An−1(1) reflection K-matrices

Abstract

We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated with the A (1) n−1 affine Lie algebra. We have classified them in two classes of solutions. The first class consists of n(n − 1)/2 K-matrix solutions with three free parameters. The second class are solutions that depend on the parity of n. For n odd there exist n reflection Kmatrices with 2 + [n/2] free parameters. It turns out that for n even there exist n/2 K-matrices with 2 + n/2 free parameters and n/2 K-matrices with 1 + n/2 free parameters.

Reflection matrices for the Uq[sl(r|2m)(2Lima-Santos1, Galleas2 2010Nuclear Physics BSelf Cite90130View full textAdd to dashboardCite

“…Before the end of our discussion on the U q [sl(m|n) (1) ] reflection matrices, we will make (by a referee suggestion), the comparision with the sl(m + n) reflection matrices [13]. The diagonal solutions and the Type-I solutions are the same for the both models.…”

“…[12] matrices written above and one Type-III matrix with four free parameters β 1,2 , β 2,1 , β 3,4 and β 1,1 : with one diagonal equal to D [13]c . The diagonals with only entries of the types f (x) and x 2 f (x −1 ) are the solutions obtained in [9].…”

“…These procedure is now standard [13]. For instance, if we are looking for K (1) (0) [14][56] , the nondiagonal elements k i,j (x), (i + j = 5 mod 6 ) in terms of k 1,4 (x) = 0 are given by…”

Reflection equation for theN= 3 Cremmer–GervaisR-matrix

“…Up to now, there are many works about the reflection equation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Taking the XXZ chain for example, the diagonal solution was obtained in [1,2], the general solution in [3].…”

Reflection equation for the N=3 Cremmer-Gervais R-matrix

“…Up to now, there are many works about the reflection equation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Taking the XXZ chain for example, the diagonal solution was obtained in [1,2], the general solution in [3].…”