Simple modules over the multiparameter quantum function algebra at roots of 1

1998 ◽  
Vol 3 (2) ◽  
pp. 115-134 ◽  
Author(s):  
M. Costantini
Keyword(s):  
Function Algebra ◽  
Simple Modules ◽  
Quantum Function

scholarly journals Multiparameter quantum function algebra at roots of 1

1996 ◽  
Vol 306 (1) ◽  
pp. 759-780 ◽  
Author(s):  
M. Costantini ◽  
M. Varagnolo
Keyword(s):  
Function Algebra ◽  
Quantum Function

On the Quantum Function Algebra at Roots of One

2004 ◽  
Vol 32 (6) ◽  
pp. 2377-2383 ◽  
Author(s):  
Mauro Costantini
Keyword(s):  
Function Algebra ◽  
Quantum Function

scholarly journals PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES

2007 ◽  
Vol 49 (3) ◽  
pp. 479-488
Author(s):  
FABIO GAVARINI
Keyword(s):  
Function Algebra ◽  
Direct Application ◽  
Hopf Subalgebra ◽  
Frobenius Morphism ◽  
Quantum Matrices ◽  
Root Of Unity ◽  
Frobenius Structures ◽  
Quantum Function

AbstractLet $G \in \{{\it Mat}_n(\C), {GL}_n(\C), {SL}_n(\C)\}$, let $\Oqg$ be the quantum function algebra – over $\Z [q,q^{-1}]$ – associated to G, and let $\Oeg$ be the specialisation of the latter at a root of unity ϵ, whose order ℓ is odd. There is a quantum Frobenius morphism that embeds $\Og,$ the function algebra of G, in $\Oeg$ as a central Hopf subalgebra, so that $\Oeg$ is a module over $\Og$. When $G = {SL}_n(\C)$, it is known by [3], [4] that (the complexification of) such a module is free, with rank ℓdim(G). In this note we prove a PBW-like theorem for $\Oqg$, and we show that – when G is Matn or GLn – it yields explicit bases of $\Oeg $ over $ \Og$ over $\Og,$. As a direct application, we prove that $\Oegl$ and $\Oem$ are free Frobenius extensions over $\Ogl$ and $\Om$, thus extending some results of [5].

scholarly journals Quantum Function Algebra at Roots of 1

1994 ◽  
Vol 108 (2) ◽  
pp. 205-262 ◽  
Author(s):  
C. Deconcini ◽  
V. Lyubashenko
Keyword(s):  
Function Algebra ◽  
Quantum Function

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

Heat equation with singular potential and singular data

2005 ◽  
Vol 135 (4) ◽  
pp. 863-886 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović ◽  
D. Rajter-Ćirić
Keyword(s):  
Polynomial Growth ◽  
Singular Potential ◽  
Function Algebra ◽  
Energy Estimates ◽  
Generalized Function ◽  
Type Condition ◽  
Classical Energy ◽  
The Cauchy Problem ◽  
White Noise Calculus ◽  
Singular Data

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

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