A Class of Weak Hopf Algebras
We introduce a class of noncommutative and noncocommutative weak Hopf algebras with infinite Ext quivers and study their structure. We decompose them into a direct sum of two algebras. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver. The weak Hopf extension of Hopf algebra has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to .
Weak Hopf algebra was introduced by Li in 1998 as a generalization of Hopf algebras . It had been proved in [1, 2]; for some sorts of finite dimensional weak Hopf algebras , the quantum quasidouble of is quasibraided equipped with some quasi-R-matrix . Hence is a solution of the Quantum Yang-Baxter Equation.
First two examples of noncommutative and noncocommutative weak Hopf algebras were given in . Up to now, many examples of weak Hopf algebras have been found [2, 4–7]. So far, all examples of weak Hopf algebras were based on some Hopf algebras and were constructed by weak extension.
In this paper, we first give a Hopf algebra, denoted by . By weak extension, we construct a weak Hopf algebra corresponding to and study their structure. has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to . And as an algebra, can be decomposed into a direct sum of two algebras, one of which is . The coalgebra structures of these weak Hopf algebras are described by their Ext quiver [8, 9].
We organize our paper as follows. In Section 2, we introduce the Hopf algebra . In Section 3, we define a class of weak Hopf algebras . In Section 4, we study the structure of and decompose into a direct sum of and some algebra of polynomials as an algebra. We give the Ext-quiver of coalgebra of and prove that has a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic to .
2. A Quiver Hopf Algebra
The Hopf Algebra is defined in . Let . As a -algebra is generated by , and subject to the relations
The coalgebra structure of is determined by
We generalize to , which is defined as follows. Let be a field, , . As a -algebra is generated by , and subject to the relations
The coalgebra structure of is determined by
The antipode is induced by
3. A Class of Weak Hopf Algebras
In this section, we construct a class of weak Hopf algebra corresponding to .
First recall the definition of weak Hopf algebra .
Definition 3.1. A -bialgebra is called a weak Hopf algebra if there exists such that and where is called a weak antipode of .
A weak Hopf algebra is called pointed if it is pointed as a coalgebra. If a weak Hopf algebra is pointed, then the set of all group-like elements is a regular monoid .
Now we construct weak Hopf algebra corresponding to . The set of group-like elements of weak Hopf algebra is a regular monoid which has generators , , , subject to .
To construct all possible weak extension we need the following discussion.
Recall, for any coalgebra , that the group-like elements in are the set ; necessarily for . Note that a simple subcoalgebra of is one-dimensional for some . A coalgebra is pointed if all of its simple subcoalgebras are one-dimensional. For , the -primitive elements in are the set ; necessarily for . Note that ; an -primitive element is nontrivial if . If , the -primitives are simply called primitive; otherwise they are called skew primitive.
The following result is a generalization of .
Lemma 3.2. Let be the weak Hopf algebra defined above. One has
Proof. Let , then . Hence, The second inclusion is proved similarly.
Corollary. For , one has
Proof. We only prove the first equation. In fact, the map , is a linear map with inverse , . Hence, and are isomorphic as vector spaces.
Since all the dimensions in Corollary 3.3 are same, we have the following corollary.
Corollary 3.4. One has
Proof. The map , is a linear map. If , for some , then , the vector space spanned by all group-like elements, because is graded. Hence, . Therefore, the linear map is an injection. Consequently, The proof of the second inequality is similar.
By the above discussion we know that weak Hopf algebra is determined by , , and . Take to be linearly independent nontrivial elements in , and linearly independent nontrivial elements in . Let
and a basis of . Then is determined by , , .
To summarize, we define weak Hopf algebra corresponding to as follows.
Definition 3.5. Let be a field. For any positive integers , and nonzero element , we define to be associative algebra over field generated by , subject to
can be endowed with coalgebra structure by
while the weak antipode is induced by
Theorem 3.6. For any positive integers , is a weak Hopf algebra.
Proof. First we must check that the coproduct is an algebra map. It suffices to prove that preserves the relations (3.7)–(3.10). It is easy to see that preserves the relations (3.7). And
Next we prove that is the weak antipode. It suffices to prove that for each generator , the action of is the same as that of , and the action of is the same as that of .
we get Since it follows that Since we get
4. The Structure of
In this section we study the algebra and coalgebra structure of .
It is easy to prove that the elements and are a pair of orthogonal central idempotents. Set , . We have the following.
Theorem 4.1. can be written as a direct sum of two-sided ideals . And one has the following. (1)As an algebra, is isomorphic to , where .(2)As an algebra, is isomorphic to the free associative algebra of generators, where .
Proof. () Since and are a pair of orthogonal central idempotents,
The isomorphism is induced by , , , , .
() Note that and . Since are generators of , the isomorphism is defined by , , .
A weak Hopf ideal of a weak Hopf algebra is a bi-ideal such that , where is the weak antipode of . It is easy to see that has a natural structure of a weak Hopf algebra.
Theorem 4.2. The ideal in generated by is a weak Hopf ideal. And the quotient weak Hopf algebra is a Hopf algebra, which is isomorphic to , where .
is a weak Hopf ideal in .
The isomorphism is defined by , , , , .
The Ext quiver of is shown in Figure 1. The multiplicity of arrow is . The multiplicity of arrow is . The multiplicity of other arrows is all .
Theorem 4.3. The sub-coalgebra related to the subquiver in Figure 2 is isomorphic to as coalgebra.
Proof. The isomorphism is induced by , , , , , .
Remark 4.4. The isomorphisms described in Theorem 4.1 are not isomorphisms of bialgebras.
This research is supported by Doctor scientific research start fund of Henan University of Science and Technology, supported by SRF of Henan University of Science and Technology (2006zy007), and partly supported by NNSF of China (10571153).