4.3 Hopf algebras

Hopf algebras are crucial structures in noncommutative geometry, blending algebra and coalgebra concepts. They provide a framework for studying symmetries and quantum groups in noncommutative settings, bridging classical and quantum mathematics.

These algebras have rich properties, including multiplication, comultiplication, and an antipode map. They're used to construct quantum spaces and analyze symmetries in various mathematical contexts, making them essential tools for exploring noncommutative geometric objects.

Definition of Hopf algebras

• Hopf algebras are a fundamental concept in noncommutative geometry that combine the structures of an algebra and a coalgebra with additional compatibility conditions
• They provide a framework for studying symmetries and quantum groups in a noncommutative setting
• Hopf algebras are used to construct and analyze various geometric objects such as quantum homogeneous spaces and Hopf-Galois extensions

Algebra over a field

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• A Hopf algebra is first an algebra over a field $k$, which means it is a vector space equipped with a multiplication operation that is associative and distributive over addition
• The multiplication is given by a linear map $m: H \otimes H \to H$ satisfying $m(m \otimes \text{id}) = m(\text{id} \otimes m)$ (associativity) and $m(h \otimes 1) = m(1 \otimes h) = h$ for all $h \in H$ (unit element)
• Examples of algebras include polynomial rings, matrix algebras, and group algebras

Coalgebra structure

• A Hopf algebra also has a coalgebra structure, which is dual to the algebra structure
• It is equipped with a comultiplication $\Delta: H \to H \otimes H$ and a counit $\varepsilon: H \to k$ satisfying certain axioms dual to those of an algebra
• The comultiplication is coassociative: $(\Delta \otimes \text{id})\Delta = (\text{id} \otimes \Delta)\Delta$ and the counit satisfies $(\varepsilon \otimes \text{id})\Delta = (\text{id} \otimes \varepsilon)\Delta = \text{id}$

Antipode map

• A Hopf algebra has an antipode map $S: H \to H$ which is an algebra anti-homomorphism and a coalgebra anti-homomorphism
• It satisfies the antipode axiom: $m(S \otimes \text{id})\Delta = m(\text{id} \otimes S)\Delta = \eta \circ \varepsilon$ where $\eta: k \to H$ is the unit map
• The antipode generalizes the concept of inverse elements in a group and allows for a notion of "opposite multiplication" in the Hopf algebra

Compatibility conditions

• The algebra, coalgebra, and antipode structures of a Hopf algebra are required to satisfy certain compatibility conditions
• The multiplication and comultiplication are required to be algebra homomorphisms: $\Delta(ab) = \Delta(a)\Delta(b)$ and $\varepsilon(ab) = \varepsilon(a)\varepsilon(b)$ for all $a,b \in H$
• The unit and counit are also required to be compatible: $\Delta(1) = 1 \otimes 1$ and $\varepsilon(1) = 1_k$
• These conditions ensure that the Hopf algebra axioms are consistent and allow for the development of a rich theory

Examples of Hopf algebras

• Several important examples of Hopf algebras arise naturally in various areas of mathematics, showcasing their wide applicability
• These examples include group algebras, universal enveloping algebras of Lie algebras, and quantum groups
• Studying these concrete examples helps to build intuition and understanding of the general theory of Hopf algebras

Group algebras

• Given a group $G$, the group algebra $kG$ is a Hopf algebra over the field $k$
• As a vector space, $kG$ has basis elements $\{g : g \in G\}$ and multiplication is given by the group operation: $(ag)(bh) = abgh$ for $a,b \in k$ and $g,h \in G$
• The comultiplication is defined by $\Delta(g) = g \otimes g$, the counit by $\varepsilon(g) = 1$, and the antipode by $S(g) = g^{-1}$ for all $g \in G$

Universal enveloping algebras

• The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is a Hopf algebra that captures the symmetries of the Lie algebra
• As an algebra, $U(\mathfrak{g})$ is generated by elements of $\mathfrak{g}$ subject to the relations $xy - yx = [x,y]$ for all $x,y \in \mathfrak{g}$
• The comultiplication, counit, and antipode are determined by $\Delta(x) = x \otimes 1 + 1 \otimes x$, $\varepsilon(x) = 0$, and $S(x) = -x$ for all $x \in \mathfrak{g}$

Quantum groups

• Quantum groups are deformations of universal enveloping algebras or function algebras on groups that arise in the study of quantum symmetries
• Examples include the quantum enveloping algebras $U_q(\mathfrak{g})$ and the quantum function algebras $\mathcal{O}_q(G)$
• These Hopf algebras play a central role in the theory of quantum integrable systems and the study of noncommutative spaces

Modules over Hopf algebras

• The representation theory of Hopf algebras is a rich and important area of study, with connections to various aspects of noncommutative geometry
• Modules and comodules are the basic objects of study, generalizing the notion of group representations
• The fundamental theorem of Hopf modules provides a powerful tool for understanding the structure of Hopf modules

Hopf modules

• A (left) Hopf module over a Hopf algebra $H$ is a vector space $M$ that is both a left $H$-module and a left $H$-comodule, with compatibility condition $\rho(h \cdot m) = \Delta(h) \cdot \rho(m)$ for all $h \in H, m \in M$
• Here, $\rho: M \to H \otimes M$ is the comodule structure map and $\cdot$ denotes the module action
• Hopf modules provide a natural framework for studying representations of quantum groups and other Hopf algebras

Comodules

• A (right) $H$-comodule is a vector space $M$ equipped with a linear map $\rho: M \to M \otimes H$ (the coaction) satisfying $(\rho \otimes \text{id})\rho = (\text{id} \otimes \Delta)\rho$ and $(\text{id} \otimes \varepsilon)\rho = \text{id}$
• Comodules are the dual notion to modules and play an important role in the study of Hopf algebras and quantum groups
• Examples of comodules include the regular comodule (given by the comultiplication) and comodules arising from group actions

Fundamental theorem of Hopf modules

• The fundamental theorem of Hopf modules states that, under certain conditions, Hopf modules can be understood as a tensor product of a module and a comodule
• Specifically, if $H$ is a finite-dimensional Hopf algebra and $M$ is a Hopf module, then $M \cong M^{\text{co}H} \otimes H$ as Hopf modules, where $M^{\text{co}H} = \{m \in M : \rho(m) = m \otimes 1\}$ is the space of coinvariants
• This theorem allows for the study of Hopf modules to be reduced to the study of modules and comodules separately, simplifying many problems

Actions of Hopf algebras

• Hopf algebras can act on various algebraic structures, such as algebras and coalgebras, providing a framework for studying symmetries and deformations
• These actions generalize the notion of group actions and are central to the study of noncommutative spaces and quantum group symmetries
• Important constructions, such as smash products and crossed products, arise from these actions

Hopf algebra representations

• A representation of a Hopf algebra $H$ on an algebra $A$ is a linear map $\alpha: H \otimes A \to A$ satisfying certain compatibility conditions with the algebra and coalgebra structures
• These conditions ensure that the action respects the algebraic structure of $A$ and is compatible with the comultiplication and antipode of $H$
• Representations of Hopf algebras generalize group representations and provide a way to study the symmetries of noncommutative algebras

Coactions on algebras

• A coaction of a Hopf algebra $H$ on an algebra $A$ is a linear map $\rho: A \to A \otimes H$ satisfying certain compatibility conditions with the algebra and coalgebra structures
• These conditions ensure that the coaction respects the algebraic structure of $A$ and is compatible with the multiplication and unit of $H$
• Coactions provide a dual notion to actions and are important in the study of quantum homogeneous spaces and Hopf-Galois extensions

Smash products

• Given an action of a Hopf algebra $H$ on an algebra $A$, the smash product $A \# H$ is a new algebra that captures the semi-direct product structure of the action
• As a vector space, $A \# H$ is the tensor product $A \otimes H$, and the multiplication is given by $(a \# h)(b \# k) = a(h_{(1)} \cdot b) \# h_{(2)}k$ for $a,b \in A$ and $h,k \in H$, where $\Delta(h) = h_{(1)} \otimes h_{(2)}$ (Sweedler notation)
• Smash products are used to construct new algebras with prescribed symmetries and to study the structure of Hopf algebra actions

Duality for Hopf algebras

• Duality is a fundamental concept in the theory of Hopf algebras, relating the algebra and coalgebra structures
• For finite-dimensional Hopf algebras, there is a natural dual Hopf algebra structure on the dual vector space
• The Drinfeld double construction provides a way to combine a Hopf algebra and its dual into a larger Hopf algebra with a rich structure

Finite-dimensional Hopf algebras

• When a Hopf algebra $H$ is finite-dimensional as a vector space, there is a natural Hopf algebra structure on the dual vector space $H^*$
• The multiplication on $H^*$ is defined by the comultiplication on $H$ (and vice versa), and the antipode on $H^*$ is the transpose of the antipode on $H$
• This duality provides a powerful tool for studying the structure and representations of finite-dimensional Hopf algebras

Dual Hopf algebra

• For a finite-dimensional Hopf algebra $H$, the dual vector space $H^*$ admits a natural Hopf algebra structure
• The multiplication on $H^*$ is given by the dual of the comultiplication on $H$: $(\phi \psi)(h) = (\phi \otimes \psi)(\Delta(h))$ for $\phi, \psi \in H^*$ and $h \in H$
• The comultiplication on $H^*$ is given by the dual of the multiplication on $H$: $\Delta(\phi)(h \otimes k) = \phi(hk)$ for $\phi \in H^*$ and $h,k \in H$

Drinfeld double construction

• The Drinfeld double $D(H)$ of a finite-dimensional Hopf algebra $H$ is a new Hopf algebra that combines $H$ and its dual $H^*$
• As a vector space, $D(H) = H^* \otimes H$, and the multiplication is given by a semi-direct product structure that uses the natural actions of $H$ on $H^*$ and vice versa
• The Drinfeld double has a rich representation theory and is used to study quantum groups and braided monoidal categories

Structure theorems

• Several important structure theorems in the theory of Hopf algebras provide insights into their algebraic and geometric properties
• These theorems, such as the integral theory, Frobenius properties, and Maschke's theorem, highlight the interplay between the algebra, coalgebra, and antipode structures
• Understanding these results is crucial for the application of Hopf algebras in noncommutative geometry and other areas of mathematics

Integral theory

• Integrals in a Hopf algebra $H$ are elements of the dual space $H^*$ that are invariant under the left or right action of $H$ on itself
• The space of left integrals and right integrals are both one-dimensional for a finite-dimensional Hopf algebra
• Integrals play a crucial role in the structure and representation theory of Hopf algebras, with applications to the construction of invariant functionals and the study of Frobenius properties

Frobenius properties

• A Hopf algebra $H$ is called Frobenius if it is finite-dimensional and the space of left (or right) integrals is one-dimensional
• Equivalently, $H$ is Frobenius if there exists a non-degenerate bilinear form on $H$ that is associative and $H$-linear
• The Frobenius property has important consequences for the structure and representation theory of Hopf algebras, such as the existence of a bijective correspondence between simple $H$-modules and simple $H^*$-comodules

Maschke's theorem for Hopf algebras

• Maschke's theorem states that, under certain conditions, the category of representations of a Hopf algebra is semisimple
• Specifically, if $H$ is a finite-dimensional Hopf algebra over a field of characteristic zero and the integral in $H$ is invertible (i.e., $H$ is semisimple), then every $H$-module is a direct sum of simple modules
• This result generalizes the classical Maschke's theorem for group representations and has important applications in the study of quantum groups and tensor categories

Applications in noncommutative geometry

• Hopf algebras play a central role in various aspects of noncommutative geometry, providing a framework for studying quantum symmetries and noncommutative spaces
• Key applications include the theory of Hopf-Galois extensions, quantum homogeneous spaces, and the study of Hopf algebroids and groupoids
• These applications highlight the deep connections between Hopf algebras and geometry, and they motivate the development of new algebraic and geometric tools

Hopf-Galois extensions

• A Hopf-Galois extension is a generalization of the classical notion of a Galois extension, where the Galois group is replaced by a Hopf algebra
• Formally, an $H$-Hopf-Galois extension is an extension of algebras $A \subseteq B$ together with a coaction of a Hopf algebra $H$ on $B$ such that a certain map $B \otimes_A B \to B \otimes H$ is an isomorphism
• Hopf-Galois extensions provide a framework for studying noncommutative principal bundles and have applications in the theory of quantum groups and noncommutative geometry

Quantum homogeneous spaces

• Quantum homogeneous spaces are noncommutative analogues of classical homogeneous spaces, such as spheres and projective spaces
• They are defined as the invariant subalgebras of a Hopf algebra action on a noncommutative algebra
• Examples include the quantum spheres, quantum projective spaces, and quantum flag varieties, which arise as quantum homogeneous spaces for quantum groups
• The study of quantum homogeneous spaces is a rich area of noncommutative geometry, with connections to representation theory, K-theory, and index theory

Hopf algebroids and groupoids

• Hopf algebroids are a generalization of Hopf algebras that allow for a more flexible base algebra structure
• They consist of a pair of algebras (the base and the total algebra) with compatible coalgebra and antipode structures
• Hopf algebroids are closely related to groupoids, which are categories where all morphisms are invertible
• The study of Hopf algebroids and groupoids provides a framework for noncommutative geometry that encompasses both Hopf algebras and groupoids, allowing for the development of new geometric tools and insights