Quantum Group - Drinfel'd-Jimbo Type Quantum Groups

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_λ (where λ is an element of the weight lattice, i.e. 2(λ, α_i)/(α_i, α_i) is an integer for all i), and e_i and f_i (for simple roots, α_i), subject to the following relations:

• ,
• ,
• ,
• ,
• ,
• If i ≠ j then:

where for all positive integers n, and 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 → 1, these relations approach the relations for the universal enveloping algebra U(G), where k_λ → 1 and as q → 1, where the element, t_λ, of the Cartan subalgebra satisfies (t_λ, h) = λ(h) for all h in the Cartan subalgebra.

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

• ,
• ,
• ,
• ,
• ,
• ,
• ,
• ,
• ,

where the set of generators has been extended, if required, to include k_λ 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 coproduct T Δ, where T is given by T(x ⊗ y) = y ⊗ x, giving three more possible versions.

The counit on U_q(A) is the same for all these coproducts: ε(k_λ) = 1, ε(e_i) = ε(f_i) = 0, and the respective antipodes for the above coproducts are given by

• ,
• ,

Alternatively, the quantum group U_q(G) can be regarded as an algebra over the field C(q), the field of all rational functions of an indeterminate q over C.

Similarly, the quantum group U_q(G) can be regarded as an algebra over the field Q(q), the field of all rational functions of an indeterminate q over Q (see below in the section on quantum groups at q = 0). The center of quantum group can be described by quantum determinant.