Weak Hopf Algebras Corresponding to Borcherds–Cartan Matrices.

Let $${\fancyscript G}$$ be a generalized Kac–Moody algebra with an integral Borcherds–Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac–Moody algebra $$ wU^{d}_{q} {\left( {\fancyscript G} \right)} $$ , which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra $$ wU^{d}_{q} {\left( {\fancyscript G} \right)} $$ and weak A-forms of $$ wU^{d}_{q} {\left( {\fancyscript G} \right)} $$ . [ABSTRACT FROM AUTHOR]