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 $mu$, 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 $mu$ is nonzero.
We show that Yaus 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.
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 $gim(M_n)$.
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.
Linear codes are considered over the ring $mathbb{Z}_4+vmathbb{Z}_4$, where $v^2=v$. Gray weight, Gray maps for linear codes are defined and MacWilliams identity for the Gray weight enumerator is given. Self-dual codes, construction of Euclidean isodual codes, unimodular complex lattices, MDS codes and MGDS codes over $mathbb{Z}_4+vmathbb{Z}_4$ are studied. Cyclic codes and quadratic residue codes are also considered. Finally, some examples for illustrating the main work are given.
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 Persssons surface and Tans surfaces with odd canonical degrees explicitly.
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 $Delta_{B_r}$, where $r geq 3$. We show that im(A^d) is isomorphic to the universal covering algebra of $so_{2r+1}(a,eta,C,chi)$, where $a$ is an associative algebra with involution $eta$, and $C$ is an $a$-module with hermitian form $chi$. We provide a description of all four of the components $a$, $eta$, $C$, and $chi$.
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.
