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Yun Gao

Publications and source records attributed to Yun Gao.

54 records · Page 3Linked to original sources
arXiv2014

A family of representations of the affine Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $μ$, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter $μ$ is nonzero.

math.RT↗
arXiv2014

Canonical maps of surfaces defined by Abelian covers

In this paper, we classified the surfaces whose canonical maps are abelian covers over $\mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $\mathbb{Z}_2^{\oplus 3}$ and give defining equations for Perssson's surface and Tan's surfaces with odd canonical degrees explicitly.

math.AG↗
arXiv2014

On the Griffiths numbers for higher dimensional singularities

We show that Yau's conjecture on the inequalities for (n-1)-th Griffiths number and (n-1)-th Hironaka number does not hold for isolated rigid Gorenstein singularities of dimension greater than 2. But his conjecture on the inequality for (n-1)-th Griffiths number is true for irregular singularities.

math.AG↗
arXiv2014

Finite-dimensional representations for a class of generalized intersection matrix Lie algebras

In this paper, we study a class of generalized intersection matrix Lie algebras $\gim(M_n)$, and prove that its every finite-dimensional semi-simple quotient is of type $M(n,{\bf a}, {\bf c},{\bf d})$. Particularly, any finite dimensional irreducible $\gim(M_n)$ module must be an irreducible module of $M(n,{\bf a}, {\bf c},{\bf d})$ and any finite dimensional irreducible $M(n,{\bf a}, {\bf c},{\bf d})$ module must be an irreducible module of $\gim(M_n)$.

math.QA↗
arXiv2014

Twisted vertex operators and unitary Lie algebras

A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.

math.QA↗
arXiv2010

Realizations of BC_r graded intersection matrix algebras with grading subalgebras of type B_r, $r \geq 3$

We study intersection matrix algebras im(A^d) that arise from affinizing a Cartan matrix A of type B_r with d arbitrary long roots in the root system $Δ_{B_r}$, where $r \geq 3$. We show that im(A^d) is isomorphic to the universal covering algebra of $so_{2r+1}(a,η,C,χ)$, where $a$ is an associative algebra with involution $η$, and $C$ is an $a$-module with hermitian form $χ$. We provide a description of all four of the components $a$, $η$, $C$, and $χ$.

math.RA↗
arXiv2008

A quantized Tits-Kantor-Koecher algebra

We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space.

math.QA↗
arXiv2008

From the entropy to the statistical structure of spike trains

We use statistical estimates of the entropy rate of spike train data in order to make inferences about the underlying structure of the spike train itself. We first examine a number of different parametric and nonparametric estimators (some known and some new), including the ``plug-in'' method, several versions of Lempel-Ziv-based compression algorithms, a maximum likelihood estimator tailored to renewal processes, and the natural estimator derived from the Context-Tree Weighting method (CTW). The theoretical properties of these estimators are examined, several new theoretical results are developed, and all estimators are systematically applied to various types of synthetic data and under different conditions. Our main focus is on the performance of these entropy estimators on the (binary) spike trains of 28 neurons recorded simultaneously for a one-hour period from the primary motor and dorsal premotor cortices of a monkey. We show how the entropy estimates can be used to test for the existence of long-term structure in the data, and we construct a hypothesis test for whether the renewal process model is appropriate for these spike trains. Further, by applying the CTW algorithm we derive the maximum a posterior (MAP) tree model of our empirical data, and comment on the underlying structure it reveals.

q-bio.NC↗
arXiv2005

Universal coverings of Steinberg Lie algebras of small characteristic

It is well-known that the second homology group $H_2(\st)$ of the Steinberg Lie algebra $\st$ is trivial when $n\geq 5$. In this paper, we will work out $H_2(\st)$ explicitly for $n=3, 4$ which are not necessarily trivial. Consequently, we obtained $H_2(sl_n(R))$ for $n=3, 4$.

math.QA↗
arXiv2005

Representations of Two-parameter Quantum Orthogonal and Symplectic Groups

We investigate the finite-dimensional representation theory of two-parameter quantum orthogonal and symplectic groups that we found in [BGH] under the assumption that $rs^{-1}$ is not a root of unity and extend some results [BW1, BW2] obtained for type $A$ to types $B$, $C$ and $D$. We construct the corresponding $R$-matrices and the quantum Casimir operators, by which we prove that the complete reducibility Theorem also holds for the categories of finite-dimensional weight modules for types $B$, $C$, $D$.

math.QA↗
arXiv2005

Drinfel'd Doubles and Lusztig's Symmetries of Two-Parameter Quantum Groups

We find the defining structures of two-parameter quantum groups $U_{r,s}(\frak g)$ corresponding to the orthogonal and the symplectic Lie algebras, which are realized as Drinfel'd doubles. We further investigate the environment conditions upon which the Lusztig's symmetries exist between $(U_{r,s}(\frak g), < ,>)$ and its associated object $(U_{s^{-1}, r^{-1}}(\frak g), < | >)$.

math.RT↗
arXiv2002

$U_q(\hat{\frak{gl}}_{{}_N})$ action on $\hat{\frak{gl}}_{{}_N}$ -modules and quantum toroidal algebras

We construct a vertex representation for the quantum toroidal algebra through the quantum general linear algebra. Using a new realization of the quantum general linear algebra we construct vertex operators for root vectors on the basic representation of the affine Lie algebra $gl_n$ and show that the simple generators give rise a realization of the quantum toroidal algebra with two parameters.

math.QA↗
arXiv2001

A unified view of some vertex operator constructions

We present a general vertex operator construction based on the Fock space for an affine Lie algebras of type $A$. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie algebras $\hat{gl}_N$ as well as for some extended affine Lie algebras coordinatized by certain quantum tori.

math.QA↗