Integrable Highest Weight Representations of Affine Lie algebras#

In this section \(\mathfrak{g}\) can be an arbitrary Kac-Moody Lie Algebra made with a symmetrizable, indecomposable Cartan matrix.

Suppose that \(V\) is a representation with a weight space decomposition as in Roots and Weights. Let \(\alpha\) be a real root, and let \(\mathfrak{g}_\alpha\) be the corresponding root space, that is, the one-dimensional weight space for \(\alpha\) in the adjoint representation of \(\mathfrak{g}\) on itself. Then \(-\alpha\) is also a root. The two one-dimensional spaces \(\mathfrak{g}_\alpha\) and \(\mathfrak{g}_{-\alpha}\) generate a Lie algebra isomorphic to \(\mathfrak{sl}_2\). The module \(V\) is called integrable if for each such \(\alpha\) the representation of \(\mathfrak{sl}_2\) obtained this way integrates to a representation of the Lie group \(\operatorname{SL}_2\). Since this group contains an element that stabilizes \(\mathfrak{h}\) and induces the corresponding simple reflection on the weight lattice, integrability implies that the weight multiplicities are invariant under the action of the Weyl group.

If the Kac-Moody Lie algebra \(\mathfrak{g}\) is finite-dimensional then the integrable highest weight representations are just the irreducible finite-dimensional ones. For a general Kac-Moody Lie algebra the integrable highest weight representations are the analogs of the finite-dimensional ones, that is, with WeylCharacterRing elements. Their theory has many aspects in common with the finite-dimensional representations of finite-dimensional simple Lie algebras, such as the parametrization by dominant weights, and generalizations of the Weyl denominator and character formulas, due to Macdonald and Kac respectively.

If \(\Lambda\) is a dominant weight, then the irreducible highest weight module \(L(\Lambda)\) defined in Roots and Weights is integrable. Moreover every highest weight integrable representation arises this way, so these representations are in bijection with the cone \(P^+\) of dominant weights.

The affine case#

Now we assume that \(\mathfrak{g}\) is affine. The integrable highest weight representations (and their crystals) are extremely interesting. Integrable highest weight representations of \(\mathfrak{g}\) arise in a variety of contexts, from string theory to the modular representation theory of the symmetric groups, and the theory of modular forms. The representation \(L(\Lambda_0)\) is particularly ubiquitous and is called the basic representation.

Therefore in [KMPS] (published in 1990) tabulated data for many of these representations. They wrote

We present a vast quantity of numerical data in tabular form, this being the only source for such information. The computations are tedious and not particularly straightforward when it is necessary to carry them out individually. We hope the appearance of this book will spur interest in a field that has become, in barely 20 years, deeply rewarding and full of promise for the future. It would indeed be gratifying if these tables were to appear to the scientists of 2040 as obsolete as the dust-gathering compilations of transcendental functions appear for us today because of their availability on every pocket calculator.

As we will explain, Sage can reproduce the contents of these tables. Moreover the tables in [KMPS] are limited to the untwisted types, but Sage also implements the twisted types.

Although Sage can reproduce the tables in the second volume of [KMPS], the work remains very useful. The first volume is a down-to-earth and very helpful exposition of the theory of integrable representations of affine Lie algebras with explicit examples and explanations of the connections with mathematical physics and vertex operators.

The support of an integrable highest weight representation#

Let \(\Lambda \in P^+\) and let \(V = L(\Lambda)\) be the integrable representation with highest weight \(\Lambda\). If \(\mu\) is another weight, let \(\operatorname{mult}(\mu)\) denote the multiplicity of the weight \(\mu\) in \(L(\Lambda)\). Define the support of the representation \(\operatorname{supp}(V)\) to be the set of \(\mu\) such that \(\operatorname{mult}(\mu) > 0\).

If \(\operatorname{mult}(\mu) > 0\) then \(\lambda-\mu\) is a linear combination of the simple roots with nonnegative integer coefficients. Moreover \(\operatorname{supp}(V)\) is contained in the paraboloid

\[(\Lambda+\rho | \Lambda+\rho) - (\mu+\rho | \mu+\rho) \geq 0\]

where \((\, | \,)\) is the invariant inner product on the weight lattice and \(\rho\) is the Weyl vector (Untwisted Affine Kac-Moody Lie Algebras). Moreover if \(\mu \in \operatorname{supp}(V)\) then \(\Lambda - \mu\) is an element of the root lattice \(Q\) ([Kac], Propositions 11.3 and 11.4).

We organize the weight multiplicities into sequences called string functions or strings as follows. By [Kac], Proposition 11.3 or Corollary 11.9, for fixed \(\mu\) the function \(\operatorname{mult}(\mu - k\delta)\) of \(k\) is an increasing sequence. We adjust \(\mu\) by a multiple of \(\delta\) to the beginning of the positive part of the sequence. Thus we define \(\mu\) to be maximal if \(\operatorname{mult}(\mu) \neq 0\) but \(\operatorname{mult}(\mu + \delta) = 0\).

Since \(\delta\) is fixed under the action of the affine Weyl group, and since the weight multiplicities are Weyl group invariant, the function \(k \mapsto \operatorname{mult}(\mu - k \delta)\) is unchanged if \(\mu\) is replaced by \(w(\mu)\) for some Weyl group element \(w\). Now every Weyl orbit contains a dominant weight. Therefore in enumerating the string we may assume that the weight \(\mu\) is dominant. There are only a finite number of dominant maximal weights. Thus there are only a finite number of such strings to be computed.

Modular Forms#

Remarkably, [KacPeterson] showed that each string is the set of Fourier coefficients of a weakly holomorphic modular form; see also [Kac] Chapters 12 and 13. Here weakly holomorphic modular means that the form is allowed to have poles at cusps.

To this end we define the modular characteristic

\[m_\Lambda = \frac{|\Lambda+\rho|^2}{2(k+h^\vee)} - \frac{|\rho|^2}{2h^\vee}.\]

Here \(k = (\Lambda | \delta)\) is the level of the representation and \(h^\vee\) is the dual Coxeter number (Labels and Coxeter Number). If \(\mu\) is a weight, define

\[m_{\Lambda,\mu} = m_\Lambda - \frac{|\mu|^2}{2k}.\]

Let \(\Lambda\) be a weight, which we may assume maximal. Then Kac and Peterson defined the string function

\[c_\mu^\Lambda = q^{m_{\Lambda,\mu}} \sum_{n\in\ZZ} \operatorname{mult}(\mu - n\delta) q^n.\]

Although these do arise as partition functions in string theory, the term “string” here does not refer to physical strings.

The string function \(c_\mu^\Lambda\) is a weakly holomorphic modular form, possibly of half-integral weight. See [Kac], Corollary 13.10, or [KacPeterson]. It can have poles at infinity, but multiplying \(c_\mu^\Lambda\) by \(\eta(\tau)^{\dim\,\mathfrak{g}^\circ}\) gives a holomorphic modular form (for some level). Here \(\eta\) is the Dedekind eta function:

\[\eta(\tau) = q^{1/24} \prod_{k=1}^\infty(1-q^k), \qquad q = e^{2\pi i \tau}.\]

The weight of this modular form \(\eta(\tau)^{\dim\,\mathfrak{g}^\circ} c^\Lambda_\lambda\) is the number of positive roots of \(\mathfrak{g}^\circ\).

Integrable Highest Weight Representations of Affine Lie algebras#

The affine case#

The support of an integrable highest weight representation#

Modular Forms#

Sage methods for integrable representations#