Quantum Group

DRINFEL'D TYPE QUANTUM GROUPS

One common structure, which is called a "quantum group", after the work of Vladimir Drinfel'd , Nicolai Reshetikhin , Michio Jimbo , and others, is a deformation of the Universal Enveloping Algebra of a Semisimple Lie Algebra or, more generally, a Kac-Moody 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 Root s,

\alpha_i

), subject to

$k_0 = 1$ ,

$k_{\lambda} k_{\mu} = k_{\lambda+\mu}$ ,

$k_{\lambda} e_i k_{\lambda}^{-1} = q^{(\lambda,\alpha_i)} e_i$ ,

$k_{\lambda} f_i k_{\lambda}^{-1} = q^{- (\lambda,\alpha_i)} f_i$ ,

{Link without Title} = \delta_{ij} rac{k_i - k_i^{-1}}{q_i - q_i^{-1}},

$_{q_i}! [n]_{q_i}!} e_i^n e_j e_i^{1 - a_{ij} - n} = 0$ , for $i$

$_{q_i}! [n]_{q_i}!} f_i^n f_j f_i^{1 - a_{ij} - n} = 0$ , for $i$

where

k_i = k_{\alpha_i}

q_i = q^{rac{1}{2}(\alpha_i,\alpha_i)}

= 1

_{q_i} = rac{q_i^m - q_i^{-m}}{q_i - q_i^{-1}}

. These are the Q-factorial and Q-series , respectively, the Q-analog s of the ordinary Factorial . The last two relations above are the ''q''-Serre relations, the deformations of the Serre Relation s.

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

rac{k_{\lambda} - k_{-\lambda}}{q - q^{-1}} 	o t_{\lambda}

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,

$\Delta_1(k_\lambda) = k_\lambda \otimes k_\lambda$ , $\Delta_1(e_i) = 1 \otimes e_i + e_i \otimes k_i$ , $\Delta_1(f_i) = k_i^{-1} \otimes f_i + f_i \otimes 1$ ,

$\Delta_2(k_\lambda) = k_\lambda \otimes k_\lambda$ , $\Delta_2(e_i) = k_i^{-1} \otimes e_i + e_i \otimes 1$ , $\Delta_2(f_i) = 1 \otimes f_i + f_i \otimes k_i$ ,

$\Delta_3(k_\lambda) = k_\lambda \otimes k_\lambda$ , $\Delta_3(e_i) = k_i^{-rac{1}{2}} \otimes e_i + e_i \otimes k_i^{rac{1}{2}}$ , $\Delta_3(f_i) = k_i^{-rac{1}{2}} \otimes f_i + f_i \otimes k_i^{rac{1}{2}}$ , 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 $\Delta_5 = T \circ \Delta_2$ ,

$\Delta_6(k_\lambda) = k_\lambda \otimes k_\lambda$ , $\Delta_6(e_i) = k_i^{rac{1}{2}} \otimes e_i + e_i \otimes k_i^{-rac{1}{2}}$ , $\Delta_6(f_i) = k_i^{rac{1}{2}} \otimes f_i + f_i \otimes k_i^{-rac{1}{2}}$ , where $\Delta_6 = T \circ \Delta_3$ .

The 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 Function s 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 Function s 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 $\mu$ ,

$c_{2\alpha_i} = 1$ for all ''i''.

Of special interest are Highest Weight Representation s, 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

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 $\mu$ ,

$c_{2\alpha_i} = 1$ 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

Quasitriangulaity

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

\beta

, 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 .

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/ 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

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 + rac{2 (\lambda,\alpha_i)}{(\alpha_i,\alpha_i)} \ge 0

, and

v_n 
e 0

only if

n - rac{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}

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

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

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

iff there exist polynomials

g(q)

and

h(q)

in the polynomial ring

{\Bbb Q}

{Link without Title} 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'', $ilde{e}_i b = b'$ iff $ilde{f}_i b' = b$ .

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

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

q = 0

. If an integrable module has a crystal base, then the module is irreducible iff 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

-algebra generated by $\alpha$ and $\beta$ ,subject to

= \beta^--- \beta, \ \alpha \beta = \mu \beta \alpha, \ \alpha \beta^--- = \mu \beta^--- \alpha, \ \alpha \alpha^--- + \mu^2 \beta^--- \beta = \alpha^--- \alpha + \beta^--- \beta = I,

& \alpha^--- \end{matrix} ight), 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 $\kappa(\alpha) = \alpha^---$ , $\kappa(\beta) = - \mu^{-1} \beta$ , $\kappa(\beta^---) = - \mu \beta^---$ , $\kappa(\alpha^---) = \alpha$ . Note that $w$ is a unitary representation. The realizations can be identified by equating $\gamma = \sqrt{\mu} \beta$ .

When

\mu = 1

, then

SU_{\mu}(2)

is equal to the concrete compact group

SU(2)

.

SEE ALSO

Lie Bialgebra

Poisson-Lie Group

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Quantum Group