Weak Quantum Enveloping Algebras of Borcherds Superalgebras

Abstract

In present paper we define a new kind of weak quantized enveloping algebra \(wU_{q}^{\tau}({\mathcal{G}})\) of Borcherds superalgebras \({\mathcal{G}}\) . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras \({\mathcal{U}}_{q}({\mathcal{G}})\) . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra \(U_{q}({\mathcal{G}})\) of \({\mathcal{G}}\) . Moreover, \({\mathcal{U}}_{q}({\mathcal{G}})\) is isomorphic to a direct sum of \(U_{q}({\mathcal{G}})\) and an other algebra as algebras.