Definitions

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.

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

• $k_0 = 1$,
• $k_\left\{lambda\right\} k_\left\{mu\right\} = k_\left\{lambda+mu\right\}$,
• $k_\left\{lambda\right\} e_i k_\left\{lambda\right\}^\left\{-1\right\} = q^\left\{\left(lambda,alpha_i\right)\right\} e_i$,
• $k_\left\{lambda\right\} f_i k_\left\{lambda\right\}^\left\{-1\right\} = q^\left\{- \left(lambda,alpha_i\right)\right\} f_i$,
• $\left[e_i,f_j\right] = delta_\left\{ij\right\} frac\left\{k_i - k_i^\left\{-1\right\}\right\}\left\{q_i - q_i^\left\{-1\right\}\right\}$,
• $sum_\left\{n=0\right\}^\left\{1 - a_\left\{ij\right\}\right\} \left(-1\right)^n frac\left\{\left[1 - a_\left\{ij\right\}\right]_\left\{q_i\right\}!\right\}\left\{\left[1 - a_\left\{ij\right\} - n\right]_\left\{q_i\right\}! \left[n\right]_\left\{q_i\right\}!\right\} e_i^n e_j e_i^\left\{1 - a_\left\{ij\right\} - n\right\} = 0$, for $i ne j$,
• $sum_\left\{n=0\right\}^\left\{1 - a_\left\{ij\right\}\right\} \left(-1\right)^n frac\left\{\left[1 - a_\left\{ij\right\}\right]_\left\{q_i\right\}!\right\}\left\{\left[1 - a_\left\{ij\right\} - n\right]_\left\{q_i\right\}! \left[n\right]_\left\{q_i\right\}!\right\} f_i^n f_j f_i^\left\{1 - a_\left\{ij\right\} - n\right\} = 0$, for $i ne j$,

where $k_i = k_\left\{alpha_i\right\}$, $q_i = q^\left\{frac\left\{1\right\}\left\{2\right\}\left(alpha_i,alpha_i\right)\right\}$, $\left[0\right]_\left\{q_i\right\}! = 1$, $\left[n\right]_\left\{q_i\right\}! = prod_\left\{m=1\right\}^n \left[m\right]_\left\{q_i\right\}$ for all positive integers $n$, and $\left[m\right]_\left\{q_i\right\} = frac\left\{q_i^m - q_i^\left\{-m\right\}\right\}\left\{q_i - q_i^\left\{-1\right\}\right\}$. 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 $U_q\left(G\right)$, where $k_\left\{lambda\right\} to 1$ and $frac\left\{k_\left\{lambda\right\} - k_\left\{-lambda\right\}\right\}\left\{q - q^\left\{-1\right\}\right\} to t_\left\{lambda\right\}$ as $q to 1$, where the element, $t_\left\{lambda\right\}$, of the Cartan subalgebra satisfies $\left(t_\left\{lambda\right\},h\right) = lambda\left(h\right)$ for all h in the Cartan subalgebra.

There are various coassociative coproducts under which the quantum groups are Hopf algebras, for example,

*$Delta_1\left(k_lambda\right) = k_lambda otimes k_lambda$, $Delta_1\left(e_i\right) = 1 otimes e_i + e_i otimes k_i$, $Delta_1\left(f_i\right) = k_i^\left\{-1\right\} otimes f_i + f_i otimes 1$,

*$Delta_2\left(k_lambda\right) = k_lambda otimes k_lambda$, $Delta_2\left(e_i\right) = k_i^\left\{-1\right\} otimes e_i + e_i otimes 1$, $Delta_2\left(f_i\right) = 1 otimes f_i + f_i otimes k_i$,

*$Delta_3\left(k_lambda\right) = k_lambda otimes k_lambda$, $Delta_3\left(e_i\right) = k_i^\left\{-frac\left\{1\right\}\left\{2\right\}\right\} otimes e_i + e_i otimes k_i^\left\{frac\left\{1\right\}\left\{2\right\}\right\}$, $Delta_3\left(f_i\right) = k_i^\left\{-frac\left\{1\right\}\left\{2\right\}\right\} otimes f_i + f_i otimes k_i^\left\{frac\left\{1\right\}\left\{2\right\}\right\}$, where the set of generators has been extended, if required, to include $k_\left\{lambda\right\}$ 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 $T : U_q\left(G\right) otimes U_q\left(G\right) to U_q\left(G\right) otimes U_q\left(G\right)$ is given by $T\left(x otimes y\right) = y otimes x$, i.e.

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

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

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

The counit on $U_q\left(A\right)$ is the same for all these coproducts: $epsilon\left(k_\left\{lambda\right\}\right) = 1$, $epsilon\left(e_i\right) = 0$, $epsilon\left(f_i\right) = 0$, and the respective antipodes for the above coproducts are given by

*$S_1\left(k_\left\{lambda\right\}\right) = k_\left\{-lambda\right\}, S_1\left(e_i\right) = - e_i k_i^\left\{-1\right\}, S_1\left(f_i\right) = - k_i f_i$,

*$S_2\left(k_\left\{lambda\right\}\right) = k_\left\{-lambda\right\}, S_2\left(e_i\right) = - k_i e_i, S_2\left(f_i\right) = - f_i k_i^\left\{-1\right\}$,

*$S_3\left(k_\left\{lambda\right\}\right) = k_\left\{-lambda\right\}, S_3\left(e_i\right) = - q_i e_i, S_3\left(f_i\right) = - q_i^\left\{-1\right\} f_i$,

*$S_4\left(k_\left\{lambda\right\}\right) = k_\left\{-lambda\right\}, S_4\left(e_i\right) = - k_i^\left\{-1\right\} e_i, S_4\left(f_i\right) = - f_i k_i$,

*$S_5\left(k_\left\{lambda\right\}\right) = k_\left\{-lambda\right\}, S_5\left(e_i\right) = - e_i k_i, S_5\left(f_i\right) = - k_i^\left\{-1\right\} f_i$,

*$S_6\left(k_\left\{lambda\right\}\right) = k_\left\{-lambda\right\}, S_6\left(e_i\right) = - q_i^\left\{-1\right\} e_i, S_6\left(f_i\right) = - q_i f_i$.

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

Similarly, the quantum group $U_q\left(G\right)$ can be regarded as an algebra over the field $\left\{Bbb Q\right\}\left(q\right)$, 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, $U_q\left(G\right)$ has an adjoint representation on itself as a module, with the action being given by $\left\{mathrm\left\{Ad\right\}\right\}_x.y = sum_\left\{\left(x\right)\right\} x_\left\{\left(1\right)\right\} y S\left(x_\left\{\left(2\right)\right\}\right)$, where $Delta\left(x\right) = sum_\left\{\left(x\right)\right\} x_\left\{\left(1\right)\right\} otimes x_\left\{\left(2\right)\right\}$.

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 $k_\left\{lambda\right\}.v = d_\left\{lambda\right\} v$ for all $lambda$, where $d_\left\{lambda\right\}$ are complex numbers for all weights $lambda$ such that

*$d_0 = 1$,

*$d_\left\{lambda\right\} d_\left\{mu\right\} = d_\left\{lambda + mu\right\}$, 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 $d_\left\{lambda\right\}$ associated with a weight vector satisfy $d_\left\{lambda\right\} = c_\left\{lambda\right\} q^\left\{\left(lambda,nu\right)\right\}$, where $nu$ is an element of the weight lattice, and $c_\left\{lambda\right\}$ are complex numbers such that

*$c_0 = 1$,

*$c_\left\{lambda\right\} c_\left\{mu\right\} = c_\left\{lambda + mu\right\}$, for all weights $lambda$ and $mu$,

*$c_\left\{2alpha_i\right\} = 1$ 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 $k_\left\{lambda\right\}.v = d_\left\{lambda\right\} v$ 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 $k_\left\{lambda\right\}.v = d_\left\{lambda\right\} v$ for all weights $lambda$, and $f_i.v = 0$ for all i.

Define a vector v to have weight $nu$ if $k_\left\{lambda\right\}.v = q^\left\{\left(lambda,nu\right)\right\} v$ for all $lambda$ in the weight lattice.

If G is a Kac-Moody algebra, then in any irreducible highest weight representation of $U_q\left(G\right)$, with highest weight $nu$, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of $U\left(G\right)$ 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 $2 \left(mu,alpha_i\right)/\left(alpha_i,alpha_i\right)$ 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 $k_\left\{lambda\right\}.v = c_\left\{lambda\right\} q^\left\{\left(lambda,nu\right)\right\} v$, where $c_\left\{lambda\right\}$ are complex numbers such that

*$c_0 = 1$,

*$c_\left\{lambda\right\} c_\left\{mu\right\} = c_\left\{lambda + mu\right\}$, for all weights $lambda$ and $mu$,

*$c_\left\{2alpha_i\right\} = 1$ for all i,

and $nu$ 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 $U_q\left(G\right)$, and for vectors v and w in the respective modules, $x.\left(v otimes w\right) = Delta\left(x\right).\left(v otimes w\right)$, so that $k_\left\{lambda\right\}.\left(v otimes w\right) = k_\left\{lambda\right\}.v otimes k_\left\{lambda\right\}.w$, and in the case of coproduct $Delta_1$, $e_i.\left(v otimes w\right) = k_i.v otimes e_i.w + e_i.v otimes w$ and $f_i.\left(v otimes w\right) = v otimes f_i.w + f_i.v otimes k_i^\left\{-1\right\}.w$.

The integrable highest weight module described above is a tensor product of a one-dimensional module (on which $k_\left\{lambda\right\} = c_\left\{lambda\right\}$ 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 $k_\left\{lambda\right\}.v_0 = q^\left\{\left(lambda,nu\right)\right\} v_0$ 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).

Quasitriangularity

Case 1: q is not a root of unity

Strictly, the quantum group $U_q\left(G\right)$ 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 $t_\left\{lambda\right\}$, where $k_\left\{lambda\right\}$ is formally identified with $q^\left\{t_\left\{lambda\right\}\right\}$. The infinite formal sum is the product of two factors, $q^\left\{eta sum_j t_\left\{lambda_j\right\} otimes t_\left\{mu_j\right\}\right\}$, and an infinite formal sum, where $\left\{lambda_j\right\}$ is a basis for the dual space to the Cartan subalgebra, and $\left\{mu_j\right\}$ 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 $beta$, then $q^\left\{eta sum_j t_\left\{lambda_j\right\} otimes t_\left\{mu_j\right\}\right\}.\left(v otimes w\right) = q^\left\{eta \left(alpha,beta\right)\right\} v otimes w$, 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 $mathrm\left\{Hom\right\}\left(V\right) otimes mathrm\left\{Hom\right\}\left(V\right)$) 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 $U_q\left(G\right)$, $U_q\left(G\right)$ can be regarded as a Hopf algebra over $\left\{Bbb Q\right\}\left(q\right)$, the field of all rational functions of an indeterminate q over $Bbb Q$.

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

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

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

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

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

*$L$ is a free $A$-submodule of $M$ such that $M = \left\{Bbb Q\right\}\left(q\right) otimes_A L$;

*$B$ is a $Bbb Q$-basis of the vector space $L/qL$ over $Bbb Q$,

*$L = oplus_\left\{lambda\right\} L_\left\{lambda\right\}$ and $B = sqcup_\left\{lambda\right\} B_\left\{lambda\right\}$, where $L_\left\{lambda\right\} = L cap M_\left\{lambda\right\}$ and $B_\left\{lambda\right\} = B cap \left(L_\left\{lambda\right\}/qL_\left\{lambda\right\}\right)$,

*$tilde\left\{e\right\}_i L subset L$ and $tilde\left\{f\right\}_i L subset L$ for all i,

*$tilde\left\{e\right\}_i B subset B cup \left\{0\right\}$ and $tilde\left\{f\right\}_i B subset B cup \left\{0\right\}$ for all i,

*for all $b in B$ and $b\text{'} in B$, and for all i, $tilde\left\{e\right\}_i b = b\text{'}$ if and only if $tilde\left\{f\right\}_i b\text{'} = b$.

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 $tilde\left\{e\right\}_i$ and $tilde\left\{f\right\}_i$ on the module are introduced so that the actions of $tilde\left\{e\right\}_i tilde\left\{f\right\}_i$ and $tilde\left\{f\right\}_i tilde\left\{e\right\}_i$ are regular at $q = 0$ on the module. There exists a $\left\{Bbb Q\right\}\left(q\right)$-basis of weight vectors $tilde\left\{B\right\}$ for $M$, with respect to which the actions of $tilde\left\{e\right\}_i$ and $tilde\left\{f\right\}_i$ 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 $tilde\left\{e\right\}_i$ and $tilde\left\{f\right\}_i$ are evaluated at $q = 0$. Furthermore, the basis can be chosen such that at $q = 0$, for all $i$, $tilde\left\{e\right\}_i$ and $tilde\left\{f\right\}_i$ 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 $b_2 = tilde\left\{f\right\}_i b_1$ (and, equivalently, that $b_1 = tilde\left\{e\right\}_i b_2$), 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 $tilde\left\{e\right\}_i$ and $tilde\left\{f\right\}_i$ 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 $\left(L,B\right)$ and $M\text{'}$ be an integrable module with crystal base $\left(L\text{'},B\text{'}\right)$. For crystal bases, the coproduct $Delta$, given by $Delta\left(k_\left\{lambda\right\}\right) = k_\left\{lambda\right\} otimes k_\left\{lambda\right\}, Delta\left(e_i\right) = e_i otimes k_i^\left\{-1\right\} + 1 otimes e_i, Delta\left(f_i\right) = f_i otimes 1 + k_i otimes f_i$, is adopted. The integrable module $M otimes_\left\{\left\{Bbb Q\right\}\left(q\right)\right\} M\text{'}$ has crystal base $\left(L otimes_A L\text{'},B otimes B\text{'}\right)$, where $B otimes B\text{'} = \left\{ b otimes_\left\{Bbb Q\right\} b\text{'} : b in B, b\text{'} in B\text{'} \right\}$. For a basis vector $b in B$, define $epsilon_i\left(b\right) = mathrm\left\{max\right\}\left\{ n ge 0 : tilde\left\{e\right\}_i^n b ne 0 \right\}$ and $phi_i\left(b\right) = mathrm\left\{max\right\}\left\{ n ge 0 : tilde\left\{f\right\}_i^n b ne 0 \right\}$. The actions of $tilde\left\{e\right\}_i$ and $tilde\left\{f\right\}_i$ on $b otimes b\text{'}$ are given by

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

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

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 $Delta : C\left(G\right) to C\left(G\right) otimes C\left(G\right)$ (where $C\left(G\right) otimes C\left(G\right)$ is the C*-algebra tensor product - the completion of the algebraic tensor product of $C\left(G\right)$ and $C\left(G\right)$), such that $Delta\left(f\right)\left(x,y\right) = f\left(xy\right)$ for all $f in C\left(G\right)$, and for all $x, y in G$ (where $\left(f otimes g\right)\left(x,y\right) = f\left(x\right) g\left(y\right)$ for all $f, g in C\left(G\right)$ and all $x, y in G$). There also exists a linear multiplicative mapping $kappa : C\left(G\right) to C\left(G\right)$, such that $kappa\left(f\right)\left(x\right) = f\left(x^\left\{-1\right\}\right)$ for all $f in C\left(G\right)$ and all $x in G$. Strictly, this does not make $C\left(G\right)$ 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 $C\left(G\right)$ which is also a Hopf *-algebra. Specifically, if $g mapsto \left(u_\left\{ij\right\}\left(g\right)\right)_\left\{i,j\right\}$ is an $n$-dimensional representation of $G$, then $u_\left\{ij\right\} in C\left(G\right)$ for all $i, j$, and $Delta\left(u_\left\{ij\right\}\right) = sum_k u_\left\{ik\right\} otimes u_\left\{kj\right\}$ for all $i, j$. It follows that the *-algebra generated by $u_\left\{ij\right\}$ for all $i, j$ and $kappa\left(u_\left\{ij\right\}\right)$ for all $i, j$ is a Hopf *-algebra: the counit is determined by $epsilon\left(u_\left\{ij\right\}\right) = delta_\left\{ij\right\}$ for all $i, j$ (where $delta_\left\{ij\right\}$ is the Kronecker delta), the antipode is $kappa$, and the unit is given by $1 = sum_k u_\left\{1k\right\} kappa\left(u_\left\{k1\right\}\right) = sum_k kappa\left(u_\left\{1k\right\}\right) u_\left\{k1\right\}.$

As a generalization, a compact matrix quantum group is defined as a pair $\left(C,u\right)$, where $C$ is a C*-algebra and $u = \left(u_\left\{ij\right\}\right)_\left\{i,j = 1,dots,n\right\}$ 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 $Delta : C to C otimes C$ (where $C otimes C$ is the C*-algebra tensor product - the completion of the algebraic tensor product of $C$ and $C$) such that $Delta\left(u_\left\{ij\right\}\right) = sum_k u_\left\{ik\right\} otimes u_\left\{kj\right\}$ for all $i, j$ ($Delta$ is called the comultiplication);

*There exists a linear antimultiplicative map $kappa : C_0 to C_0$ (the coinverse) such that $kappa\left(kappa\left(v*\right)*\right) = v$ for all $v in C_0$ and $sum_k kappa\left(u_\left\{ik\right\}\right) u_\left\{kj\right\} = sum_k u_\left\{ik\right\} kappa\left(u_\left\{kj\right\}\right) = delta_\left\{ij\right\} I,$ where $I$ is the identity element of $C$. Since $kappa$ is antimultiplicative, then $kappa\left(vw\right) = kappa\left(w\right) kappa\left(v\right)$ 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 $v = \left(v_\left\{ij\right\}\right)_\left\{i,j = 1,dots,n\right\}$ with entries in $A$ (so $v in M_n\left(A\right)$) such that $Delta\left(v_\left\{ij\right\}\right) = sum_\left\{k=1\right\}^n v_\left\{ik\right\} otimes v_\left\{kj\right\}$ for all $i, j$ and $epsilon\left(v_\left\{ij\right\}\right) = delta_\left\{ij\right\}$ for all $i, j$). Furthermore, a representation, v, is called unitary if the matrix for v is unitary (or equivalently, if $kappa\left(v_\left\{ij\right\}\right) = v^*_\left\{ji\right\}$ for all i, j).

An example of a compact matrix quantum group is $SU_\left\{mu\right\}\left(2\right)$, where the parameter $mu$ is a positive real number. So $SU_\left\{mu\right\}\left(2\right) = \left(C\left(SU_\left\{mu\right\}\left(2\right),u\right)$, where $C\left(SU_\left\{mu\right\}\left(2\right)\right)$ is the C*-algebra generated by $alpha$ and $gamma$,subject to

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

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

Equivalently, $SU_\left\{mu\right\}\left(2\right) = \left(C\left(SU_\left\{mu\right\}\left(2\right),w\right)$, where $C\left(SU_\left\{mu\right\}\left(2\right)\right)$ is the C*-algebra generated by $alpha$ and $beta$,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 $w = left\left(begin\left\{matrix\right\} alpha & mu beta - beta^* & alpha^* end\left\{matrix\right\} right\right),$ so that the comultiplication is determined by $Delta\left(alpha\right) = alpha otimes alpha - mu beta otimes beta^*$, $Delta\left(beta\right) = alpha otimes beta + beta otimes alpha^*$, and the coinverse is determined by $kappa\left(alpha\right) = alpha^*$, $kappa\left(beta\right) = - mu^\left\{-1\right\} beta$, $kappa\left(beta^*\right) = - mu beta^*$, $kappa\left(alpha^*\right) = alpha$. Note that $w$ is a unitary representation. The realizations can be identified by equating $gamma = sqrt\left\{mu\right\} beta$.

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