Browsing by Author "Lu, Kang"
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Item Bethe Ansatz Equations for Orthosymplectic Lie Superalgebras and Self-dual Superspaces(Springer, 2021-12) Lu, Kang; Mukhin, Evgeny; Mathematical Sciences, School of ScienceWe study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $$\mathfrak {osp}_{2m+1|2n}$$and $$\mathfrak {osp}_{2m|2n}$$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $$\mathcal R$$. Under some technical assumptions, we show that the superkernel W of $$\mathcal R$$is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in W and with the set of symmetric complete factorizations of $$\mathcal R$$. In particular, our results apply to the case of even Lie algebras of type D$${}_m$$corresponding to $$\mathfrak {osp}_{2m|0}=\mathfrak {so}_{2m}$$.Item Gaudin models associated to classical Lie algebras(2020-08) Lu, Kang; Mukhin, Evgeny; Its, Alexander; Roeder, Roland; Tarasov, VitalyWe study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.Item Jacobi–Trudi Identity and Drinfeld Functor for Super Yangian(Oxford, 2021-11) Lu, Kang; Mukhin, Evgeny; Mathematical Sciences, School of ScienceWe show that the quantum Berezinian that gives a generating function of the integrals of motions of XXX spin chains associated to super Yangian $\textrm{Y}(\mathfrak{g}\mathfrak{l}_{m|n})$ can be written as a ratio of two difference operators of orders $m$ and $n$ whose coefficients are ratios of transfer matrices corresponding to explicit skew Young diagrams. In the process, we develop several missing parts of the representation theory of $\textrm{Y}(\mathfrak{g}\mathfrak{l}_{m|n})$ such as $q$-character theory, Jacobi–Trudi identity, Drinfeld functor, extended T-systems, and Harish-Chandra map.Item Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus(National Academy of Science of Ukraine, 2018) Lu, Kang; Mathematical Sciences, School of ScienceThe self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.Item On the Gaudin model associated to Lie algebras of classical types(AIP, 2016-10) Lu, Kang; Mukhin, Eugene; Varchenko, A.; Department of Mathematical Sciences, School of ScienceWe derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.Item On the Gaudin model of type G2(World Scientific, 2018) Lu, Kang; Mukhin, Evgeny; Mathematical Sciences, School of ScienceWe derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G2. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G2 are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.Item On the Supersymmetric XXX Spin Chains Associated to gl1|1(Springer, 2021-09) Lu, Kang; Mukhin, Evgeny; Mathematical Sciences, School of ScienceYangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.