On the differential geometry of

Abstract

Abstract. The differential calculus on the quantum supergroup GL q (1|1) was introduced by Schmidke et al. (1990 Z. Phys. C 48 249). We construct a differential calculus on the quantum supergroup GL q (1|1) in a different way and we obtain its quantum superalgebra. The main structures are derived without an R-matrix. It is seen that the found results can be written with help of a matrixR

Differential geometry of GLh(1|1)

“…To do this, we shall use the method of ref. 13. For this reason, we decompose the algebra A into subalgebras.…”

“…and T ′ into (13). After rather complicated and tedious calculations by using the consistency of calculus, as the final result one has the following commutation relations aα = αa + h(αβ − ba), ab = ba − hbβ,…”

“…To determine them we use the consistency of calculus (see, for details, ref. 13). Continuing in this way, we can obtain the other relations.…”

Differential geometry of GLp,q(1|1)

“…We have seen, in the previous section, that A is an associative algebra generated by the matrix elements of (1) with the relations (2). A differential algebra on A is a z 2 -graded associative algebra Γ L equipped with a linear operator δ L given (8). Also the algebra Γ L has to be generated by A ∪ δ L A.…”

“…It is necessary to point out that in the work of ref. 8, the generating elements of Gr q (1|1) 10 have been interpreted as differentials of coordinate functions on the quantum supergroup GL q (1|1) (T = δ R T , in the ref. 8 notation).…”