Quantum group

In mathematics and theoretical physics, quantum groups are certain noncommutative algebras that first appeared in the theory of quantum integrable systems, and which were then formalized by Vladimir Drinfel'd and Michio Jimbo. There is no single, all-encompassing definition of quantum group.

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. Distinct but related objects, also called quantum groups, are deformations of the algebra of functions on a semisimple algebraic group or a compact Lie group.

Since the discovery of quantum groups, it has become fashionable to introduce the attribute "quantum" into the names of many other mathematical objects, such as quantum plane or quantum grassmanian. They may also be loosely referred to as aspects of "quantum groups".

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, cannot be deformed. One of the ideas behind quantum groups is that if we consider in some sense equivalent but larger structure, namely a group algebra or a universal enveloping algebra, then it "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 others) and related work by the Japanese School.Fact|date=March 2007

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 = (a_{ij}) be the Cartan matrix of the Kac-Moody algebra, and let "q" be a nonzero complex number distinct from 1, then the quantum group, U_q(G), where "G" is the Lie algebra whose Cartan matrix is "A", is defined as the unital associative algebra with generators k_{lambda} (where lambda is an element of the weight lattice, "i.e." 2 (lambda,alpha_i)/(alpha_i,alpha_i) in mathbb{Z} for all "i"), and e_i and f_i (for simple roots, alpha_i), subject to the following relations:

*k_0 = 1,

*k_{lambda} k_{mu} = k_{lambda+mu},

*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 e 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 e 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 o 1, these relations approach the relations for the universal enveloping algebra U_q(G), where k_{lambda} o 1 and frac{k_{lambda} - k_{-lambda{q - q^{-1 o t_{lambda} as q o 1, where the element, t_{lambda}, of the Cartan subalgebra satisfies (t_{lambda},h) = lambda(h) for all "h" in the Cartan subalgebra.

There are various coassociative coproducts under which the quantum groups 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 k_{lambda} for λ which is expressible as the sum of an element of the weight lattice and half an element of the root lattice,

along with the reverse coproducts T circ Delta, where T : U_q(G) otimes U_q(G) o U_q(G) otimes U_q(G) is given by T(x otimes y) = y otimes x,"i.e."

:*Delta_4(k_lambda) = k_lambda otimes k_lambda, Delta_4(e_i) = k_i otimes e_i + e_i otimes 1, Delta_4(f_i) = 1 otimes f_i + f_i otimes k_i^{-1}, where Delta_4 = T circ Delta_1,

:*Delta_5(k_lambda) = k_lambda otimes k_lambda, Delta_5(e_i) = 1 otimes e_i + e_i otimes k_i^{-1}, Delta_5(f_i) = k_i otimes f_i + f_i otimes 1, where Delta_5 = T circ Delta_2,

:*Delta_6(k_lambda) = k_lambda otimes k_lambda, Delta_6(e_i) = k_i^{frac{1}{2 otimes e_i + e_i otimes k_i^{-frac{1}{2, Delta_6(f_i) = k_i^{frac{1}{2 otimes f_i + f_i otimes k_i^{-frac{1}{2, where Delta_6 = T circ Delta_3.

The counit on U_q(A) is the same for all these coproducts: epsilon(k_{lambda}) = 1, epsilon(e_i) = 0, epsilon(f_i) = 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,

:*S_4(k_{lambda}) = k_{-lambda}, S_4(e_i) = - k_i^{-1} e_i, S_4(f_i) = - f_i k_i,

:*S_5(k_{lambda}) = k_{-lambda}, S_5(e_i) = - e_i k_i, S_5(f_i) = - k_i^{-1} f_i,

:*S_6(k_{lambda}) = k_{-lambda}, S_6(e_i) = - q_i^{-1} e_i, S_6(f_i) = - q_i f_i.

Alternatively, the quantum group U_q(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 U_q(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 representation 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, U_q(G) has an adjoint representation on itself as a module, with the action being given by {mathrm{Ad_x.y = sum_{(x)} x_{(1)} y S(x_{(2)}), 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 k_{lambda}.v = d_{lambda} v for all lambda, where d_{lambda} are complex numbers for all weights lambda such that

:*d_0 = 1,

:*d_{lambda} d_{mu} = d_{lambda + mu}, for all weights lambda and mu.

A weight module is called integrable if the actions of e_i and f_i 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 d_{lambda} associated with a weight vector satisfy d_{lambda} = c_{lambda} q^{(lambda, u)}, where u is an element of the weight lattice, and c_{lambda} are complex numbers such that

:*c_0 = 1,

:*c_{lambda} c_{mu} = c_{lambda + mu}, for all weights lambda and mu,

:*c_{2alpha_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 k_{lambda}.v = d_{lambda} v for all weights lambda, and e_i.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 k_{lambda}.v = d_{lambda} v for all weights lambda, and f_i.v = 0 for all "i".

Define a vector "v" to have weight u if k_{lambda}.v = q^{(lambda, u)} v for all lambda in the weight lattice.

If "G" is a Kac-Moody algebra, then in any irreducible highest weight representation of U_q(G), with highest weight u, 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 mu is dominant and integral if mu satisfies the condition that 2 (mu,alpha_i)/(alpha_i,alpha_i) 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 k_{lambda}.v = c_{lambda} q^{(lambda, u)} v, where c_{lambda} are complex numbers such that

:*c_0 = 1,

:*c_{lambda} c_{mu} = c_{lambda + mu}, for all weights lambda and mu,

:*c_{2alpha_i} = 1 for all "i",

and u 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 U_q(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 Delta_1, 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 k_{lambda} = c_{lambda} for all lambda, and e_i = f_i = 0 for all "i") and a highest weight module generated by a nonzero vector v_0, subject to k_{lambda}.v_0 = q^{(lambda, u)} v_0 for all weights lambda, and e_i.v_0 = 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).

Case 2: "q" is a root of unity

Quasitriangularity

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

Strictly, the quantum group U_q(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 e_i and f_i, and Cartan generators t_{lambda}, where k_{lambda} 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 {lambda_j} is a basis for the dual space to the Cartan subalgebra, and {mu_j} is the dual basis, and eta 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 alpha and "w" has weight eta, then q^{eta sum_j t_{lambda_j} otimes t_{mu_j.(v otimes w) = q^{eta (alpha,eta)} 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.

Case 2: "q" is a root of unity


=Quantum groups at "q" = 0=

Masaki Kashiwara has researched the limiting behaviour of quantum groups as q o 0.

As a consequence of the defining relations for the quantum group U_q(G), U_q(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 alpha_i 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 lambda, a vector u in M_{lambda} ("i.e." a vector u in M with weight lambda) 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 e 0 only if n + frac{2 (lambda,alpha_i)}{(alpha_i,alpha_i)} ge 0, and v_n e 0 only if n - frac{2 (lambda,alpha_i)}{(alpha_i,alpha_i)} ge 0. Linear mappings ilde{e}_i : M o M and ilde{f}_i : M o M can be defined on M_{lambda} by

* ilde{e}_i u = sum_{n=1}^infty f_i^{(n-1)} u_n = sum_{n=0}^infty e_i^{(n+1)} v_n,

* ilde{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) e 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}),

:* ilde{e}_i L subset L and ilde{f}_i L subset L for all "i",

:* ilde{e}_i B subset B cup {0} and ilde{f}_i B subset B cup {0} for all "i",

:*for all b in B and b' in B, and for all "i", ilde{e}_i b = b' if and only if ilde{f}_i b' = b.

To put this into a more informal setting, the actions of e_i f_i and f_i e_i are generally singular at q = 0 on an integrable module M. The linear mappings ilde{e}_i and ilde{f}_i on the module are introduced so that the actions of ilde{e}_i ilde{f}_i and ilde{f}_i ilde{e}_i are regular at q = 0 on the module. There exists a {Bbb Q}(q)-basis of weight vectors ilde{B} for M, with respect to which the actions of ilde{e}_i and ilde{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 ilde{e}_i and ilde{f}_i are evaluated at q = 0. Furthermore, the basis can be chosen such that at q = 0, for all i, ilde{e}_i and ilde{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 v_1 to vertex v_2, represents that b_2 = ilde{f}_i b_1 (and, equivalently, that b_1 = ilde{e}_i b_2), where b_1 is the basis element represented by v_1, and b_2 is the basis element represented by v_2. The graph completely determines the actions of ilde{e}_i and ilde{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 V_1 and V_2 such that there are no edges joining any vertex in V_1 to any vertex in V_2).

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 Delta, 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 : ilde{e}_i^n b e 0 } and phi_i(b) = mathrm{max}{ n ge 0 : ilde{f}_i^n b e 0 }. The actions of ilde{e}_i and ilde{f}_i on b otimes b' are given by

:* ilde{e}_i (b otimes b') = left{ egin{matrix} ilde{e}_i b otimes b', & mathrm{if} phi_i(b) ge epsilon_i(b'), \ b otimes ilde{e}_i b', & mathrm{if} phi_i(b) < epsilon_i(b'), end{matrix} ight.

:* ilde{f}_i (b otimes b') = left{ egin{matrix} ilde{f}_i b otimes b', & mathrm{if} phi_i(b) > epsilon_i(b'), \ b otimes ilde{f}_i b', & mathrm{if} phi_i(b) le epsilon_i(b'). end{matrix} ight.

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) o 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 Delta(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) o C(G), such that kappa(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 u_{ij} for all i, j and kappa(u_{ij}) for all i, j is a Hopf *-algebra: the counit is determined by epsilon(u_{ij}) = delta_{ij} for all i, j (where delta_{ij} is the Kronecker delta), the antipode is kappa, and the unit is given by 1 = sum_k u_{1k} kappa(u_{k1}) = sum_k kappa(u_{1k}) u_{k1}.

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, C_0, of C, which is generated by the matrix elements of u, is dense in C;

:*There exists a C*-algebra homomorphism Delta : C o 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 (Delta is called the comultiplication);

:*There exists a linear antimultiplicative map kappa : C_0 o C_0 (the coinverse) such that kappa(kappa(v*)*) = v for all v in C_0 and sum_k kappa(u_{ik}) u_{kj} = sum_k u_{ik} kappa(u_{kj}) = delta_{ij} I, where I is the identity element of C. Since kappa is antimultiplicative, then kappa(vw) = kappa(w) kappa(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 C_0 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 epsilon(v_{ij}) = delta_{ij} 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 SU_{mu}(2), where the parameter mu is a positive real number. So SU_{mu}(2) = (C(SU_{mu}(2),u), where C(SU_{mu}(2)) is the C*-algebra generated by alpha and gamma,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( egin{matrix} alpha & gamma \ - gamma^* & alpha^* end{matrix} ight), 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 kappa(alpha) = alpha^*, kappa(gamma) = - mu^{-1} gamma, kappa(gamma^*) = - mu gamma^*, kappa(alpha^*) = alpha. Note that u is a representation, but not a unitary representation. u is equivalent to the unitary representation v = left( egin{matrix} alpha & sqrt{mu} gamma \ - frac{1}{sqrt{mu gamma^* & alpha^* end{matrix} ight).

Equivalently, SU_{mu}(2) = (C(SU_{mu}(2),w), where C(SU_{mu}(2)) is the C*-algebra generated by alpha and eta,subject to

:eta eta^* = eta^* eta, alpha eta = mu eta alpha, alpha eta^* = mu eta^* alpha, alpha alpha^* + mu^2 eta^* eta = alpha^* alpha + eta^* eta = I,

and w = left( egin{matrix} alpha & mu eta \ - eta^* & alpha^* end{matrix} ight), so that the comultiplication is determined by Delta(alpha) = alpha otimes alpha - mu eta otimes eta^*, Delta(eta) = alpha otimes eta + eta otimes alpha^*, and the coinverse is determined by kappa(alpha) = alpha^*, kappa(eta) = - mu^{-1} eta, kappa(eta^*) = - mu eta^*, kappa(alpha^*) = alpha. Note that w is a unitary representation. The realizations can be identified by equating gamma = sqrt{mu} eta.

When mu = 1, then SU_{mu}(2) is equal to the concrete compact group SU(2).

ee also

* Lie bialgebra
* Poisson-Lie group
* Affine quantum group

References

* [http://arxiv.org/abs/q-alg/9704002 Elementary introduction to quantum groups]
* Christian Kassel. "Quantum Groups" (Springer: 1994). ISBN 0-387-94370-6.
* Shahn Majid, N. J. Hitchin (series editor). "A Quantum Groups Primer" (Cambridge University Press: 2002). ISBN 0-521-01041-1.
* Ross Street, "Quantum Groups: A Path to Current Algebra" (Cambridge University Press: 2007). ISBN 0-521-69524-4
* cite journal
last = Majid
first = Shahn
title = What Is...a Quantum Group?
journal = Notices of the American Mathematical Society
year = 2006
month = January
volume = 53
issue = 1
pages = pp.30&ndash;31
url = http://www.ams.org/notices/200601/what-is.pdf
format = PDF
accessdate = 2008-01-16


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  • Locally compact quantum group — The locally compact (l.c.) quantum group is a relatively new C* algebraic formalism for quantum groups, generalizing the Kac algebra, compact quantum group and Hopf algebra approaches. Earlier attempts of a unifying definition of quantum groups… …   Wikipedia

  • Affine quantum group — is a common name of several objects in representation theory, which include Yangians and quantized universal enveloping algebras of affine Kac Moody Lie algebras (quantized affine algebras).Affine quantum groups were introduced by Vladimir… …   Wikipedia

  • Supersymmetry as a quantum group — The concept in theoretical physics of supersymmetry can be reinterpretated in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity. ( 1)F Let s look at… …   Wikipedia

  • Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia

  • Quantum number — Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc.Since any quantum system… …   Wikipedia

  • Quantum mind — theories are based on the premise that quantum mechanics is necessary to fully understand the mind and brain, particularly concerning an explanation of consciousness. This approach is considered a minority opinion in science, although it does… …   Wikipedia

  • Quantum information science — concerns information science that depends on quantum effects in physics. It includes theoretical issues in computational models as well as more experimental topics in quantum physics including what can and cannot be done with quantum information …   Wikipedia

  • Quantum error correction — is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault tolerant quantum computation that can deal not only with noise on …   Wikipedia

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