QUANTUM LIE ALGEBRAS FOR Bl, Cl, AND Dl*)

CHRISTOPHER GARDNER**)

Department of Mathematics, King's College LondonStrand, London WC2R 2LS, Great Britain

Received 4 August 1997

We give the quantum structure constants for the analogues of the classical Lie algebrasso(n) and sp(n) for any n, as well as their quantum Killing form. We also include asummary of the method used to obtain them.

1 Introduction

Quantum Lie algebras are generalizations of Lie algebras whose structure con-stants depend on a quantum parameter q and which are related to the quantizedenveloping algebras (quantum groups [1, 2]) Uh(g) in a way similar to the wayordinary Lie algebras are related to their enveloping algebras U(g) (cf. [3]).

Our approach to the determination of the structure constants of quantum Liealgebras relies on the observation [4] that the quantum Lie bracket is an intertwinerfrom adjoint X adjoint —> adjoint where by adjoint we mean the adjoint representa-tion of Uh(g). Thus the structure constants are given by the corresponding inversequantum Clebsch-Gordan coefficients.

The outline of this paper, which is a summary of [5], is as follows. We begin insection 2 by recalling the definition and some properties of quantum Lie algebrasand then explain our method for determining their structure constants in section3. In section 4 we introduce an invariant Killing form on the quantum Lie alge-bras. Section 5 contains our results for the structure constants of the quantumLie algebras. By using the Killing form to lower indices we are able to give conciseexpressions. In section 6 we derive some further relations between the structure con-stants, which we have verified using Mathematica [6] for the quantum Lie algebrasof low rank.

To fix our notation we write the coproduct D on Uh(g) (associated to thefinite-dimensional simple complex Lie algebra g), whose generators are denotedby xi+, xi-, hi, 1 < i < rank(g), as

*) Presented at the 6th International Colloquium on Quantum Groups: "Quantum Groups andIntegrable Systems", Prague, 19-21 June 1997

**) E-mail: cgardner@mth.kcl.ac.uk

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Here qi E C[[h]], the ring over which Uh(g) is defined.

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The quantum Lie bracket is given by the adjoint action of Uh (g) restricted to Lh(g),which in Sweedler's notation [8], is (adx)y = Ex(1)yS(x(2)) Vx,y E Lh(g). Andobviously in general (adx)y = (ady)x and so we have to introduce two differenttypes of roots, namely lA and rA.

The basis can be chosen so that quantum structure constants possess the sym-metries [3, 4]:

C. Gardner

2 Quantum Lie algebras

A classical complex Lie algebra g is a submodule of its' universal envelopingalgebra U(g) with respect to the adjoint action (or equivalently Lie bracket). So anatural definition of a quantum Lie algebra Lh(g) would be as an ad-submoduleof the quantized enveloping algebra Uh(g) associated to g with the quantum Liebracket given by the adjoint action of Uh(g).

Of course there are many such ad-submodules inside Uh(g). So which ad-submo-dule is the correct candidate for a quantum Lie algebra? It should be indecom-posable and a deformation of the classical Lie algebra, namely there is an alge-bra isomorphism g ~ Lh(g)|h=0. However there are still infinitely many such ad-submodules, but one can show for Bl, Cl and Dl that they are all isomorphic.Such a module transforms as the adjoint representation under the adjoint action.It is well known that this representation, like all highest weight representations ofU h (g) , is just a deformation of the classical adjoint representation. One immediateconsequence of this is that Lh(g) has the same grading a g.

Choosing as a basis for Lh(g), {XA | A E R}U{Hi | i = 1,...rank(g)} (R is theclassical root space), the most general Lie bracket relations compatible with thisgrading are

Quantum Lie algebras ...

where B is the Killing form, to be discussed in section 4, p is half the sum ofthe positive roots, and ~ denotes q-conjugation, i.e., the operation h —> –h orequivalently q —> 1/q.

3 Calculation of quantum Clebsch-Gordan coefficients

Employing the observation that the quantum Lie bracket is an intertwiner fromadjoint X adjoint —> adjoint, the structure constants are the corresponding inversequantum Clebsch-Gordon coefficents. We build them from the quantum Clebsch-Gordan coefficients for vector X vector into vector and singlet. We denote by Vu therepresentation space with highest weight u, e.g., Vs is the adjoint representation,VE1 the vector representation and V0 the singlet representation.

The direct product representation VE1 X VE1 has the decomposition: VE1XVE1 —>V2E1 + VE1+E2 + V0, with VS = VE1+E2 for Bl and Dl, and VS = V2E1 for Cl.We denote by {uu

a}a=1,...,dim Vu the basis of a representation space Vu. Then thedecomposition is described by the inverse Clebsch-Gordan coefficients as follows

It is very convenient to introduce a graphical notation for the Clebsch-Gordancoefficients and their inverses as in figure 1.

Fig. 1. Graphical representation of Clebsch-Gordan coefficients.

The composition of any number of intertwiners is again an intertwiner. Weexploit this fact to build the intertwiner for adjoint X adjoint —> adjoint from theintertwiners for vector X vector into adjoint and singlet as shown in the followingdiagram:

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C. Gardner

Here u is the the element of Uh(g) satisfying the properties u a u-1 = S2(a)VA E U h(g) and D ( u ) = u X u. The TrS denotes the trace over the adjoint repre-sentation. The definition is motivated by the following (proved in [5]):

Proposition 4.1. The Killing form is ad-invariant, non-degenerate, bilinear, andis not symmetric, i.e.,

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Since the adjoint representation appears inside vector X vector with unit multi-plicity such an intertwiner (and therefore the quantum Lie bracket) is unique uptorescaling.

4 The Killing form

In this section we define a quantum analogue to the classical Killing form.As mentioned in the introduction, the Killing form can be used to simplify theexpressions for the structure constants. We actually calculate the intertwiner fromadjoint X adjoint into the singlet, which is proportional to the quantum Killingform [5]. This intertwiner is unique upto rescaling.

Definition 4.1. The quantum Killing form is the map B : L h ( g ) X L h ( g ) —> C[[h]]given by

The intertwiner from adjoint X adjoint —> singlet, which is proportional to B,is calculated according to the formula

The Killing form on the roots has the simple form

The Killing form on the Cartan subalgebra for the algebras Bl, Cl and Dl respec-tively, is given by

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5.1 The quantum Lie algebra (B l)h .

The quantum roots are

These expressions are surprisingly simple q-deformations of the classical ones. Inparticular B(Hi, Hj) = 0 if and only if Ai . Aj = 0.

5 The quantum structure constants

We only give the minimal set of the structure constants. The remaining onescan be calculated using the symmetry properties, see equations (2.2)-(2.6). Thestructure constants fij

k for the Cartan subalgebra simplify dramatically if onelowers the last index using the Killing form, i.e., fijk := fij

mB(Hm,Hk). Sincethe Killing form is symmetric on the Cartan subalgebra it is easy to see, using(2.2),(2.3) and (2.5), that fijk is completely symmetric. We also observed thatfijk = 0 iff Ai . Aj = 0. This implies in particular that the structure constants fijk

are non-zero only if at least two of the indices are the same. So below we give onlya few f's, all others can be obtained by symmetry or are zero. We give also onlyenough of the NA,B so that the others can be obtained using (2.2)-(2.4).

Quantum Lie algebras ...

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where i < j < l and k < m < l.

The remaining structure constants are determined by

where j < k < l, i < l. The structure constants for the Cartan subalgebra are

where i < j < l and k < m < l.

5.2 The quantum Lie algebra (C l)h

The quantum roots are

The remaining structure constants are determined by

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where j < k < l, i < l. The structure constants for the Cartan subalgebra are

In other words, the Lie brackets of the Cartan subalgebra elements are given bythe amount of the split between left and right quantum roots.

6 Some more relations between the structure constants

We have already observed that the product of any number of intertwiners isstill an intertwiner. We use this observation to derive some interesting relationsbetween the structure constants by using the intertwiners from adjoint X adjoint—> adjoint and singlet —> adjoint X adjoint. This second intertwiner is the inverseof the Killing form, denoted by Bpq and defined by BprB

rq = Dpq.

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where i < j < l and k < m < l.

5.4 The structure of the Cartan subalgebra

For quantum Lie algebras the quantum Lie bracket is non-vanishing between ele-ments of the Cartan subalgebra. Our explicit results for the corresponding structureconstants fijk given above have lead us to the following observation:

where i < l, j < l — 1. Because of the Dynkin diagram automorphism T whichinterchanges Hl and Hl-1 we don't need to give the structure constants involvingHl, they are equal to those involving Hl-1. The f's involving both l and l — 1 arezero. The remaining structure constants are determined by

where j < k < l, i < l. The structure constants for the Cartan subalgebra are

5.3 The quantum Lie algebra (Dl)h

The quantum roots are

Quantum Lie algebras ...

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These two new relations between the structure constants have been checked forquantum Lie algebras of low rank.

7 Discussion

We have shown how to calculate the structure constants of the quantum Liealgebras associated to Bl, Cl and Dl. The structure constants satisfy the symme-tries discovered in [3]. We have introduced an ad-invariant Killing form which isproportional to the intertwiner from adjoint X adjoint —> singlet.

As is well known, the structure constants of the simple complex Lie algebrasare determined entirely in terms of their simple roots. Eventually we would hopeto arrive at a similar result for the quantum Lie algebras. In this paper we havecome one step closer to this goal by our observation that the structure constantsfijk of the Cartan subalgebra are completely determined in terms of the quantumroots according to Eq. (5.10). An understanding of how the NA,B's and the higherquantum roots are related to the simple roots is still missing.

The matrices (4.5)-(4.7) describing the quantum Killing form on the Cartansubalgebras are surprisingly simple. In particular we find that only those entries inthe matrices are non-zero which are also non-zero classically.

There still remain a lot of unanswered questions. In particular: What is a goodaxiomatic setting for the theory of quantum Lie algebras. How should one q-deform

The intertwiner from adjoint to adjoint given in Figure 2b is proportional to theidentity map. Setting p — q = A we have

Fig. 2. Graphical representation of the relations. a) There is no non-zero intertwinerfrom singlet to adjoint. b) The intertwiner from adjoint to adjoint is proportional to the

identity.

The intertwiner from adjoint to singlet given in Figure 2a is zero. Setting p = kwe have (after some manipulations)

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Quantum Lie algebras ...

the Jacobi identity? What characterizes the quantum root system? What are q-Weylreflections? How does one define representations of these non-associative algebras?And many more. Those interested in the available literature on quantum Lie alge-bras are welcome to look up the webpages at

http://www.mth.kcl.ac.uk/~delius/q-lie/

References

[1] Drinfel'd V.G.: Sov. Math. Dokl. 32 (1985) 254.

[2] Drinfel'd V.G.: in Quantum Groups, Proc. Int. Congr. Math., Berkeley 1986, p. 798.

[3] Delius G.W. and Huffmann A.: J. Phys. A 29 (1996) 1703; q-alg/9506017.

[4] Delius G.W. and Gould M.: Quantum Lie algebras, their existence, uniqueness andq-antisymmetry, q-alg/9605025; Commun. Math. Phys (in print).

[5] Delius G.W., Gardner C,, and Gould M.: The structure of quantum Lie algebras forthe classical series Bl, Cl and Dl, q-alg/9706029.

[6] Wolfram S.: Mathematica, 2nd ed., Addison-Wesley Publishing Co., New York, 1991.

[7] Delius G.W., Huffmann A., Gould M.D., and Zhang Y.-Z.: J. Phys. A 29 (1996) 5611;q-alg/9508013.

[8] Sweedler M.E.: Hopf algebras, Benjamin, New York, 1996.

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