In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Heinz Hopf (November 19, 1894 – June 3, 1971) was a mathematician born in Gräbschen, Germany. ... In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ...

The map S is called the antipode map of the Hopf algebra. If an antipode exists, it must be unique. S is sometimes required to have a K-linear inverse, which is automatic in the finite-dimensional case, or if H is commutative or cocommutative (or more generally quasitriangular). If S² = Id, then the Hopf algebra is said to be involutive, which is always true if H is commutative or cocommutative. In general, S is an antihomomorphism, so S² is a homomorphism, which is therefore an automorphism if S was invertible (as may be required). The definition of Hopf algebra is self-dual, so if one can define a dual of H (which is always possible of H is finite-dimensional), then it is automatically a Hopf algebra. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ... In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of such that for all , where is the coproduct on H, and the linear map is given by , , , where , , and , where , , and , are algebra morphisms determined by R is called the R-matrix. ... In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. ... In abstract algebra, a homomorphism is a structure-preserving map. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

Examples

Group algebra. Suppose G is a group. The group algebra KG is a unital associative algebra over K. It turns into a Hopf algebra if we define In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

• Δ : KG → KG ⊗ KG by Δ(g) = g ⊗ g for all g in G
• ε : KG → K by ε(g) = 1 for all g in G
• S : KG → KG by S(g) = g ^-1 for all g in G.

Functions on a finite group. Suppose now that G is a finite group. Then the set K^G of all functions from G to K with pointwise addition and multiplication is a unital associative algebra over K, and K^G ⊗ K^G is naturally isomorphic to K^GxG (for G infinite, K^G ⊗ K^G is a proper subset of K^GxG). The set K^G becomes a Hopf algebra if we define

• Δ : K^G → K^GxG by Δ(f)(x,y) = f(xy) for all f in K^G and all x,y in G
• ε : K^G → K by ε(f) = f(e) for every f in K^G [here e is the identity element of G]
• S : K^G → K^G by S(f)(x) = f(x^-1) for all f in K^G and all x in G.

Regular functions on an algebraic group. Generalizing the previous example, we can use the same formulas to show that for a given algebraic group G over K, the set of all regular functions on G forms a Hopf algebra. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined. ...

Universal enveloping algebra. Suppose g is a Lie algebra over the field K and U is its universal enveloping algebra. U becomes a Hopf algebra if we define In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ...

• Δ : U → U ⊗ U by Δ(x) = x ⊗ 1 + 1 ⊗ x for every x in g (this rule is compatible with commutators and can therefore be uniquely extended to all of U).
• ε : U → K by ε(x) = 0 for all x in g (again, extended to U)
• S : U → U by S(x) = -x for all x in g.

In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

Quantum groups and non-commutative geometry

All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. Δ = T Δ where T: H ⊗ H → H ⊗ H is defined by T(x ⊗ y) = y ⊗ x). Other interesting Hopf algebras are certain "deformations" or "quantizations" of those from example 3 which are neither commutative nor co-commutative. These Hopf algebras are often called quantum groups, a term that is only loosely defined. They are important in noncommutative geometry, the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there does not seem to be a direct way to define or manipulate these non-standard objects, one can still work with their Hopf algebras, and indeed one identifies them with their Hopf algebras. Hence the name "quantum group". In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ...

Related concepts

Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure of the totality of all homology or cohomology groups of a space. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

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. The locally compact (l. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...

Quasi-Hopf algebras are also generalizations of Hopf algebras, where coassociativity only holds up to a twist. A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. ...

See also

• Quasitriangular Hopf algebra
• Algebra/set analogy
• Representation of a Hopf algebra
• Ribbon Hopf algebra
• Superalgebra
• Supergroup
• Anyonic Lie algebra

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of such that for all , where is the coproduct on H, and the linear map is given by , , , where , , and , where , , and , are algebra morphisms determined by R is called the R-matrix. ... In mathematics, a strict monoidal category is a category with a product operation × on objects that has properties analogous to those of the tensor product. ... In mathematics, there is a concept of a representation of a Hopf algebra. ... A Ribbon Hopf algebra is a Quasi-triangular Hopf algebra which possess an invertible central element more commonly known as the ribbon element, such that the following conditions hold: such that . ... In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ... The concept of supergroup is a generalization of a that of group. ... An anyonic Lie algebra is a U(1) graded vector space L over C equipped with a bilinear operator [.,.] and linear maps ε:L->C and Δ:L->L⊗L satisfying ε([X,Y])=ε(X)ε(Y) for pure graded elements X, Y and Z. See also Lie...

References

• Jurgen Fuchs, Affine Lie Algebras and Quantum Groups, (1992), Cambridge University Press. ISBN 0-521-48412-X
• Ross Moore, Sam Williams and Ross Talent:Quantum Groups: an entrée to modern algebra