Quantum groups

The phrase "quantum group" is more or less a synonym for "Hopf algebra" . More precisely, the category of quantum groups is defined in [a1] to be dual to the category of Hopf algebras. This is natural for the following reason. There is the following general principle: The functor $X \mapsto \{ \textrm{ the algebra of functions on } X \}$ is an anti-equivalence between the category of "spaces" and the category of commutative associative unital algebras, perhaps with some additional structures or properties (this principle becomes a theorem if "space" is understood to be "affine scheme" or "compact topological space" , and "algebra" is understood to mean "C*-algebra" ). So one can translate the definition of a group into the language of algebras: instead of a space $G$ with an associative operation $G \times G \rightarrow G$ one obtains a commutative algebra $A$ over a commutative ring $k$ with a homomorphism $\Delta : \ A \rightarrow A \otimes A$ , called comultiplication; the unit $e \in G$ gives rise to a homomorphism $\epsilon : \ A \rightarrow k$ , called co-unit, and the mapping $g \mapsto g ^{-1}$ , $g \in G$ , gives rise to a bijective $k$ - linear mapping $S : \ A \rightarrow A \otimes A$ , called antipode. The group axioms are equivalent to the commutativity of the following diagrams: $$\begin{array}{crcrl} & & {A \otimes A} & & \\ {} &\nearrow ^ \Delta & &\searrow ^ { \mathop{\rm id} \otimes \Delta} &{} \\ A &{} &{} &{} & {A \otimes A \otimes A} \\ {} &\searrow _ \Delta & &\nearrow _ {\Delta \otimes \mathop{\rm id}} &{} \\ & & {A \otimes A} & & \\ \end{array}$$

$$\begin{array}{ccc} A & \stackrel{ { \mathop{\rm id}\nolimits}} \rightarrow & A \\ {\scriptsize \Delta} \downarrow &{} &\| \\ {A \otimes A} & \stackrel { \mathop{\rm id}\otimes\epsilon}\rightarrow &{A \otimes k} \\ \end{array} \qquad\qquad \begin{array}{ccc} A & \stackrel{ { \mathop{\rm id}\nolimits}} \rightarrow & A \\ {\scriptsize \Delta} \downarrow &{} &\| \\ {A \otimes A} & \stackrel {\epsilon \otimes \mathop{\rm id} } \rightarrow &{k \otimes A} \\ \end{array}$$

$$\begin{array}{ccccc} A & \stackrel \Delta \rightarrow A \otimes A &\stackrel{ { \mathop{\rm id} \otimes S}} \rightarrow &A \otimes A \stackrel{m}\rightarrow & A \\ {} &\searrow _ \epsilon &{} &\nearrow _{i} &{} \\ {} &{} & k &{} &{} \\ \end{array}$$

$$\begin{array}{ccccc} A & \stackrel \Delta \rightarrow A \otimes A & \stackrel{ {S \otimes \mathop{\rm id}}} \rightarrow &A \otimes A \stackrel{m}\rightarrow & A \\ {} &\searrow _ \epsilon &{} &\nearrow _{i} &{} \\ {} &{} & k &{} &{} \\ \end{array}$$

Here $m(a \otimes b) = ab$ , $i(c) = c \cdot 1 _{A}$ . The commutativity of these diagrams means that $(A ,\ \Delta ,\ \epsilon ,\ S )$ is a commutative Hopf algebra. Since the category of groups is anti-equivalent to the category of commutative Hopf algebras, it is natural to define a quantum group as an object of the category dual to the category of (not necessarily commutative) Hopf algebras.

A simple class of non-commutative Hopf algebras is formed by the group algebras of non-commutative groups. These Hopf algebras are commutative, i.e. $\Delta (A)$ is contained in the symmetric part of $A \otimes A$ . Essentially, all cocommutative Hopf algebras are group algebras.

Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix $n \in \mathbf N$ and $q \in k$ , where $k$ is a commutative ring. Denote by $A$ the associative $k$ - algebra with generators $x _{ij}$ , $1 \leq i,\ j \leq n$ , and defining relations $x _{ij} x _{il} = q x _{il} x _{ij}$ if $j < l$ , $x _{ij} x _{kj} = q x _{kj} x _{ij}$ if $i < k$ , $x _{il} x _{kj} = x _{kj} x _{il}$ if $i < k$ , $l > j$ , $[ x _{il} ,\ x _{kj} ] = (q ^{-1} -q) x _{ij} x _{kl}$ if $i > k$ , $l > j$ , $\sum _ {i _{1} \dots i _{n}} x _ {1i _{1}} \dots x _ {ni _{n}} \cdot (-q) ^ {l(i _{1} \dots i _{n} )} = 1$ , where $l ( i _{1} \dots i _{n} )$ is the number of inversions in the permutation $( i _{1} \dots i _{n} )$ . Then $A$ has a Hopf algebra structure defined by $\Delta (x _{ij} ) = \sum _{k} x _{ik} \otimes x _{kj}$ . If $q =1$ , then $A$ is the algebra of polynomial functions on $\mathop{\rm SL}\nolimits (n)$ . So, in the general case it is natural to consider elements of $A$ as "functions on the quantized SLn" .

The quantized $\mathop{\rm SL}\nolimits (n)$ is one of the simplest quantum groups which appear naturally in the theory of quantum integrable systems and, especially, in the quantum inverse-scattering method [a2]. The development of this method has led to the following quantization technique for constructing non-commutative non-cocommutative Hopf algebras. It is natural to construct them as deformations of commutative Hopf algebras. If a non-commutative deformation $A$ of a commutative Hopf algebra $A _{0}$ is given, then a Poisson bracket on $A _{0}$ is defined by $\{ a,\ b \} = \mathop{\rm lim}\nolimits _ {h \rightarrow 0} \ h ^{-1} (ab-ba)$ , where $h$ is the deformation parameter and $ab$ means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, $\{ a,\ bc \} = \{ a,\ b \} c + \{ a,\ c \} b$ ) and is compatible with comultiplication. In other words, $A _{0}$ is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra $A _{0}$ and then try to quantize it, i.e. to construct a Hopf algebra deformation of $A _{0}$ which induces the given Poisson bracket on $A _{0}$ .

Technically it is more convenient to deform not commutative Hopf algebras but cocommutative ones and to start not with a Poisson–Hopf algebra (or a Poisson–Lie group [a1], which is more or less the same) but with its infinitesimal version, called a Lie bi-algebra . A Lie bi-algebra is a Lie algebra $\mathfrak g$ with a linear mapping $\phi : \ \mathfrak g \rightarrow \mathfrak g \otimes \mathfrak g$ such that: 1) $\phi ^{*} : \ \mathfrak g ^{*} \otimes \mathfrak g ^{*} \rightarrow \mathfrak g ^{*}$ defines a Lie algebra structure on $\mathfrak g ^{*}$ ; and 2) $\phi$ is a $1$ - cocycle ($\mathfrak g$ acts on $\mathfrak g \otimes \mathfrak g$ by means of the adjoint representation). By definition, a quantization of $( \mathfrak g ,\ \phi )$ is a Hopf algebra deformation of the universal enveloping algebra $U \mathfrak g$ such that $\delta \mid _ {\mathfrak g} = \phi$ , where $\delta : \ U \mathfrak g \rightarrow U \mathfrak g \otimes U \mathfrak g$ is the Poisson cobracket, defined by $\delta (a) = \mathop{\rm lim}\nolimits _ {h \rightarrow 0} \ h ^{-1} ( \Delta (a) - \Delta ^ \prime (a))$ . Here $h$ is the deformation parameter, $\Delta$ the deformed comultiplication and $\Delta ^ \prime$ the opposite comultiplication.

It is not known whether every Lie bi-algebra can be quantized, and usually quantization is not unique. But in several important cases (cf. [a1], §3, §6) there exists a canonical quantization. In particular, on a Kac–Moody algebra $\mathfrak g$ with a fixed scalar product $( \ ,\ )$ there is a canonical Lie bi-algebra structure and this bi-algebra has a canonical quantization $U _{h} \mathfrak g$ , as was discovered in [a3], [a4], [a5]. Let $\mathfrak h$ be the Cartan subalgebra of $\mathfrak g$ , $H _{i} \in \mathfrak g$ the images of the simple roots $\alpha _{i} \in \mathfrak h ^{*}$ . Then $U _{h} \mathfrak g$ is generated by $\mathfrak h$ and $X _{i} ^{+}$ , $X _{i} ^{-}$ with the following defining relations:$$[a _{1} ,\ a _{2} ] = 0 \textrm{ for } a _{1} ,\ a _{2} \in \mathfrak h ;$$ $$[a ,\ X _{i} ^ \pm ] = \pm \alpha _{i} (a ) X _{i} ^ \pm \textrm{ for } a \in \mathfrak h ;$$ $$[ X _{i} ^{+} ,\ X _{j} ^{-} ] = 2 \delta _{ij} h ^{-1} \ \mathop{\rm sinh}\nolimits ( h H _{i} / 2) .$$ Setting $n = 1-A _{ij}$ , $q = \mathop{\rm exp}\nolimits \ h (H _{i} ,\ H _{j} )/2$ one has also$$\sum _{k=0} ^ n (-1) ^{k} \binom{n}{k} _{q} q ^ {-k(n-k)/2} (X _{i} ^ \pm ) ^{k} \cdot X _{j} ^ \pm \cdot (X _{i} ^ \pm ) ^{n-k} = 0 .$$ Here $(A _{ij} )$ is the Cartan matrix and $( {} _{k} ^{n} ) _{q}$ is the Gauss polynomial, i.e.,$$\binom{n}{k} _{q} = \frac{(q ^{n} -1 ) \dots (q ^{n-k+1} -1)}{( q ^{k} -1 ) \dots (q-1)} .$$ The comultiplication in $U _{h} \mathfrak g$ is such that $\Delta (a) = a \otimes 1 + 1 \otimes a$ for $a \in \mathfrak h$ and$$\Delta (X _{i} ^ \pm ) = X _{i} ^ \pm \otimes \mathop{\rm exp}\nolimits \left ( \frac{hH _{i}}{4} \right ) + \mathop{\rm exp}\nolimits \left ( \frac{-hH _{i}}{4} \right ) \otimes X _{i} ^ \pm .$$ If $\mathfrak g$ is a finite-dimensional simple Lie algebra (cf. Lie algebra, semi-simple), then the algebra of regular functions on the corresponding simply-connected algebraic group $G$ is isomorphic to the subalgebra of $( U \mathfrak g ) ^{*}$ generated by the matrix elements of the finite-dimensional representations of $U \mathfrak g$ . Therefore the subalgebra of $( U _{h} \mathfrak g ) ^{*}$ generated by the matrix elements of the finite-dimensional representations of $U \mathfrak g$ can be considered as the algebra of functions on a certain quantization of $G$ . For instance, the quantized $\mathop{\rm SL}\nolimits ( n )$ ( cf. above) can be obtained in this way.

There is an important notion of a quasitriangular Hopf algebra. This is a pair $(A,\ R)$ where $A$ is a Hopf algebra and $R$ is an invertible element of $A \otimes A$ such that $( \Delta \otimes \mathop{\rm id}\nolimits ) (R) = R ^{13} R ^{23}$ , $( \mathop{\rm id}\nolimits \otimes \Delta )(R) = R ^{13} R ^{12}$ , $\Delta ^ \prime (a) = R \cdot \Delta ( a ) \cdot R ^{-1}$ for $a \in A$ . Here $\Delta ^ \prime$ is the opposite comultiplication and $R ^{12}$ , $R ^{13}$ , $R ^{23}$ are defined as follows: If $R = \sum _{i} x _{i} \otimes y _{i}$ , where $x _{i} ,\ y _{i} \in A$ , then $R ^{12} = \sum _{i} x _{i} \otimes y _{i} \otimes 1$ , $R ^{13} = \sum _{i} x _{i} \otimes 1 \otimes y _{i}$ , $R ^{23} = \sum _{i} 1 \otimes x _{i} \otimes y _{i}$ . If $(A,\ R )$ is a quasitriangular Hopf algebra, then $R$ satisfies the quantum Yang–Baxter equation (cf. also Yang–Baxter equation), i.e., $R ^{12} R ^{13} R ^{23} = R ^{23} R ^{13} R ^{12}$ . It is known (cf. [a1], §13) that if $\mathfrak g$ is a finite-dimensional simple Lie algebra, then $U _{h} \mathfrak g$ has a canonical quasitriangular structure, while if $\mathfrak g$ is an infinite-dimensional Kac–Moody algebra, then $U _{h} \mathfrak g$ has an "almost quasitriangular" structure.

If $(A,\ R)$ is a quasitriangular Hopf algebra over $k$ and $\rho$ is a representation $A \rightarrow \mathop{\rm Mat}\nolimits (n,\ k)$ , then ${\mathcal R} = ( \rho \otimes \rho )(R) \in \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} )$ satisfies the quantum Yang–Baxter equation. There is an inverse construction (cf. [a6], [a7]), which goes back to the quantum inverse-scattering method: to a matrix solution of the quantum Yang–Baxter equation satisfying a non-degeneracy condition there corresponds a Hopf algebra. Without this condition one can only construct an associative bi-algebra (the difference between a Hopf algebra and an associative bi-algebra is that in the second case there may be no antipode). This bi-algebra is generated by elements $t _{ij}$ , $1 \leq i,\ j \leq n$ , with defining relations ${\mathcal R} T _{1} T _{2} = T _{2} T _{1} {\mathcal R}$ , where $T _{1} = T \otimes 1 \in \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} )$ , $T _{2} = 1 \otimes T \in \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} )$ , $T$ is the matrix $(t _{ij} )$ , and $\Delta$ is defined by $\Delta (t _{ij} ) = \sum _{k} t _{ik} \otimes t _{kj}$ .

Quasitriangular Hopf algebras are a natural tool for the quantum inverse-scattering in method ([a1], §11). On the other hand, they can be used (cf. [a8]) to construct invariants of knots (and of more general objects such as links and tangles) generalizing the Jones polynomial [a9]. More precisely, to an oriented knot $\gamma \subset \mathbf R ^{3}$ and a quasitriangular Hopf algebra $(A ,\ R)$ there corresponds a central element $z _ \gamma \in A$ .

The usual notion of a group has several versions: abstract group, Lie group, topological group, etc. The same is true for quantum groups. The quantum analogue of the notion of a compact group was introduced in [a10] (the idea is to use $C ^{*}$ - algebras instead of abstract algebras). The quantized $\mathop{\rm SU}\nolimits (2)$ ( cf. [a11], [a12]) is a typical example. The notion of a ring group (cf. , [a14]) and the equivalent notion of a Kac algebra (cf. [a15], [a16]) were introduced as an attempt to define a locally compact quantum group. However, these notions are not general enough (the axioms of , [a14], [a15] imply that the square of the antipode is the identity mapping, and therefore the quantized $\mathop{\rm SU}\nolimits (2)$ is not a ring group).

References

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How to Cite This Entry:
Quantum groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_groups&oldid=44258
This article was adapted from an original article by V.G. Drinfel'd (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article