# Quantum group: Wikis

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Group theory
Group theory

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra. There is no single, all-encompassing definition, but instead a family of broadly similar objects.

The term "quantum group" often denotes a kind of noncommutative algebra with additional structure that first appeared in the theory of quantum integrable systems, and which was then formalized by Vladimir Drinfel'd and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a `bicrossproduct' class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo.

In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.

Just as groups often appear as symmetries, quantum groups act on many other mathematical objects and it has become fashionable to introduce the adjective quantum in such cases; for example there are quantum planes and quantum Grassmannians.

## Intuitive meaning

The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of Alain Connes' noncommutative geometry. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang-Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgenii Sklyanin, Nicolai Reshetikhin and Korepin) and related work by the Japanese School.[1] The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity[2].

## Drinfel'd-Jimbo type quantum groups

One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfel'd and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac-Moody algebra, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra.

Let A = (aij) be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, Uq(G), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators kλ (where λ is an element of the weight lattice, i.e. $2 (\lambda,\alpha_i)/(\alpha_i,\alpha_i) \in \mathbb{Z}$ for all i), and ei and fi (for simple roots, αi), subject to the following relations:

• k0 = 1,
• kλkμ = kλ + μ,
• $k_{\lambda} e_i k_{\lambda}^{-1} = q^{(\lambda,\alpha_i)} e_i$,
• $k_{\lambda} f_i k_{\lambda}^{-1} = q^{- (\lambda,\alpha_i)} f_i$,
• $[e_i,f_j] = \delta_{ij} \frac{k_i - k_i^{-1}}{q_i - q_i^{-1}}$,
• $\sum_{n=0}^{1 - a_{ij}} (-1)^n \frac{[1 - a_{ij}]_{q_i}!}{[1 - a_{ij} - n]_{q_i}! [n]_{q_i}!} e_i^n e_j e_i^{1 - a_{ij} - n} = 0$, for $i \ne j$,
• $\sum_{n=0}^{1 - a_{ij}} (-1)^n \frac{[1 - a_{ij}]_{q_i}!}{[1 - a_{ij} - n]_{q_i}! [n]_{q_i}!} f_i^n f_j f_i^{1 - a_{ij} - n} = 0$, for $i \ne j$,

where $k_i = k_{\alpha_i}$, $q_i = q^{\frac{1}{2}(\alpha_i,\alpha_i)}$, $[0]_{q_i}! = 1$, $[n]_{q_i}! = \prod_{m=1}^n [m]_{q_i}$ for all positive integers n, and $[m]_{q_i} = \frac{q_i^m - q_i^{-m}}{q_i - q_i^{-1}}.$ These are the q-factorial and q-number, respectively, the q-analogs of the ordinary factorial. The last two relations above are the q-Serre relations, the deformations of the Serre relations.

In the limit as $q \to 1$, these relations approach the relations for the universal enveloping algebra U(G), where $k_{\lambda} \to 1$ and $\frac{k_{\lambda} - k_{-\lambda}}{q - q^{-1}} \to t_{\lambda}$ as $q \to 1$, where the element, tλ, of the Cartan subalgebra satisfies (tλ,h) = λ(h) for all h in the Cartan subalgebra.

There are various coassociative coproducts under which these algebras are Hopf algebras, for example,

• $\Delta_1(k_\lambda) = k_\lambda \otimes k_\lambda$, $\Delta_1(e_i) = 1 \otimes e_i + e_i \otimes k_i$, $\Delta_1(f_i) = k_i^{-1} \otimes f_i + f_i \otimes 1$,
• $\Delta_2(k_\lambda) = k_\lambda \otimes k_\lambda$, $\Delta_2(e_i) = k_i^{-1} \otimes e_i + e_i \otimes 1$, $\Delta_2(f_i) = 1 \otimes f_i + f_i \otimes k_i$,
• $\Delta_3(k_\lambda) = k_\lambda \otimes k_\lambda$, $\Delta_3(e_i) = k_i^{-\frac{1}{2}} \otimes e_i + e_i \otimes k_i^{\frac{1}{2}}$, $\Delta_3(f_i) = k_i^{-\frac{1}{2}} \otimes f_i + f_i \otimes k_i^{\frac{1}{2}}$, where the set of generators has been extended, if required, to include kλ for λ which is expressible as the sum of an element of the weight lattice and half an element of the root lattice.

In addition, any Hopf algebra leads to another with reversed copproduct $T \circ \Delta$, where T is given by $T(x \otimes y) = y \otimes x$, giving three more possible versions.

The counit on Uq(A) is the same for all these coproducts: ε(kλ) = 1, ε(ei) = 0, ε(fi) = 0, and the respective antipodes for the above coproducts are given by

• $S_1(k_{\lambda}) = k_{-\lambda},\ S_1(e_i) = - e_i k_i^{-1},\ S_1(f_i) = - k_i f_i$,
• $S_2(k_{\lambda}) = k_{-\lambda},\ S_2(e_i) = - k_i e_i,\ S_2(f_i) = - f_i k_i^{-1}$,
• $S_3(k_{\lambda}) = k_{-\lambda},\ S_3(e_i) = - q_i e_i,\ S_3(f_i) = - q_i^{-1} f_i.$

Alternatively, the quantum group Uq(G) can be regarded as an algebra over the field ${\Bbb C}(q)$, the field of all rational functions of an indeterminate q over $\Bbb C$.

Similarly, the quantum group Uq(G) can be regarded as an algebra over the field ${\Bbb Q}(q)$, the field of all rational functions of an indeterminate q over $\Bbb Q$ (see below in the section on quantum groups at q = 0).

### Representation theory

Just as there are many different types of representations for Kac-Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups.

As is the case for all Hopf algebras, Uq(G) has an adjoint representation on itself as a module, with the action being given by

where $\Delta(x) = \sum_{(x)} x_{(1)} \otimes x_{(2)}$.

#### Case 1: q is not a root of unity

One important type of representation is a weight representation, and the corresponding module is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector v such that kλ.v = dλv for all λ, where dλ are complex numbers for all weights λ such that

• d0 = 1,
• dλdμ = dλ + μ, for all weights λ and μ.

A weight module is called integrable if the actions of ei and fi are locally nilpotent (i.e. for any vector v in the module, there exists a positive integer k, possibly dependent on v, such that $e_i^k.v = f_i^k.v = 0$ for all i). In the case of integrable modules, the complex numbers dλ associated with a weight vector satisfy dλ = cλq(λ,ν), where ν is an element of the weight lattice, and cλ are complex numbers such that

• $c_0 = 1,\,$
• cλcμ = cλ + μ, for all weights λ and μ,
• $c_{2\alpha_i} = 1$ for all i.

Of special interest are highest weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vector v, subject to kλ.v = dλv for all weights λ, and ei.v = 0 for all i. Similarly, a quantum group can have a lowest weight representation and lowest weight module, i.e. a module generated by a weight vector v, subject to kλ.v = dλv for all weights λ, and fi.v = 0 for all i.

Define a vector v to have weight ν if kλ.v = q(λ,ν)v for all λ in the weight lattice.

If G is a Kac-Moody algebra, then in any irreducible highest weight representation of Uq(G), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of U(G) with equal highest weight. If the highest weight is dominant and integral (a weight μ is dominant and integral if μ satisfies the condition that 2(μ,αi) / (αii) is a non-negative integer for all i), then the weight spectrum of the irreducible representation is invariant under the Weyl group for G, and the representation is integrable.

Conversely, if a highest weight module is integrable, then its highest weight vector v satisfies kλ.v = cλq(λ,ν)v, where cλ are complex numbers such that

• c0 = 1,
• cλcμ = cλ + μ, for all weights λ and μ,
• $c_{2\alpha_i} = 1$ for all i,

and ν is dominant and integral.

As is the case for all Hopf algebras, the tensor product of two modules is another module. For an element x of Uq(G), and for vectors v and w in the respective modules, $x.(v \otimes w) = \Delta(x).(v \otimes w)$, so that $k_{\lambda}.(v \otimes w) = k_{\lambda}.v \otimes k_{\lambda}.w$, and in the case of coproduct Δ1, $e_i.(v \otimes w) = k_i.v \otimes e_i.w + e_i.v \otimes w$ and $f_i.(v \otimes w) = v \otimes f_i.w + f_i.v \otimes k_i^{-1}.w$.

The integrable highest weight module described above is a tensor product of a one-dimensional module (on which kλ = cλ for all λ, and ei = fi = 0 for all i) and a highest weight module generated by a nonzero vector v0, subject to kλ.v0 = q(λ,ν)v0 for all weights λ, and ei.v0 = 0 for all i.

In the specific case where G is a finite-dimensional Lie algebra (as a special case of a Kac-Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional.

In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac-Moody algebra (the highest weights are the same, as are their multiplicities).

### Quasitriangularity

#### Case 1: q is not a root of unity

Strictly, the quantum group Uq(G) is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an R-matrix. This infinite formal sum is expressible in terms of generators ei and fi, and Cartan generators tλ, where kλ is formally identified with $q^{t_{\lambda}}$. The infinite formal sum is the product of two factors, $q^{\eta \sum_j t_{\lambda_j} \otimes t_{\mu_j}}$, and an infinite formal sum, where j} is a basis for the dual space to the Cartan subalgebra, and j} is the dual basis, and η is a sign (+1 or -1).

The formal infinite sum which plays the part of the R-matrix has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product if two lowest weight modules. Specifically, if v has weight α and w has weight β, then $q^{\eta \sum_j t_{\lambda_j} \otimes t_{\mu_j}}.(v \otimes w) = q^{\eta (\alpha,\beta)} v \otimes w$, and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on $v \otimes w$ to a finite sum.

Specifically, if V is a highest weight module, then the formal infinite sum, R, has a well-defined, and invertible, action on $V \otimes V$, and this value of R (as an element of $\mathrm{Hom}(V) \otimes \mathrm{Hom}(V)$) satisfies the Yang-Baxter equation, and therefore allows us to determine a representation of the braid group, and to define quasi-invariants for knots, links and braids.

### Quantum groups at q = 0

Masaki Kashiwara has researched the limiting behaviour of quantum groups as $q \to 0$.

As a consequence of the defining relations for the quantum group Uq(G), Uq(G) can be regarded as a Hopf algebra over ${\Bbb Q}(q)$, the field of all rational functions of an indeterminate q over $\Bbb Q$.

For simple root αi and non-negative integer n, define $e_i^{(n)} = e_i^n/[n]_{q_i}!$ and $f_i^{(n)} = f_i^n/[n]_{q_i}!$ (specifically, $e_i^{(0)} = f_i^{(0)} = 1$). In an integrable module M, and for weight λ, a vector $u \in M_{\lambda}$ (i.e. a vector u in M with weight λ) can be uniquely decomposed into the sums

• $u = \sum_{n=0}^\infty f_i^{(n)} u_n = \sum_{n=0}^\infty e_i^{(n)} v_n$,

where $u_n \in \mathrm{ker}(e_i) \cap M_{\lambda + n \alpha_i}$, $v_n \in \mathrm{ker}(f_i) \cap M_{\lambda - n \alpha_i}$, $u_n \ne 0$ only if $n + \frac{2 (\lambda,\alpha_i)}{(\alpha_i,\alpha_i)} \ge 0$, and $v_n \ne 0$ only if $n - \frac{2 (\lambda,\alpha_i)}{(\alpha_i,\alpha_i)} \ge 0$. Linear mappings $\tilde{e}_i : M \to M$ and $\tilde{f}_i : M \to M$ can be defined on Mλ by

• $\tilde{e}_i u = \sum_{n=1}^\infty f_i^{(n-1)} u_n = \sum_{n=0}^\infty e_i^{(n+1)} v_n$,
• $\tilde{f}_i u = \sum_{n=0}^\infty f_i^{(n+1)} u_n = \sum_{n=1}^\infty e_i^{(n-1)} v_n$.

Let A be the integral domain of all rational functions in ${\Bbb Q}(q)$ which are regular at q = 0 (i.e. a rational function f(q) is an element of A if and only if there exist polynomials g(q) and h(q) in the polynomial ring ${\Bbb Q}[q]$ such that $h(0) \ne 0$, and f(q) = g(q) / h(q)). A crystal base for M is an ordered pair (L,B), such that

• L is a free A-submodule of M such that $M = {\Bbb Q}(q) \otimes_A L$;
• B is a $\Bbb Q$-basis of the vector space L / qL over $\Bbb Q$,
• $L = \oplus_{\lambda} L_{\lambda}$ and $B = \sqcup_{\lambda} B_{\lambda}$, where $L_{\lambda} = L \cap M_{\lambda}$ and $B_{\lambda} = B \cap (L_{\lambda}/qL_{\lambda})$,
• $\tilde{e}_i L \subset L$ and $\tilde{f}_i L \subset L$ for all i,
• $\tilde{e}_i B \subset B \cup \{0\}$ and $\tilde{f}_i B \subset B \cup \{0\}$ for all i,
• for all $b \in B$ and $b' \in B$, and for all i, $\tilde{e}_i b = b'$ if and only if $\tilde{f}_i b' = b$.

To put this into a more informal setting, the actions of eifi and fiei are generally singular at q = 0 on an integrable module M. The linear mappings $\tilde{e}_i$ and $\tilde{f}_i$ on the module are introduced so that the actions of $\tilde{e}_i \tilde{f}_i$ and $\tilde{f}_i \tilde{e}_i$ are regular at q = 0 on the module. There exists a ${\Bbb Q}(q)$-basis of weight vectors $\tilde{B}$ for M, with respect to which the actions of $\tilde{e}_i$ and $\tilde{f}_i$ are regular at q = 0 for all i. The module is then restricted to the free A-module generated by the basis, and the basis vectors, the A-submodule and the actions of $\tilde{e}_i$ and $\tilde{f}_i$ are evaluated at q = 0. Furthermore, the basis can be chosen such that at q = 0, for all i, $\tilde{e}_i$ and $\tilde{f}_i$ are represented by mutual transposes, and map basis vectors to basis vectors or 0.

A crystal base can be represented by a directed graph with labelled edges. Each vertex of the graph represents an element of the $\Bbb Q$-basis B of L / qL, and a directed edge, labelled by i, and directed from vertex v1 to vertex v2, represents that $b_2 = \tilde{f}_i b_1$ (and, equivalently, that $b_1 = \tilde{e}_i b_2$), where b1 is the basis element represented by v1, and b2 is the basis element represented by v2. The graph completely determines the actions of $\tilde{e}_i$ and $\tilde{f}_i$ at q = 0. If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets V1 and V2 such that there are no edges joining any vertex in V1 to any vertex in V2).

For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate Kac-Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac-Moody algebra.

It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base.

#### Tensor products of crystal bases

Let M be an integrable module with crystal base (L,B) and M' be an integrable module with crystal base (L',B'). For crystal bases, the coproduct Δ, given by $\Delta(k_{\lambda}) = k_{\lambda} \otimes k_{\lambda},\ \Delta(e_i) = e_i \otimes k_i^{-1} + 1 \otimes e_i,\ \Delta(f_i) = f_i \otimes 1 + k_i \otimes f_i$, is adopted. The integrable module $M \otimes_{{\Bbb Q}(q)} M'$ has crystal base $(L \otimes_A L',B \otimes B')$, where $B \otimes B' = \{ b \otimes_{\Bbb Q} b' : b \in B,\ b' \in B' \}$. For a basis vector $b \in B$, define $\epsilon_i(b) = \mathrm{max}\{ n \ge 0 : \tilde{e}_i^n b \ne 0 \}$ and $\phi_i(b) = \mathrm{max}\{ n \ge 0 : \tilde{f}_i^n b \ne 0 \}$. The actions of $\tilde{e}_i$ and $\tilde{f}_i$ on $b \otimes b'$ are given by

• $\tilde{e}_i (b \otimes b') = \left\{ \begin{matrix} \tilde{e}_i b \otimes b', & \mathrm{if} \ \phi_i(b) \ge \epsilon_i(b'), \\ b \otimes \tilde{e}_i b', & \mathrm{if} \ \phi_i(b) < \epsilon_i(b'), \end{matrix} \right.$
• $\tilde{f}_i (b \otimes b') = \left\{ \begin{matrix} \tilde{f}_i b \otimes b', & \mathrm{if} \ \phi_i(b) > \epsilon_i(b'), \\ b \otimes \tilde{f}_i b', & \mathrm{if} \ \phi_i(b) \le \epsilon_i(b'). \end{matrix} \right.$

The decomposition of the product two integrable highest weight modules into irreducible submodules is determined by the decomposition of the graph of the crystal base into its connected components (i.e. the highest weights of the submodules are determined, and the multiplicity of each highest weight is determined).

## Compact matrix quantum groups

S.L. Woronowicz introduced compact matrix quantum groups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.

The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. By the Gelfand theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.

For a compact topological group, G, there exists a C*-algebra homomorphism $\Delta : C(G) \to C(G) \otimes C(G)$ (where $C(G) \otimes C(G)$ is the C*-algebra tensor product - the completion of the algebraic tensor product of C(G) and C(G)), such that Δ(f)(x,y) = f(xy) for all $f \in C(G)$, and for all $x, y \in G$ (where $(f \otimes g)(x,y) = f(x) g(y)$ for all $f, g \in C(G)$ and all $x, y \in G$). There also exists a linear multiplicative mapping $\kappa : C(G) \to C(G)$, such that κ(f)(x) = f(x − 1) for all $f \in C(G)$ and all $x \in G$. Strictly, this does not make C(G) a Hopf algebra, unless G is finite. On the other hand, a finite-dimensional representation of G can be used to generate a *-subalgebra of C(G) which is also a Hopf *-algebra. Specifically, if $g \mapsto (u_{ij}(g))_{i,j}$ is an n-dimensional representation of G, then $u_{ij} \in C(G)$ for all i,j, and $\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}$ for all i,j. It follows that the *-algebra generated by uij for all i,j and κ(uij) for all i,j is a Hopf *-algebra: the counit is determined by ε(uij) = δij for all i,j (where δij is the Kronecker delta), the antipode is κ, and the unit is given by

 1 = ∑ u1kκ(uk1) = ∑ κ(u1k)uk1. k k

As a generalization, a compact matrix quantum group is defined as a pair (C,u), where C is a C*-algebra and $u = (u_{ij})_{i,j = 1,\dots,n}$ is a matrix with entries in C such that

• The *-subalgebra, C0, of C, which is generated by the matrix elements of u, is dense in C;
• There exists a C*-algebra homomorphism $\Delta : C \to C \otimes C$ (where $C \otimes C$ is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that $\Delta(u_{ij}) = \sum_k u_{ik} \otimes u_{kj}$ for all i,j (Δ is called the comultiplication);
• There exists a linear antimultiplicative map $\kappa : C_0 \to C_0$ (the coinverse) such that κ(κ(v * ) * ) = v for all $v \in C_0$ and  ∑ κ(uik)uk j = ∑ uikκ(uk j) = δijI, k k
where I is the identity element of C. Since κ is antimultiplicative, then κ(vw) = κ(w)κ(v) for all $v, w \in C_0$.

As a consequence of continuity, the comultiplication on C is coassociative.

In general, C is not a bialgebra, and C0 is a Hopf *-algebra.

Informally, C can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.

A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassiative coalgebra A is a square matrix $v = (v_{ij})_{i,j = 1,\dots,n}$ with entries in A (so $v \in M_n(A)$) such that $\Delta(v_{ij}) = \sum_{k=1}^n v_{ik} \otimes v_{kj}$ for all i,j and ε(vij) = δij for all i,j). Furthermore, a representation, v, is called unitary if the matrix for v is unitary (or equivalently, if $\kappa(v_{ij}) = v^*_{ji}$ for all i, j).

An example of a compact matrix quantum group is SUμ(2), where the parameter μ is a positive real number. So SUμ(2) = (C(SUμ(2),u), where C(SUμ(2)) is the C*-algebra generated by α and γ,subject to

$\gamma \gamma^* = \gamma^* \gamma, \ \alpha \gamma = \mu \gamma \alpha, \ \alpha \gamma^* = \mu \gamma^* \alpha, \ \alpha \alpha^* + \mu \gamma^* \gamma = \alpha^* \alpha + \mu^{-1} \gamma^* \gamma = I,$

and $u = \left( \begin{matrix} \alpha & \gamma \\ - \gamma^* & \alpha^* \end{matrix} \right),$ so that the comultiplication is determined by $\Delta(\alpha) = \alpha \otimes \alpha - \gamma \otimes \gamma^*$, $\Delta(\gamma) = \alpha \otimes \gamma + \gamma \otimes \alpha^*$, and the coinverse is determined by κ(α) = α * , κ(γ) = − μ − 1γ, κ(γ * ) = − μγ * , κ(α * ) = α. Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation $v = \left( \begin{matrix} \alpha & \sqrt{\mu} \gamma \\ - \frac{1}{\sqrt{\mu}} \gamma^* & \alpha^* \end{matrix} \right).$

Equivalently, SUμ(2) = (C(SUμ(2)),w), where C(SUμ(2)) is the C*-algebra generated by α and β,subject to

$\beta \beta^* = \beta^* \beta, \ \alpha \beta = \mu \beta \alpha, \ \alpha \beta^* = \mu \beta^* \alpha, \ \alpha \alpha^* + \mu^2 \beta^* \beta = \alpha^* \alpha + \beta^* \beta = I,$

and $w = \left( \begin{matrix} \alpha & \mu \beta \\ - \beta^* & \alpha^* \end{matrix} \right),$ so that the comultiplication is determined by $\Delta(\alpha) = \alpha \otimes \alpha - \mu \beta \otimes \beta^*$, $\Delta(\beta) = \alpha \otimes \beta + \beta \otimes \alpha^*$, and the coinverse is determined by κ(α) = α * , κ(β) = − μ − 1β, κ(β * ) = − μβ * , κ(α * ) = α. Note that w is a unitary representation. The realizations can be identified by equating $\gamma = \sqrt{\mu} \beta$.

When μ = 1, then SUμ(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).

## Bicrossproduct quantum groups

Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first. The very simplest nontrivial example corresponds to two copies of $\Bbb R$ locally acting on each other and results in a quantum group (given here in an algebraic form) with generators p,K,K − 1, say, and coproduct

[p,K] = hK(K − 1), $\Delta p=p\otimes K+1\otimes p$, $\Delta K=K\otimes K$

where h is the deformation parameter. This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra (the Iwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated to g. For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions.

## Notes

1. ^ Schwiebert, Christian (1994), Generalized quantum inverse scattering, arΧiv:hep-th/9412237v3
2. ^ Majid, Shahn (1988), "Hopf algebras for physics at the Planck scale", Classical and Quantum Gravity 5: 1587-1607