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Zhuohui Zhang - One of the best experts on this subject based on the ideXlab platform.
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Intertwining Operator for $Sp(4,\mathbb{R})$ and Orthogonal Polynomials
arXiv: Representation Theory, 2019Co-Authors: Zhuohui ZhangAbstract:We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an improvement to the method to calculate the Intertwining Operators. We have shown that the matrix entries of the simple Intertwining Operators for $Sp(4,\mathbb{R})$-principal series are Hahn polynomials, and the matrix entries of the long Intertwining Operator can be expressed as the constant term of the Laurent expansion of some hypergeometric function.
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Intertwining Operator for sp 4 mathbb r and orthogonal polynomials
arXiv: Representation Theory, 2019Co-Authors: Zhuohui ZhangAbstract:We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an improvement to the method to calculate the Intertwining Operators. We have shown that the matrix entries of the simple Intertwining Operators for $Sp(4,\mathbb{R})$-principal series are Hahn polynomials, and the matrix entries of the long Intertwining Operator can be expressed as the constant term of the Laurent expansion of some hypergeometric function.
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Principal Series Representation of $SU(2,1)$ and Its Intertwining Operator
arXiv: Representation Theory, 2019Co-Authors: Zhuohui ZhangAbstract:In this paper, following a similar procedure developed by Buttcane and Miller in \cite{MillerButtcane} for $SL(3,\RR)$, the $(\frakg,K)$-module structure of the minimal principal series of real reductive Lie groups $SU(2,1)$ is described explicitly by realizing the representations in the space of $K$-finite functions on $U(2)$. Moreover, by combining combinatorial techniques and contour integrations, this paper introduces a method of calculating Intertwining Operators on the principal series. Upon restriction to each $K$-type, the matrix entries of Intertwining Operators are represented by $\Gamma$-functions and Laurent series coefficients of hypergeometric series. The calculation of the $(\frakg,K)$-module structure of principal series can be generalized to real reductive Lie groups whose maximal compact subgroup is a product of $SU(2)$'s and $U(1)$'s.
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principal series representation of su 2 1 and its Intertwining Operator
arXiv: Representation Theory, 2019Co-Authors: Zhuohui ZhangAbstract:In this paper, following a similar procedure developed by Buttcane and Miller in \cite{MillerButtcane} for $SL(3,\RR)$, the $(\frakg,K)$-module structure of the minimal principal series of real reductive Lie groups $SU(2,1)$ is described explicitly by realizing the representations in the space of $K$-finite functions on $U(2)$. Moreover, by combining combinatorial techniques and contour integrations, this paper introduces a method of calculating Intertwining Operators on the principal series. Upon restriction to each $K$-type, the matrix entries of Intertwining Operators are represented by $\Gamma$-functions and Laurent series coefficients of hypergeometric series. The calculation of the $(\frakg,K)$-module structure of principal series can be generalized to real reductive Lie groups whose maximal compact subgroup is a product of $SU(2)$'s and $U(1)$'s.
Yi-zhi Huang - One of the best experts on this subject based on the ideXlab platform.
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Intertwining Operators among twisted modules associated to not-necessarily-commuting automorphisms
Journal of Algebra, 2017Co-Authors: Yi-zhi HuangAbstract:Abstract We introduce Intertwining Operators among twisted modules or twisted Intertwining Operators associated to not-necessarily-commuting automorphisms of a vertex Operator algebra. Let V be a vertex Operator algebra and let g 1 , g 2 and g 3 be automorphisms of V. We prove that for g 1 -, g 2 - and g 3 -twisted V-modules W 1 , W 2 and W 3 , respectively, such that the vertex Operator map for W 3 is injective, if there exists a twisted Intertwining Operator of type ( W 3 W 1 W 2 ) such that the images of its component Operators span W 3 , then g 3 = g 1 g 2 . We also construct what we call the skew-symmetry and contragredient isomorphisms between spaces of twisted Intertwining Operators among twisted modules of suitable types. The proofs of these results involve careful analysis of the analytic extensions corresponding to the actions of the not-necessarily-commuting automorphisms of the vertex Operator algebra.
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Intertwining Operators among twisted modules associated to not-necessarily-commuting automorphisms
arXiv: Quantum Algebra, 2017Co-Authors: Yi-zhi HuangAbstract:We introduce Intertwining Operators among twisted modules or twisted Intertwining Operators associated to not-necessarily-commuting automorphisms of a vertex Operator algebra. Let $V$ be a vertex Operator algebra and let $g_{1}$, $g_{2}$ and $g_{3}$ be automorphisms of $V$. We prove that for $g_{1}$-, $g_{2}$- and $g_{3}$-twisted $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$, respectively, such that the vertex Operator map for $W_{3}$ is injective, if there exists a twisted Intertwining Operator of type ${W_{3}\choose W_{1}W_{2}}$ such that the images of its component Operators span $W_{3}$, then $g_{3}=g_{1}g_{2}$. We also construct what we call the skew-symmetry and contragredient isomorphisms between spaces of twisted Intertwining Operators among twisted modules of suitable types. The proofs of these results involve careful analysis of the analytic extensions corresponding to the actions of the not-necessarily-commuting automorphisms of the vertex Operator algebra.
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Full Field Algebras
Communications in Mathematical Physics, 2007Co-Authors: Yi-zhi Huang, Liang KongAbstract:We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex Operator algebras V ^ L and V ^ R , $${V^{L}\otimes V^{R}}$$ is naturally a full field algebra and we introduce a notion of full field algebra over $${V^{L}\otimes V^{R}}$$ . We study the structure of full field algebras over $${V^{L}\otimes V^{R}}$$ using modules and Intertwining Operators for V ^ L and V ^ R . For a simple vertex Operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of Intertwining Operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of Intertwining Operator algebras but also on the modular invariance for Intertwining Operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over $${V\otimes V}$$ and an invariant bilinear form on this algebra.
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on the concepts of Intertwining Operator and tensor product module in vertex Operator algebra theory
arXiv: Quantum Algebra, 2004Co-Authors: Yi-zhi Huang, James Lepowsky, Lin ZhangAbstract:We produce counterexamples to show that in the definition of the notion of Intertwining Operator for modules for a vertex Operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We further give a sufficient condition for the commutator formula to imply the Jacobi identity in this definition. Using these results we illuminate the crucial role of the condition called the ``compatibility condition'' in the construction of the tensor product module in vertex Operator algebra theory, as carried out in work of Huang and Lepowsky. In particular, we prove by means of suitable counterexamples that the compatibility condition was indeed needed in this theory.
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Intertwining Operator superalgebras and vertex tensor categories for superconformal algebras i
Communications in Contemporary Mathematics, 2002Co-Authors: Yi-zhi Huang, Antun MilasAbstract:We apply the general theory of tensor products of modules for a vertex Operator algebra (developed by Lepowsky and the first author) and the general theory of Intertwining Operator algebras (developed by the first author) to the case of the N=1 superconformal minimal models and related models in superconformal field theory. We show that for the category of modules for a vertex Operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex Operator superalgebras associated to the N =1 Neveu–Schwarz Lie superalgebra, the Intertwining Operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of a braided tensor category structure on the category of finite direct sums of minimal modules of central charge for the N = 1 Neveu–Schwarz Lie superalgebra for any fixed integers p, q larger than 1 such that p - q ∈ 2ℤ and (p - q)/2 and q relatively prime to each other.
Antun Milas - One of the best experts on this subject based on the ideXlab platform.
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Intertwining Operator superalgebras and vertex tensor categories for superconformal algebras i
Communications in Contemporary Mathematics, 2002Co-Authors: Yi-zhi Huang, Antun MilasAbstract:We apply the general theory of tensor products of modules for a vertex Operator algebra (developed by Lepowsky and the first author) and the general theory of Intertwining Operator algebras (developed by the first author) to the case of the N=1 superconformal minimal models and related models in superconformal field theory. We show that for the category of modules for a vertex Operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex Operator superalgebras associated to the N =1 Neveu–Schwarz Lie superalgebra, the Intertwining Operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of a braided tensor category structure on the category of finite direct sums of minimal modules of central charge for the N = 1 Neveu–Schwarz Lie superalgebra for any fixed integers p, q larger than 1 such that p - q ∈ 2ℤ and (p - q)/2 and q relatively prime to each other.
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Intertwining Operator superalgebras and vertex tensor categories for superconformal algebras, II
Transactions of the American Mathematical Society, 2001Co-Authors: Yi-zhi Huang, Antun MilasAbstract:We construct the Intertwining Operator superalgebras and vertex tensor categories for the N = 2 superconformal unitary minimal models and other related models.
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Intertwining Operator superalgebras and vertex tensor categories for superconformal algebras, II
arXiv: Quantum Algebra, 1999Co-Authors: Yi-zhi Huang, Antun MilasAbstract:This is Part I of a series of papers constructing Intertwining Operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the Intertwining Operator superalgebras and vertex tensor categories in the N=1 case.
Demni Nizar - One of the best experts on this subject based on the ideXlab platform.
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Markov semi-groups associated with the complex unimodular group $Sl(2,\mathbb{C})$
2019Co-Authors: Demni NizarAbstract:In this paper, we derive the explicit expressions of two Markov semi-groups constructed by P. Biane in \cite{Bia1} from the restriction of a particular positive definite function on the complex unimodular group $Sl(2,\mathbb{C})$ to two commutative subalgebras of its universal $C^{\star}$-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $-1$, and yield absolutely-convergent double series representations of the semi-group densities. In the last part of the paper, we discuss the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $Mp(4,\mathbb{R})$ and to the Landau Operator in the complex plane.Comment: The Intertwining Operator is derived in the case of principal serie
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Markov semi-groups associated with the complex unimodular group $Sl(2,\mathbb{C})$
Springer Verlag, 2019Co-Authors: Demni NizarAbstract:The Intertwining Operator is derived in the case of principal seriesInternational audienceIn this paper, we derive the explicit expressions of two Markov semi-groups constructed by P. Biane in \cite{Bia1} from the restriction of a particular positive definite function on the complex unimodular group $Sl(2,\mathbb{C})$ to two commutative subalgebras of its universal $C^{\star}$-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $-1$, and yield absolutely-convergent double series representations of the semi-group densities. In the last part of the paper, we discuss the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $Mp(4,\mathbb{R})$ and to the Landau Operator in the complex plane
Bin Gui - One of the best experts on this subject based on the ideXlab platform.
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Regular vertex Operator subalgebras and compressions of Intertwining Operators
Journal of Algebra, 2020Co-Authors: Bin GuiAbstract:Abstract Let V be a vertex Operator subalgebra of U. Assume that U, V, and its commutant V c in U are CFT-type, self-dual, and regular VOAs. Assume also that the double commutant V c c equals V. We prove that any Intertwining Operator of V is a compression of Intertwining Operators of U.
