The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
Brendan Nolan - One of the best experts on this subject based on the ideXlab platform.
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A strong Dixmier–Moeglin equivalence for quantum Schubert cells
Journal of Algebra, 2017Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Abstract Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier–Moeglin equivalence. We define quantities which measure how “close” an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the Algebra satisfies the strong Dixmier–Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl 2 ( C ) , we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence. For a simple complex Lie Algebra g , a non-root of unity q ≠ 0 in an infinite field K , and an element w of the Weyl group of g , De Concini, Kac, and Procesi have constructed a subAlgebra U q [ w ] of the quantised Enveloping K -Algebra U q ( g ) . These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.
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A strong Dixmier-Moeglin equivalence for quantum Schubert cells
arXiv: Quantum Algebra, 2015Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to each of these three properties, then we say that the Algebra satisfies the strong Dixmier-Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl_2(C), we show that the strong Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie Algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subAlgebra U_q[w] of the quantised Enveloping K-Algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the strong Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier-Moeglin equivalence.
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A generalised Dixmier-Moeglin equivalence for quantum Schubert cells
arXiv: Quantum Algebra, 2015Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to these three properties, then we say that the Algebra satisfies the generalised Dixmier-Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl_2(C), we show that the generalised Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie Algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subAlgebra U_q[w] of the quantised Enveloping K-Algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the generalised Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the generalised Dixmier-Moeglin equivalence.
Jason P. Bell - One of the best experts on this subject based on the ideXlab platform.
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A strong Dixmier–Moeglin equivalence for quantum Schubert cells
Journal of Algebra, 2017Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Abstract Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier–Moeglin equivalence. We define quantities which measure how “close” an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the Algebra satisfies the strong Dixmier–Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl 2 ( C ) , we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence. For a simple complex Lie Algebra g , a non-root of unity q ≠ 0 in an infinite field K , and an element w of the Weyl group of g , De Concini, Kac, and Procesi have constructed a subAlgebra U q [ w ] of the quantised Enveloping K -Algebra U q ( g ) . These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.
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A strong Dixmier-Moeglin equivalence for quantum Schubert cells
arXiv: Quantum Algebra, 2015Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to each of these three properties, then we say that the Algebra satisfies the strong Dixmier-Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl_2(C), we show that the strong Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie Algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subAlgebra U_q[w] of the quantised Enveloping K-Algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the strong Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier-Moeglin equivalence.
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A generalised Dixmier-Moeglin equivalence for quantum Schubert cells
arXiv: Quantum Algebra, 2015Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to these three properties, then we say that the Algebra satisfies the generalised Dixmier-Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl_2(C), we show that the generalised Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie Algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subAlgebra U_q[w] of the quantised Enveloping K-Algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the generalised Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the generalised Dixmier-Moeglin equivalence.
Fokko J. Van De Bult - One of the best experts on this subject based on the ideXlab platform.
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Ruijsenaars' hypergeometric function and the modular double of U-q(sl(2)(C))
Advances in Mathematics, 2006Co-Authors: Fokko J. Van De BultAbstract:Abstract Simultaneous eigenfunctions of two Askey–Wilson second-order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized Universal Enveloping Algebra U q ( sl 2 ( C ) ) . These eigenfunctions are shown to be equal to Ruijsenaars’ hypergeometric function under a proper parameter correspondence.
Edward Salamanca - One of the best experts on this subject based on the ideXlab platform.
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on the combinatorics of the Universal Enveloping Algebra widehat u _h mathfrak sl _2
The São Paulo Journal of Mathematical Sciences, 2019Co-Authors: Rafael Diaz, Edward SalamancaAbstract:Using combinatorial methods we study the structural coefficients of the formal homogeneous Universal Enveloping Algebra \(\widehat{U}_h({\mathfrak {sl}}_2) \) of the special linear Algebra \( {\mathfrak {sl}}_2\) over a characteristic zero field. We provide explicit formulae for the product of generic elements in \( \widehat{U}_h({\mathfrak {sl}}_2),\) and construct combinatorial objects giving flesh to these formulae; in particular, we provide explicit formulae and combinatorial interpretations for the structural coefficients of divided power Poincare–Birkhoff–Witt basis.
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On the combinatorics of the Universal Enveloping Algebra \(\widehat{U}_h({{\mathfrak {sl}}}_2)\)
São Paulo Journal of Mathematical Sciences, 2018Co-Authors: Rafael Diaz, Edward SalamancaAbstract:Using combinatorial methods we study the structural coefficients of the formal homogeneous Universal Enveloping Algebra \(\widehat{U}_h({\mathfrak {sl}}_2) \) of the special linear Algebra \( {\mathfrak {sl}}_2\) over a characteristic zero field. We provide explicit formulae for the product of generic elements in \( \widehat{U}_h({\mathfrak {sl}}_2),\) and construct combinatorial objects giving flesh to these formulae; in particular, we provide explicit formulae and combinatorial interpretations for the structural coefficients of divided power Poincare–Birkhoff–Witt basis.
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On the Combinatorics of the Universal Enveloping Algebra Uh(sl2)
arXiv: Quantum Algebra, 2016Co-Authors: Rafael Diaz, Edward SalamancaAbstract:We study using combinatorial methods the structural coefficients of the formal homogeneous Universal Enveloping Algebra Uh(sl2) of the special linear Algebra sl2 over a field of characteristic zero. We provide explicit formulae for the product of generic elements in Uh(sl2), and construct combinatorial objects giving flesh to these formulae.
S Launois - One of the best experts on this subject based on the ideXlab platform.
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A strong Dixmier–Moeglin equivalence for quantum Schubert cells
Journal of Algebra, 2017Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Abstract Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier–Moeglin equivalence. We define quantities which measure how “close” an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the Algebra satisfies the strong Dixmier–Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl 2 ( C ) , we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence. For a simple complex Lie Algebra g , a non-root of unity q ≠ 0 in an infinite field K , and an element w of the Weyl group of g , De Concini, Kac, and Procesi have constructed a subAlgebra U q [ w ] of the quantised Enveloping K -Algebra U q ( g ) . These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.
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A strong Dixmier-Moeglin equivalence for quantum Schubert cells
arXiv: Quantum Algebra, 2015Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to each of these three properties, then we say that the Algebra satisfies the strong Dixmier-Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl_2(C), we show that the strong Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie Algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subAlgebra U_q[w] of the quantised Enveloping K-Algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the strong Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier-Moeglin equivalence.
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A generalised Dixmier-Moeglin equivalence for quantum Schubert cells
arXiv: Quantum Algebra, 2015Co-Authors: Jason P. Bell, S Launois, Brendan NolanAbstract:Dixmier and Moeglin gave an Algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the Universal Enveloping Algebra of a finite-dimensional complex Lie Algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian Algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to these three properties, then we say that the Algebra satisfies the generalised Dixmier-Moeglin equivalence. Using the example of the Universal Enveloping Algebra of sl_2(C), we show that the generalised Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie Algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subAlgebra U_q[w] of the quantised Enveloping K-Algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the generalised Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the generalised Dixmier-Moeglin equivalence.
