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In present work, we find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix algebras. Moreover, such Lie algebras generated by semi-positive definite matrices can be classified by the modified Dynkin diagrams., Comment: 14 pages. arXiv admin note: text overlap with arXiv:0904.2432 by other authors

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 $\gim(M_n)$., Comment: 19 pages

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)\cdot\ldots\cdot(x_lg)$ where $g\in G$ and $x_1, \ldots, x_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(x_1+\cdots+x_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=\langle g\rangle$ is a cyclic group of prime power order $n=p^\mu$ with $p \geq 7$ and $\mu\geq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $\gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then $\ind(S)=2$ if and only if $S=(mg)(mg)(m\frac{n-1}{2}g)(m\frac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $\gcd(m,n)=1$.

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$. A conjecture on the index of length four sequences says that every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. The conjecture was confirmed recently for the case when $|G|$ is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. Based on earlier work on this problem, we show that if $G=\langle g\rangle$ is a finite cyclic group of order $|G|=n$ such that $\gcd(n,6)=1$ and $S=(x_1g)(x_2g)(x_3g)(x_4g)$ is a minimal zero-sum sequence over $G$ such that $x_1,\cdots,x_4\in[1,n-1]$ with $\gcd(n,x_1,x_2,x_3,x_4)=1$, and $\gcd(n,x_i)>1$ for some $i\in[1,4]$, then $\ind(S)=1$. By using an innovative method developed in this paper, we are able to give a new (and much shorter) proof to the index conjecture for the case when $|G|$ is a product of two prime powers.

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind S$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\hbox{\rm ord}(g)$ over all possible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$ is finite such that $\gcd(|G|,6)=1$, then $\ind(S)=1$ for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$ is reduced and the (A1) condition is satisfied(see [19])., Comment: arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676 by other authors

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\hbox{\rm ord}(g)$ over all possible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$ is finite such that $\gcd(|G|,6)=1$, then $\ind(S)=1$ for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$ is reduced and at least one $n_i$ coprime to $|G|$., Comment: International Journal of Number Theory (2013). arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676 by other authors

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_kg)$ where $g\in G$ and $n_1,\cdots,n_k\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind S$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_k)/\hbox{\rm ord}(g)$ over all possible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$ is finite such that $\gcd(|G|,6)=1$, then $\ind(S)=1$ for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $|G|$ has two prime factors., Comment: Journal of Number Theory (2013). arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676, arXiv:1401.7981, arXiv:1401.7979

The purposes of this work are to construct a class of homogeneous vertex representations of $C_l^{(1)} \ (l\geq2)$, and to derive a series of product-sum identities. These identities have fine interpretation in number theory.