Construct weak Hopf algebras by using Borcherds matrix

Abstract

We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a new generator J satisfying J m = J for some integer m. We denote this algebra by wUThis algebra is a weak Hopf algebra if and only if m = 2, 3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra Uq(G) of a generalized Kac-Moody algebra G.

Some Graded Bialgebras Related to Borcherds Superalgebras

“…[2], we can also replace the group G(U q (G)) of grouplike elements by some regular monoid as in Ref. [25][26][27]. Our new generator J satisfies J m−r+1 = J for some integers 1 ≤ r ≤ m. In this way, we obtain a subclass of graded bialgebra wU τ q (G).…”

“…5). Our main aim of our paper is to generalize the results in [25][26][27] to the case about a Borcherds matrix with a coloring matrix. Thanks to the definition of quantized enveloping algebra U q (G) associated a Borcherds superalgebra G defined in Ref.…”

“…Similarly one can define graded weak Hopf algebra by replacing the graded antipode by a weak graded antipode. One of our aims is to unify the results in [27,28] and to give more nontrivial examples of weak Hopf algebras. These weak Hopf algebras are graded weak Hopf algebras (the definition of graded weak Hopf algebras is given in Sect.…”

“…Under some condition, the bialgebra wU τ q (G) becomes a graded weak Hopf algebra. In the cases r = 1, m − 1 and the coloring matrix C = (θ ij ) satisfying θ ij = 1, wU q (G) was defined in [25,27]. This graded bialgebra wU τ q (G) has a Ore set S. Localizing wU τ q (G) with this Ore set S, we obtain an algebra U τ q (G).…”

“…Some semigroup algebras over a field are a kind of weak Hopf algebras in this sense. Another examples of weak Hopf algebras in the sense of [21] are weak quantized enveloping algebras of generalized Kac-Moody algebras [1,[25][26][27][28]. Similarly one can define graded weak Hopf algebra by replacing the graded antipode by a weak graded antipode.…”