Nonlocal Matrix Generalizations of N=2 Super Virasoro Algebra

We study the generalization of second Gelfand-Dickey bracket to the superdifferential operators with matrix-valued coefficients. The associated Miura transformation is derived. Using this bracket we work out a nonlocal and nonlinear N=2 superalgebra which contains the N=2 super Virasoro algebra as a subalgebra. The bosonic limit of this algebra is considered. We show that when the spin-1 fields in this bosonic algebra are set to zero the resulting Dirac bracket gives precisely the recently derived $V_{2,2}$ algebra.Comment: 14 pages (Plain TeX), NHCU-HEP-94-2

The flipping puzzle on a graph

Let $S$ be a connected graph which contains an induced path of $n-1$ vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A configuration of the puzzle is simply an $n$-dimensional column vector over $\{0, 1\}$ with coordinates of the vector indexed by the vertex set $S$. For each configuration $u$ with a coordinate $u_s=1$, there exists a move that sends $u$ to the new configuration which flips the entries of the coordinates adjacent to $s$ in $u.$ We completely determine if one configuration can move to another in a sequence of finite steps.Comment: 18 pages, 1 figure and 1 tabl

The edge-flipping group of a graph

Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge $\epsilon\in E$ and change the colors of all adjacent edges of $\epsilon.$ Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on $X,$ and it corresponds to a group action. This group is called the edge-flipping group $\mathbf{W}_E(X)$ of $X.$ This paper shows that if $X$ has at least three vertices, $\mathbf{W}_E(X)$ is isomorphic to a semidirect product of $(\mathbb{Z}/2\mathbb{Z})^k$ and the symmetric group $S_n$ of degree $n,$ where $k=(n-1)(m-n+1)$ if $n$ is odd, $k=(n-2)(m-n+1)$ if $n$ is even, and $\mathbb{Z}$ is the additive group of integers.Comment: 19 page

The universal DAHA of type (C1∨,C1)(C_1^\vee,C_1) and Leonard triples

Assume that $\mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra generated by $A,B, C$ and the relations state that each of $$A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}}$$ is central in $\triangle_q$. The universal DAHA $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations state that \begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\ \hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\ t_0t_1t_2t_3=q^{-1}. \end{gather*} It was given an $\mathbb F$-algebra homomorphism $\triangle_q\to \mathfrak H_q$ that sends \begin{eqnarray*} A &\mapsto & t_1 t_0+(t_1 t_0)^{-1}, \\ B &\mapsto & t_3 t_0+(t_3 t_0)^{-1}, \\ C &\mapsto & t_2 t_0+(t_2 t_0)^{-1}. \end{eqnarray*} Therefore any $\mathfrak H_q$-module can be considered as a $\triangle_q$-module. Let $V$ denote a finite-dimensional irreducible $\mathfrak H_q$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $\triangle_q$-module $V$.Comment: arXiv admin note: text overlap with arXiv:2003.0625