scholarly journals Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

2019 ◽  
Vol 21 (06) ◽  
pp. 1850015
Author(s):  
Laiachi El Kaoutit ◽  
Paolo Saracco
Keyword(s):  
Tensor Product ◽  
Continuous Homomorphism ◽  
Lie Algebroid ◽  
Connected Manifold ◽  
Convolution Algebra ◽  
Hopf Algebroid ◽  
Convolution Algebras ◽  
Hopf Algebroids ◽  
Topological Tensor ◽  
Topological Tensor Product

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.

scholarly journals Homomorphisms of the algebra of locally integrable functions on the half line

2006 ◽  
Vol 81 (2) ◽  
pp. 253-278 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Dual Space ◽  
Continuous Homomorphism ◽  
Half Line ◽  
Unique Extension ◽  
Multiplier Algebra ◽  
Convolution Algebra ◽  
Convolution Algebras ◽  
Integrable Functions ◽  
Bounded Sets ◽  
Nonzero Homomorphism

AbstractLet φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

scholarly journals WEAK-STAR PROPERTIES OF HOMOMORPHISMS FROM WEIGHTED CONVOLUTION ALGEBRAS ON THE HALF-LINE

2010 ◽  
Vol 89 (1) ◽  
pp. 75-90 ◽  
Author(s):  
THOMAS VILS PEDERSEN
Keyword(s):  
Banach Algebras ◽  
Continuous Extension ◽  
Continuous Homomorphism ◽  
Convolution Algebra ◽  
Convolution Algebras ◽  
Weak Star ◽  
Commutative Banach Algebras ◽  
Beurling Algebras ◽  
Weighted Measure ◽  
Nonzero Homomorphism

AbstractLet L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension $\widetilde {\Phi }:M(\omega _1)\to M(\omega _2)$ to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra $A(\overline {\mathbb D})$ every nonzero homomorphism $\Phi :L^1({\omega })\to A(\overline {\mathbb D})$ extends uniquely to a continuous homomorphism $\widetilde {\Phi }:M(\omega )\to H^{\infty }(\mathbb D)$ which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on $\overline {\mathbb D}$ (including the algebra of absolutely convergent Taylor series on $\overline {\mathbb D}$) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism $\widetilde {\Phi }:M(\omega )\to A^+_v$ which is also continuous with respect to the weak-star topologies.

scholarly journals The EllipticGL(n)Dynamical Quantum Group as an𝔥-Hopf Algebroid

2009 ◽  
Vol 2009 ◽  
pp. 1-41 ◽  
Author(s):  
Jonas T. Hartwig
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Spectral Parameter ◽  
Exterior Algebra ◽  
Baxter Equation ◽  
Matrix Elements ◽  
Elliptic Solution ◽  
Hopf Algebroid ◽  
Quantum Dynamical ◽  
Hopf Algebroids

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

Sign in / Sign up

Export Citation Format

Share Document