Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure on the direct sum of all homology or cohomology groups of an H-space.
Locally compact quantum groups generalize Hopf algebras and carry a topology. The algebra of all continuous functions on a Lie group is a locally compact quantum group.
Quasi-Hopf algebras are generalizations of Hopf algebras, where coassociativity only holds up to a twist.
Weak Hopf algebras, or quantum groupoids, are generalizations of Hopf algebras. Like Hopf algebras, weak Hopf algebras form a self-dual class of algebras; i.e., if H is a (weak) Hopf algebra, so is H*, the dual space of linear forms on H (with respect to the algebra-coalgebra structure obtained from the natural pairing with H and its coalgebra-algebra structure). A weak Hopf algebra H is usually taken to be a
- finite dimensional algebra and coalgebra with coproduct Δ: H → H ⊗ H and counit ε: H → k satisfying all the axioms of Hopf algebra except possibly or for some a,b in H. Instead one requires the following:
for all a,b, and c in H.
- H has a weakened antipode S: H → H satisfying the axioms:
(a) for all a in H (the right-hand side is the interesting projection usually denoted by or with image a separable subalgebra denoted by HR or Hs);
(b) for all a in H (another interesting projection usually denoted by or with image a separable algebra HL or Ht, anti-isomorphic to HL via S);
(c) for all a in H.
Note that if Δ(1) = 1 ⊗ 1, these conditions reduce to the two usual conditions on the antipode of a Hopf algebra.
The axioms are partly chosen so that the category of H-modules is a rigid monoidal category. The unit H-module is the separable algebra HL mentioned above.
For example, a finite groupoid algebra is a weak Hopf algebra. In particular, the groupoid algebra on with one pair of invertible arrows and between i and j in is isomorphic to the algebra H of n x n matrices. The weak Hopf algebra structure on this particular H is given by coproduct, counit and antipode . The separable subalgebras HL and HR coincide and are non-central commutative algebras in this particular case (the subalgebra of diagonal matrices).
Early theoretical contributions to weak Hopf algebras are to be found in as well as
Hopf algebroids introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000): Hopf algebroids generalize weak Hopf algebras and certain skew Hopf algebras. They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The antipode axioms have been changed by G. Böhm and K. Szlachanyi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.
A left Hopf algebroid (H,R) is a left bialgebroid together with an antipode: the bialgebroid (H,R) consists of a total algebra H and a base algebra R and two mappings, an algebra homomorphism s: R → H called a source map, an algebra anti-homomorphism t: R → H called a target map, such that the commutativity condition is satisfied for all . The axioms resemble those of a Hopf algebra but are complicated by the possibility that R is a noncommutative algebra or its images under s and t are not in the center of H. In particular a left bialgebroid (H,R) has an R-R-bimodule structure on H which prefers the left side as follows: for all h in H, . There is a coproduct Δ: H → H ⊗R H and counit ε: H → R that make (H, R, Δ, ε) an R-coring (with axioms like that of a coalgebra such that all mappings are R-R-bimodule homomorphisms and all tensors over R). Additionally the bialgebroid (H,R) must satisfy Δ(ab) = Δ(a)Δ(b) for all a,b in H, and a condition to make sure this last condition makes sense: every image point Δ(a) satisfies for all r in R. Also Δ(1) = 1 ⊗ 1. The counit is required to satisfy and the condition .
The antipode S: H → H is usually taken to be an algebra anti-automorphism satisfying conditions of exchanging the source and target maps and satisfying two axioms like Hopf algebra antipode axioms; see the references in Lu or in Böhm-Szlachanyi for a more example-category friendly, though somewhat more complicated, set of axioms for the antipode S. The latter set of axioms depend on the axioms of a right bialgebroid as well, which are a straightforward switching of left to right, s with t, of the axioms for a left bialgebroid given above.
As an example of left bialgebroid, take R to be any algebra over a field k. Let H be its algebra of linear self-mappings. Let s(r) be left multication by r on R; let t(r) be right multiplication by r on R. H is a left bialgebroid over R, which may be seen as follows. From the fact that one may define a coproduct by Δ(f)(r ⊗ u) = f(ru) for each linear transformation f from R to itself and all r,u in R. Coassociativity of the coproduct follows from associativity of the product on R. A counit is given by ε(f) = f(1). The counit axioms of a coring follow from the identity element condition on multiplication in R. The reader will be amused, or at least edified, to check that (H,R) is a left bialgebroid. In case R is an Azumaya algebra, in which case H is isomorphic to R ⊗ R, an antipode comes from transposing tensors, which makes H a Hopf algebroid over R.
Multiplier Hopf algebras introduced by Alfons Van Daele in 1994 are generalizations of Hopf algebras where comultiplication from an algebra (with or withthout unit) to the multiplier algebra of tensor product algebra of the algebra with itself.
Hopf group-(co)algebras introduced by V.G.Turaev in 2000 are also generalizations of Hopf algebras.
Read more about this topic: Hopf Algebras
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