Last update: 2 June 2014
This is a typed copy of the LOMI preprint Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I by N.Yu. Reshetikhin.
Recommended for publication by the Scientific Council of Steklov Mathematical Institute, Leningrad Department (LOMI) 6, June, 1987.
The of the universal enveloping algebras (or quantum universal enveloping algebras QUEA) arose when studying some special class of integrable systems. The definition of for any simple Lie algebra was given in [Dri1985,Dri1986-3,Jim1985]. One can define these algebras by generators and relations.
Definition 1.1 [Dri1985,Dri1986-3,Jim1985]. QUEA is the algebra with generators and with the relations: where is the Cartan matrix of is the scalar product of the roots
The generators play the role of Chevalley basis in The elements form the commutative subalgebra This subalgebra is the analog of Cartan subalgebra in The elements correspond to the simple roots of Let be the root of a corresponding element of we denote as
Theorem 1.1 [Dri1985,Dri1986-3,Jim1985]. The algebra is the Hopf algebra [Abe1980] (see also Appendix A) with comultiplication and with the antipode defined on the generators by the following formulae: where is the element of and is the positive roots of
The parameter is the deformation parameter. At the algebra is the universal enveloping algebra of For simple Lie algebras of the algebras are simple for general The special values of are situated in the rational points on the unit circle.
Let be the Borel subalgebras in The algebras and with generators and are the Hopf subalgebras in There is a duality between and Let is the basis in and be the dual basis in
It is not difficult to prove that the map where is the permutation in also defines the structure of Hopf algebra on with antipode
Theorem 1.2 [Dri1985,Dri1986-3]. The comultiplication and are connected by the following relation where is the permutation operator in and
The proof of this theorem the following one is based on the construction of double Hopf algebra [Jim1985].
Theorem 1.3. The element satisfies the relations Here
Let and be the matrices of multiplication, comultiplication and antipode in algebra The matrices and are the matrices of multiplication, comultiplication and antipode in From (1.7) we have the commutation relations between the elements and in From this commutation relations and from the definition of follows (1.8c). To prove (1.8d) we use the definition of antipode (A.): Similarly The following formulae prove the relations (1.8a) and (1.8b) The theorem is proved.
The algebras and have a common Cartan subalgebra generated by the elements This means that the theories of the finite dimensional representations of and are quite parallel. The following two statements will play the main role in the text.
Proposition 1.1. The finite dimensional representations of are completely irreducible.
|(a)||The irreducible finlte-dimensional representations (IFR) of are parametrized by the highest weights of the algebra|
|(b)||Any irreducible finite dimensional module is decomposed into the sum of the weight spaces and the dimensions of are the same as for IFR of|
From the complete irreducibility of finite dimensional representations of it follows that any IFR can be considered as some irreducible component of corresponding tensorial power of basic representations. The list of basic representation is given in table 1.
For algebras and it is natural to separate from the finite dimensional representations the category of tensorial representations (with integers highest weights). In this category the basic representation is vectorial representation with h.w. in both cases.
It is very important for our purposes that all finite dimensional representations of are contained in tensorial powers of the basic representations.
Below we study the restrictions of universal on the IFR's. If and are IFR of and is the permutation operator on then we denote the matrix as and we call it the in representation (or This matrix map on and satisfies the following relations: The left and the right sides of these relations act from to
To find the matrices one can use two ways. The first one is the explicit projection of the universal by factorising both copies of over corresponding ideals. The second way uses the complete irreducibility of IFR of Because all of IFR are contained in tensorial powers of basic representations of one can construct the matrices from a tensorial product of the acting in basic representations. The second way gives also the important relations between the matrices and we will use this way below.
Consider a Cartan antiinvolution in and automorphism corresponding to on authomorphism of Dynkin diagram:
We define a of the element with the maximal length of the Weyl group by the formula
It is not difficult to check that in basic representations the element has the form where is the corresponding element of the Weyl group for
(b) The element satisfies the relation
Definition 1.2. If the projection matrix is called the matrix of Klebsch-Gordan coefficient (KGC) (here index numbers the components in if the multiplicity of is more than one).
The matrices and are orthogonal if They are normalized when
Theorem 1.4. The matrices and for satisfy the following relations
Here is the parity of in and is the transposition.
Following theorems are the keys ones to study the matrices
Theorem 1.5. Let and then where is the parity of in and is the value of Casimir operator of on the irreducible representation with highest weight
From the definition of universal follows the equality where The representation is reducible and are the projectors on the irreducible components: From (1.23) and (1.24) we have Comparing with (1.25) we obtain where is some function of To find we consider the limit Let be the highest weight vector in From (1.27) follows the equality At we have Here is the so-called classical [Bax1982,Fad1984]. is the h.w.v. of and and some unknown vectors and constants.
Theorem 1.4 implies symmetry relations for and Comparing the coefficients at in (1.28) we obtain the equation for Multiplying this equality by from the left and using the symmetry relations (1.33) we obtain the following expression for
Using the symmetry relations (1.33) and the property of highest weight vectors the value of one can express through the Casimir operators
From the symmetry relations (1.15) and from the definition of we have
Moreover, the function has no poles and zeros for finite values of From Liouville theorem follows that there is only one function satisfying the relations (1.29), (1.34)-(1.36)
The equality (1.20) is the transposition of (1.19). The Theorem is proved.
Consequence. When all irreducible components in have the multiplicity equal to one the matrices and have the following decompositions: where These will be called with simple spectrum.
Let us consider the space and the matrices and similar matrices
Theorem 1.6. The matrices introduced above satisfy the following relations
The restriction or the relation (1.6) on the irreducible representation gives: Multiplying this formula by the permutation operator from the left and using the equality we obtain (1.41). The relation (1.40) is proved similarly.
Crossing-symmetry of the universal (1.8d) implies crossing-symmetry of the matrices and
Theorem 1.7. The matrices and satisfy the following crossing-symmetry relations: where is transposition over the first space, is the highest weight (h.w.) vector conjugated with by Carton automorphism,
Using the explicit formula for coproduct in it is not difficult to prove the first statement of the following theorem.
|(a)||The matrix is the projector on the one-dimensional subrepresentation in|
|(b)||The matrices and satisfy the relations Here is the matrix trace over the second space in|
The proof of the second statement of this theorem is given in the next section, where we introduce graphical representation for the and for KGC.