The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal

We give an explicit basis [Formula: see text] of the quotient of the Kauffman bracket skein algebra [Formula: see text] on a surface [Formula: see text] by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus [Formula: see text] into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.