# Research

### Research Interests

Generally speaking, I am interested in Representation Theory and Quantum Algebras.
More specifically, my current research interests are: Bethe ansatz method, Representation Theory of Quantum Groups, Lie Algebras and Superalgebras, and Integrable System.

### Publications

1. On the Gaudin model of type G$_2$
(with E. Mukhin), accepted by Commun. Contemp. Math. (arXiv:1711.02511)

Abstract: We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G$_2$. 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 G$_2$ 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.
2. Lower bounds for numbers of real self-dual spaces in problems of Schubert calculus
18 Oct. 2017, (arXiv:1710.06534)

Abstract: The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebra 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 $\mathrm{Gr}(N,d)$. 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.
3. Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, $\mathfrak{sp}_{2r}$, $\mathfrak{so}_{2r+1}$
(with E. Mukhin, A. Varchenko), 8 May. 2017, (arXiv:1705.02048)

Abstract: We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{Gr}(N,d)$. We introduce and study the new object: the self-dual Grassmannian $\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar $\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra of types B and C.
4. Multiplicity free gradings on semisimple Lie and Jordan algebras and skew root systems
(with Gang Han, Yucheng Liu), 12 Nov. 2016, (arXiv:1611.03943)

Abstract: A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type respectively, and use them to construct multiplicity free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp. Jordan) algebras are simple. Three families of skew root systems of Lie type (resp. of Jordan type) are constructed and the corresponding Lie (resp. Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.
5. On the Gaudin model associated to Lie algebras of classical types
(with E. Mukhin, A. Varchenko), J. Math. Phys. 57, 101703 (2016) (arXiv:1512.08524)

Abstract: 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.
6. Fine gradings of complex simple Lie algebras and Finite Root Systems
(with Gang Han, Jun Yu), 29 Oct. 2014. (arXiv:1410.7945)

Abstract: Analogous to classical root systems, we define a finite root system $R$ to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to $R$ a semisimple Lie algebra $L(R)$ together with a quasi-good grading on it. Thus one can construct nice basis of $L(R)$ by means of finite root systems. We classify finite maximal abelian subgroups $T$ in $\mathrm{Aut}(L)$ for complex simple Lie algebras $L$ such that the grading induced by the action of $T$ on $L$ is quasi-good, and show that the set of roots of $T$ in $L$ is always a finite root system. There are five series of such finite maximal abelian subgroups, which occur only if $L$ is a classical simple Lie algebra.