for $n \times n$ matrices of real polynomials is proved.

math.OA↗
arXiv2007

Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere

A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.

math.QA↗
arXiv2007

Representations of cross product algebras of Podles quantum spheres

Hilbert space representations of the cross product *-algebras of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qr}) of Podles spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf *-algebra O(SU_q(2)) of the quantum group SU_q(2) decomposes into an orthogonal sum of projective Hopf modules corresponding to irreducible integrable *-representations of the cross product algebras and that each irreducible integrable *-representation appears with multiplicity one. The projections of these projective modules are computed. The decompositions of tensor products of irreducible integrable *-representations with spin l representations of U_q(su_2) are given. The invariant state h on O(S^2_{qr}) is studied in detail. By passing to function algebras over the quantum spheres S^2_{qr}, we give chart descriptions of quantum line bundles and describe the representations from the first approach by means of the second approach.

math.QA↗
arXiv2005

A Strict Positivstellensatz for the Weyl Algebra

Let $c$ be an element of the Weyl algebra $W(d)$ which is given by a strictly positive operator in the Schr"odinger representation. It is shown that, under some conditions, there exist elements $b_1,...,b_d$ in $W(d)$ such that $b_1 c b_1^* + ... + b_d c b_d^*$ is a finite sum of squares.

math.AG↗
arXiv2004

A Strict Positivstellensatz for Enveloping Algebras

Let G be a connected and simply connected real Lie group with Lie algebra g. Semialgebraic subsets of the unitary dual of G are defined and a strict Positivstellensatz for positive elements of the universal enveloping algebra of g is proved.

math.AG↗
arXiv2003

Hilbert space representations of cross product algebras II

In this paper, we study and classify Hilbert space representations of cross product *-algebras of the quantized enveloping algebra $U_q(e_2)$ with the coordinate algebras $O(E_q(2))$ of the quantum motion group and $O(\C_q)$ of the complex plane, and of the quantized enveloping algebra $U_q(su_{1,1})$ with the coordinate algebras $O(SU_q(1,1))$ of the quantum group $SU_q(1,1)$ and $O(U_q)$ of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitely described.

math.QA↗
arXiv2002

Hilbert Space Representations of Cross Product Algebras

Hilbert space representations of cross product *-algebras of the Hopf *-algebras U_q(gl_2) with the coordinate algebras O(C^2_q) and O(R^3_q) of quantum vector spaces and U_q(su_2) with coordinate algebras O(SU_q(2)) and O(S^2_q) of corresponding quantum spheres are investigated and classified. Invariant states on the coordinate *-algebras are described by two variants of the quantum trace.

math.QA↗
arXiv2002

On the Quantum Quarter Plane and the Real Quantum Plane

Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert space and the Weyl calculus of pseudodifferential operators we construct *-algebras of ``functions'' on the quantum quarter plane R^{++}_q and on the real quantum plane R^2_q which are left module *-algebras for the Hopf *-algebra U_q(gl_2(R)). We define a family h_k, k\in Z^2, of covariant positive linear functionals on these *-algebras and study the actions of the *-algebras O(R^2_q) and U_q(gl_2(R)) on the associated Hilbert spaces. Quantum analogs of the partial Fourier transforms and the Fourier transform are found. A differential calculus on the ``function'' *-algebras is also developed and investigated. A shorter version appeared in International Journal of Mathematics 13 (2002), 279-321.

math.OA↗
arXiv2002

On the moment problem of closed semi-algebraic sets

Let $K_f$ be a closed semi-algebraic set in $\dR^d$ such that there exist bounded real polynomials $h_1,{...},h_n$ on $K_f$. It is proved that the moment problem for $K_f$ is solvable provided it is for all sets $K_f\cap C_λ$, where $C_λ=\{x:h_1(x)=λ_1,{...},h_n(x)=λ_n\}$ and $\inf\{h_j(x); x\in K_f\}\leλ_j\le\sup\{h_j(x);x\in K_f\}$. New classes of non-compact closed semi-algebraic sets $K_f$ are found for which the moment problem is solvable. To appear in Journal fuer die Reine und Angewandte Mathematik

math.FA↗
arXiv2001

Commutator Representations of Covariant Differential Calculi on Quantum Groups

Let (G,d) be a first order differential *-calculus on a *-algebra A. We say that a pair (π,F) of a *-representation πof A on a dense domain D of a Hilbert space and a symmetric operator F on D gives a commutator representation of G if there exists a linear mapping t:G -> L(D) such that t(adb)=π(a)i[F,π(b)], a,b in A. Among others, it is shown that each left-covariant *-calculus G of a compact quantum group Hopf *-algebra A has a faithful commutator representation. For a class of bicovariant *-calculi on A there is a commutator representation such that F is the image of a central element of the quantum tangent space. If A is the Hopf *-algebra of the compact form of one of the quantum groups SL_q(n+1), O_q(n), Sp_q(2n) with real transcendental q, then this commutator representation is faithful. Keywords: Quantum Groups; Noncommuative Geometry; Differential Calculus

math.QA↗
arXiv2000

On Well-behaved Unbounded Representations of *-Algebras

A general approach to the well-behaved unbounded *-representations of a *-algebra X is proposed. Let B be a normed *-algebra equipped with a left action |> of X on B such that (x |> a)^+ b=a^+(x^+ |> b) for a,b\in B and x\in X. Then the pair (X,B) is called a compatible pair. For any continuous non-degenerate *-representation ρof B there exists a closed *-representation ρ' of X such that ρ'(x)ρ(b)=ρ(x |> b), where x\in X and b\in B. The *-representations ρ' are called the well-behaved *-representations associated with the compatible pair (X,B). A number of examples are developed in detail.

math.OA↗
arXiv1999

On holomorphic functions on a strip in the complex plane

Let $f$ be a holomorphic function on the strip $\{z\in C: -α 0$, belonging to the class $H(α,-α;ε)$ defined below. It is shown that there exist holomorphic functions $w_1$ on $\{z\in C: 0<Im z <2 α\}$ and $w_2$ on $\{z\in C: -2 α<Im z<2 α\}$ such that $w_1$ and $w_2$ have boundary values of modulus one on the real axis and satisfy the relation $w_1(z)=f(z-αi)w_2(z-2 αi)$ and $w_2(z+2 αi)= \bar{f}(z+αi)w_1(z)$ for $0<Im z<2$, where $\bar{f}(z):=\bar{f(\bar{z})}$. This leads to a "polar decomposition" $f(z)=u_f(z+αi)g_f(z)$ of the function $f(z)$, where $u_f(z+αi)$ and $g_f(z)$ are holomorphic functions for $-α<Im z<α$ such that $|u_f(x)|=1$ and $g_f(x)\ge 0$ a.e. on the real axis. As a byproduct, an operator representation of a $q$-deformed Heisenberg algebra is developed.

math.CV↗
arXiv1998

Commutator Representations of Differential Calculi on the Quantum Group SU_q(2)

Let (Γ,d) be the 3D-calculus or the 4D_{\pm}-calculus on the quantum group SU_q(2). We describe all pairs (π, F) of a *-representation πof O(SU_q(2)) and of a symmetric operator F on the representation space satisfying a technical condition concerning its domain such that there exist a homomorphism of first order differential calculi which maps dx into the commutator [iF,π(x)] for x\in O(SU_q(2)). As an application commutator representations of the 2-dimensional left-covariant calculus on Podles' quantum 2-sphere S^2_{qc} with c=0 are given.

math.QA↗
arXiv1998

On Coquasitriangular Bialgebras

The paper deals with three topics on coquasitriangular bialgebras. A characterization of universal r-forms in terms of Yetter-Drinfeld modules is given. All universal r-forms for the coordinate Hopf algebras of the quantum groups GL_q(N), SL_q(N), O_q(N) and Sp_q(N) are described. It is well-known that the square of the antipode of a coquasitriangular Hopf algebra A with universal r-form r can be expressed by means of the linear functional f_r(a)=r(a_{(1)}\otimes S(a_{(2)})), a\in A. For the standard compact matrix groups the functional f_r is related to the Woronowicz modular character f_{-2}.

math.QA↗