The matrix quantum unitary Cayley-Klein groups

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The purpose of this report is to generalize the R-matrix theory of the quan- tum groups [1] to the case of all quantum matrix unitary Cayley-Klein groups.
J. Phys. A: Math. Gen., 1993, v. 26, L5–L8.

The matrix quantum unitary Cayley-Klein groups. N.A.Gromov Komi Scientific Centre, Russian Academy of Sciences, Syktyvkar 167000, Russia

The purpose of this report is to generalize the R-matrix theory of the quantum groups [1] to the case of all quantum matrix unitary Cayley-Klein groups which also include the quantum groups of the degenerate hermitic forms. Let Rq matrix is as in the case of An (or su(n + 1)) [1] , i.e. Rq = q

n X

n X

ekk ⊗ ekk +

ekk ⊗ emm + (q − q −1 )

(ekm )ij = δik δjm ,

ekm ⊗ emk ,

k,m=0 k>m

k,m=0 k6=m

k=0

n X

k, m, i, j, = 0, 1, . . . , n.

(1)

Let us define the algebra Aq (j) as the associative C-algebra generated by the noncommutative elements k ≥ m;

(T (j))km = tkm , Jkm = 1, k ≥ m,

Jkm =

m Y

jr , k < m,

2 (T (j))km = tkm Jkm ,

k