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: $$ \begin{array}{crcrl} & & {A \otimes A} & & \\ {} &\nearrow ^ \Delta & &\searrow ^ { \mathop{\rm id} \otimes \Delta} &{} \\ A &{} &{} &{} & {A \otimes A \otimes A} \\ {} &\searrow _ \Delta & &\nearrow _ {\Delta \otimes \mathop{\rm id}} &{} \\ & & {A \otimes A} & & \\ \end{array} $$

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

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

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

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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