Related papers: The Trigonometric $E_8$ R-matrix
We describe a representation for $U_q(\widehat{sl(n)})$, when $q$ is not a root of unity, based on the fundamental representation of $sl(n)$. As $U_q(sl(n))$ has a Hopf algebra structure with a non-commutative co-product, we look for a…
The well known formulas express the curvature and the torsion of a curve in $R^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in $R^n$. It follows that a curve in…
We present a relation between the Witt invariants of 3-manifolds and the $\hat{Z}$-invariants. It provides an alternative approach to compute the Witt invariants of 3-manifolds, which were originally defined geometrically in four…
Properties of certain $q$-orthogonal polynomials are connected to the $q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to arise as matrix elements of $q$-exponentials of the generators in a representation of this…
A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices…
We introduce a universal R matrix for the Jordanian deformation of $\U{ \sl(2)}$. Using $\Uh{\so(4)}=\Uh{\sl(2)} \oplus {\rm U}_{-h}(\sl(2))$, we obtain the universal R matrix for $\Uh{\so(4)}$. Applying the graded contractions on the…
Reciprocal matrices are tridiagonal matrices $(a_{ij})_{i,j=1}^n$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,\ldots,n-1$. For these matrices, criteria are established under which their Kippenhahn curves…
A formula is given for the Seiberg-Witten invariants of a 4-manifold that is cut along certain kinds of 3-dimensional tori. The formula involves a Seiberg-Witten invariant for each of the resulting pieces.
The representation dimension of an artin algebra as introduced by M.Auslander in his Queen Mary Notes is the minimal possible global dimension of the endomorphism ring of a generator-cogenerator. The paper is based on two texts written in…
The general formula for R-matrices of slq(2,C) for the highest weight repre- sentations both for general q and for q being a root of unity by generalizing G.Gomez's and G.Sierra's one for semiperiodic representations of slq(2,C) at roots of…
In this paper, we derive eight basic identities of symmetry in three variables related to $q$-Euler polynomials and the $q$-analogue of alternating power sums. These and most of their corollaries are new, since there have been results only…
The Lagrangian for a non-abelian gauge theory with an $SU(N_{\! c})$ symmetry and a linear covariant gauge fixing is constructed in eight dimensions. The renormalization group functions are computed at one loop with the special cases of…
In this research paper, structured bi-matrix variate, matrix quadratic equations are considered. Some lemmas related to determining the eigenvalues of unknown matrices are proved. Also, a method of determining the diagonalizabe unknown…
We show that the two sets of self-dual Yang-Mills equations in 8-dimensions proposed in (E.Corrigan, C.Devchand, D.B.Fairlie and J.Nuyts, {\it Nuclear Physics} {\bf B214}, 452-464, (1983)) form respectively elliptic and overdetermined…
In this note, we mainly consider the extended Weyl algebra of two generators (u,v), that is, the algebra generated by u,v with the fundamental commutation relation. Weyl algebra is realized on the space of polynomials of u and v by defining…
We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 x 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2,4). The 15 matrices with eigenvalue one correspond to a…
The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical $R$-matrix formalism. This generalizes to the case of $U_q(sl_n)$ the result obtained by O. Babelon, D. Bernard and E. Billey for…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
A finite-dimensional matrix representation of the Jackson $q$-differential operator $D_q$, defined by $D_qf(x)$ $=$ $(f(qx)-f(x))/(x(q-1))$, is written down following Calogero. Such a representation of $D_q$ should have applications in…
For any real diagonalizable matrix with complex eigenvalues we provide a real, coordinate free decomposition with a clear geometric interpretation.