The Experts below are selected from a list of 252 Experts worldwide ranked by ideXlab platform
Christiane Quesne - One of the best experts on this subject based on the ideXlab platform.
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Non-Hermitian Hamiltonians with real and Complex eigenvalues: An sl(2,C) approach
arXiv: Mathematical Physics, 2002Co-Authors: Bijan Bagchi, Christiane QuesneAbstract:Potential Algebras are extended from Hermitian to non-Hermitian Hamiltonians and shown to provide an elegant method for studying the transition from real to Complex eigenvalues for a class of non-Hermitian Hamiltonians associated with the Complex Lie Algebra A$_1$.
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sl(2, C) as a Complex Lie Algebra and the associated nonHermitian Hamiltonians with real eigenvalues
Physics Letters B, 2000Co-Authors: Bijan Bagchi, Christiane QuesneAbstract:The powerful group theoretical formalism of potential Algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by Complexifying so(2,1), thereby getting the Complex Algebra sl(2,\C) or $A_1$. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.
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sl 2 c as a Complex Lie Algebra and the associated non hermitian hamiltonians with real eigenvalues
arXiv: Mathematical Physics, 2000Co-Authors: Bijan Bagchi, Christiane QuesneAbstract:The powerful group theoretical formalism of potential Algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by Complexifying so(2,1), thereby getting the Complex Algebra sl(2,\C) or $A_1$. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.
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, Stéphane 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, Stéphane 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, Stéphane 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.
Catharina Stroppel - One of the best experts on this subject based on the ideXlab platform.
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Projective-injective modules, Serre functors and symmetric Algebras
Crelle's Journal, 2020Co-Authors: Volodymyr Mazorchuk, Catharina StroppelAbstract:We describe Serre functors for (generalisations of) the category O associated with a semisimple Complex Lie Algebra. In our approach, projective-injective modules, that is modules which are both, ...
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A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products
Selecta Mathematica, 2007Co-Authors: Igor Frenkel, Mikhail Khovanov, Catharina StroppelAbstract:The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for $${\mathfrak{sl}_2}$$ . The main categorification is obtained using certain Harish-Chandra bimodules for the Complex Lie Algebra $${\mathfrak{gl}_n}$$ . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.
Alexander Stolin - One of the best experts on this subject based on the ideXlab platform.
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Classification of Quantum Groups and Belavin–Drinfeld Cohomologies
Communications in Mathematical Physics, 2016Co-Authors: Boris Kadets, Eugene Karolinsky, Alexander StolinAbstract:In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple Complex Lie Algebra $${\mathfrak{g}}$$ g . This problem is reduced to the classification of all Lie biAlgebra structures on $${\mathfrak{g}(\mathbb{K})}$$ g ( K ) , where $${\mathbb{K}=\mathbb{C}((\hbar))}$$ K = C ( ( ħ ) ) . The associated classical double is of the form $${\mathfrak{g}(\mathbb{K})\otimes_{\mathbb{K}} A}$$ g ( K ) ⊗ K A , where A is one of the following: $${\mathbb{K}[\varepsilon]}$$ K [ ε ] , where $${\varepsilon^{2}=0}$$ ε 2 = 0 , $${\mathbb{K}\oplus\mathbb{K}}$$ K ⊕ K or $${\mathbb{K}[j]}$$ K [ j ] , where $${j^{2}=\hbar}$$ j 2 = ħ . The first case is related to quasi-Frobenius Lie Algebras. In the second and third cases we introduce a theory of Belavin–Drinfeld cohomology associated to any non-skewsymmetric r -matrix on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984 ). We prove a one-to-one correspondence between gauge equivalence classes of Lie biAlgebra structures on $${\mathfrak{g}(\mathbb{K})}$$ g ( K ) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r -matrix.
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Irreducible highest weight modules and equivariant quantization
Advances in Mathematics, 2007Co-Authors: E. Karolinsky, Alexander Stolin, Vitaly TarasovAbstract:We consider the relationship between the Shapovalov form on an irreducible highest weight module of a semisimple Complex Lie Algebra, fusion elements, and equivariant quantization. We also discuss some limiting properties of fusion elements.
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Constant solutions of Yang-Baxter equation for $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$.
Mathematica Scandinavica, 1991Co-Authors: Alexander StolinAbstract:All constant solutions of the classical Yang-Baxter equation (CYBE) are listed for the function with values in sl(2) and sl(3) and an algorithm which allows one to obtain all constant solutions for a simple Complex Lie Algebra g is given.
Bijan Bagchi - One of the best experts on this subject based on the ideXlab platform.
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Non-Hermitian Hamiltonians with real and Complex eigenvalues: An sl(2,C) approach
arXiv: Mathematical Physics, 2002Co-Authors: Bijan Bagchi, Christiane QuesneAbstract:Potential Algebras are extended from Hermitian to non-Hermitian Hamiltonians and shown to provide an elegant method for studying the transition from real to Complex eigenvalues for a class of non-Hermitian Hamiltonians associated with the Complex Lie Algebra A$_1$.
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sl(2, C) as a Complex Lie Algebra and the associated nonHermitian Hamiltonians with real eigenvalues
Physics Letters B, 2000Co-Authors: Bijan Bagchi, Christiane QuesneAbstract:The powerful group theoretical formalism of potential Algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by Complexifying so(2,1), thereby getting the Complex Algebra sl(2,\C) or $A_1$. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.
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sl 2 c as a Complex Lie Algebra and the associated non hermitian hamiltonians with real eigenvalues
arXiv: Mathematical Physics, 2000Co-Authors: Bijan Bagchi, Christiane QuesneAbstract:The powerful group theoretical formalism of potential Algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by Complexifying so(2,1), thereby getting the Complex Algebra sl(2,\C) or $A_1$. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.