Borcherds Lie algebra

Borcherds algebra

While a Kac–Moody algebra is generated in a fairly simple way from copies of $\operatorname{sl}_2$, a Borcherds or generalized Kac–Moody algebra [a1], [a7], [a9], [a11] can also involve copies of the $3$-dimensional Heisenberg algebra. Nevertheless, it inherits many of the Kac–Moody properties. Borcherds algebras played a key role in the proof of the Monstrous Moonshine conjectures [a4], and also led to the development of a theory of automorphic products [a5].

First recall the definition of a Kac–Moody algebra. By a (symmetrizable) Cartan matrix $A = ( a _ { ij} )$, one means an integral $( \text{l} \times \text{l} )$-matrix obeying

C1) $a _ { i i } = 2$ and $a _ { i j } \leq 0$ for all $i \neq j$; and

C2) there is a diagonal matrix $D$ with each $d_{ ii} > 0$ such that $D A$ is symmetric. A (symmetrizable) Kac–Moody algebra $\mathfrak { g } = \mathfrak { g } ( A )$ [a10], [a12] is the Lie algebra on $3\text{l}$ generators $e _ { i } , f _ { i } , h _ { i }$, obeying the relations:

R1) $[ e _ { i } f _ { j } ] = \delta _ { i j } h _ { i }$, $[ h _ { i } e _ { j } ] = a _ { ij } e _ { j }$, $[ h _ { i } f _ { j } ] = - a _ { ij } f _ { j }$, and $[ h _ { i } h _ { j } ] = 0$, for all $i , j$; and

R2) $\operatorname{ad}(e_i)^{1-\alpha_{ij}} (e_j)=0$ and $\operatorname{ad}(f_i)^{1-\alpha_{ij}} (f_j)=0$ for all $i \neq j$.

A Borcherds algebra is defined similarly. By a generalized Cartan matrix $A$ one means a (possibly infinite) matrix $A = ( a _ { ij} )$, $a _ { i j } \in \mathbf{R}$, obeying

GC1) either $a _ { i i } = 2$ or $a _ { i i } \leq 0$;

GC2) $a _ { i j } \leq 0$ for $i \neq j$, and $a _ { i j } \in \mathbf{Z}$ when $a _ { i i } = 2$; and

GC3) there is a diagonal matrix $D$ with each $d_{ ii} > 0$ such that $D A$ is symmetric. By the (symmetrizable) universal Borcherds algebra $\hat { \mathfrak { g } } = \hat{\mathfrak { g } }( A )$ one means the Lie algebra (over $\mathbf{R}$ say) with generators $e _ { i } , f _ { i } , h _ { i j }$, subject to the relations [a3]:

GR1) $[ e _ { i } f _ { j } ] = h _ { i j}$, $[ h _ { i j } e _ { k } ] = \delta _ { i j } a _ { i k } e _ { k }$ and $[ h _ { i j } f _ { k } ] = - \delta _ { i j } a _ { i k } f _ { k }$, for all $i , j$;

GR2) $\operatorname{ad}(e_i)^{1-\alpha_{ij}} (e_j)=0$ and $\operatorname{ad}(f_i)^{1-\alpha_{ij}} (f_j)=0$, whenever both $a _ { i i } = 2$ and $i \neq j$; and

GR3) $[ e _ { i } e _ { j } ] = [ f _ { i } f _ { j } ] = 0$ whenever $a _ { i j } = 0$.

Note that for each $i$, $\operatorname { span } \{ e _ { i } , f _ { i } , h _ { i i } \}$ is isomorphic to $\operatorname{sl} _ { 2 } ( {\bf R} )$ when $a _ { i i } \neq 0$, and to the $3$-dimensional Heisenberg algebra when $a _ { i i } = 0$. Immediate consequences of the definition are that:

i) $[ h _ { i j } , h _ { m n } ] = 0$;

ii) $h _ { i j } = 0$ unless the $i$th and $j$th column of $A$ are identical;

iii) the $h _ { i j }$ for $i \neq j$ lie in the centre of $\hat { \mathfrak{g} }$. Setting all $h _ { i j } = 0$ for $i \neq j$ gives the definition of the (symmetrizable) Borcherds algebra $\mathfrak { g } = \mathfrak { g } ( A )$ [a1]. This central extension $\hat { \mathfrak{g} }$ of $\frak g$ is introduced for its role in the characterization of Borcherds algebras below. If $A$ has no zero columns, then $\hat { \mathfrak{g} }$ equals its own universal central extension [a3]. An important technical point is that both $\mathfrak { g } ( A )$ and $\hat { \mathfrak { g } } ( A )$ have trivial radical.

The basic structure theorem [a1] is that of Kac–Moody algebras. Let $\mathfrak { g } = \mathfrak { g } ( A )$ be a symmetrizable Borcherds algebra. Then:

a) $\frak g$ has triangular decomposition $\mathfrak { g } = \mathfrak { g } _ { + } \oplus \mathfrak { h } \oplus \mathfrak { g } _ { - }$, where $\mathfrak { g }_{ +}$ is the subalgebra generated by the $e _ { i }$, $\mathfrak{g}_{-}$ is generated by the $f_i$, and $\mathfrak { h } = \operatorname { span } \{ h _ { i } \}$ is the Cartan subalgebra. Also, $[ \mathfrak { g } _ { + } , \mathfrak { g } _ { - } ] \subset \mathfrak { h }$ and $[ \mathfrak { h } , \mathfrak { g } _ { \pm } ] \subset \mathfrak { g } _ { \pm }$.

b) $\frak g$ has a root space decomposition: formally calling $e_i$ degree $\alpha_i$ and $f_i$ degree $-\alpha_{i}$, and defining $\mathfrak { g } ^ { \alpha }$ to be the subspace of degree $\alpha \in \mathbf{Z} \alpha _ { 1 } + \mathbf{Z} \alpha _ { 2 } + \dots$, one gets $\mathfrak { h } = \mathfrak { g } ^ { 0 }$ and $\mathfrak { g } _ { \pm } = \oplus _ { \alpha \in \Delta _ { \pm } } \mathfrak { g } ^ { \alpha }$, where $[ \mathfrak { g } ^ { \alpha } , \mathfrak { g } ^ { \beta } ] \subset \mathfrak { g } ^ { \alpha + \beta}$ and $\Delta _ { - } = - \Delta _ { + }$;

c) there is an involution $\omega$ on $\frak g$ for which $\omega e _ { i } = f _ { i }$, $\omega h _ { i } = - h_i$, and $\omega \mathfrak { g } ^ { \alpha } = \mathfrak { g } ^ { - \alpha}$;

d) $\operatorname{ dim} \mathfrak { g } ^ { \alpha } < \infty$ and $\operatorname { dim } \mathfrak { g } ^ { \pm \alpha _ { i }} = 1$;

e) there is an invariant symmetric bilinear form $( . | . )$ on $\frak g$ such that for each root $\alpha \neq 0$, the restriction of $( . | . )$ to $\mathfrak { g } ^ { \alpha } \times \mathfrak { g } ^ { - \alpha }$ is non-degenerate, and $( \mathfrak { g } ^ { \alpha } | \mathfrak { g } ^ { \beta } ) = 0$ whenever $\beta \neq - \alpha$;

f) there is a linear assignment $\alpha \mapsto x _ { \alpha } \in \mathfrak{h}$ such that for all $a \in \mathfrak { g } ^ { \alpha }$, $b \in \mathfrak { g } ^ { - \alpha }$, one has $[ a , b ] = ( a | b ) x _ { \alpha }$.

The condition that $\frak g$ be symmetrizable (i.e. condition GC3)) is necessary for the existence of the bilinear form in e). For representation theory it is common to add derivations, so that the roots $\alpha$ will lie in a dual space ${\frak h} ^ { e ^ { * } }$. In particular, define $D _ { i } ( a ) = n _ { i } a$ for any $a \in \mathfrak { g } ^ { n_1 \alpha _ { 1 } + \ldots }$; then each linear mapping $D_i$ is a derivation, and adjoining these to $\mathfrak h $ defines an Abelian algebra $\mathfrak{h} ^ {e }$. The simple root $\alpha_i$ can be interpreted as the element of ${\frak h} ^ { e ^ { * } }$ obeying $\alpha _ { j } ( h _ { i } ) = a _ {i j }$ and $\alpha _ { j } ( D _ { i } ) = \delta _ { i j }$. Construct the induced bilinear form $( . | . )$ on ${\frak h} ^ { e ^ { * } }$, obeying $( \alpha _ { i } | \alpha _ { j } ) = d _ { i } a _ {i j }$.

The properties a)–f) characterize Borcherds algebras. Let $G$ be a Lie algebra (over $\mathbf{R}$) satisfying the following conditions:

1) $G$ has a $\mathbf{Z}$-grading $\oplus _ { i } G_ {i}$ (cf. also Lie algebra, graded), and $\operatorname{dim} G _ { i } < \infty$ for all $i \neq 0$;

2) $G$ has an involution $\omega$ sending $G_i$ to $G_{ - i}$ and acting as $- 1$ on $G_0$;

3) $G$ has an invariant bilinear form $( . | . )$ invariant under $\omega$ such that $( G _ { i } | G _ { j } ) = 0$ if $i \neq - j$, and such that $- ( a | \omega ( a ) ) > 0$ if $0 \neq a \in G _ { i }$ for $i \neq 0$. Then there is a homomorphism $\pi$ from some $\hat { \mathfrak { g } } ( A )$ to $G$ whose kernel is contained in the centre of $\hat { \mathfrak{g} }$, and $G$ is the semi-direct product of the image of $\pi$ with a subalgebra of the Abelian subalgebra $G_0$. That is, $G$ is obtained from $\hat { \mathfrak{g} }$ by modding out some of the centre and adding some commuting derivations. See e.g. [a4] for details.

Define $\Pi ^ { \text { re } }$ to be the set of all real simple roots, i.e. all $\alpha_i$ with $a _ { i i } = 2$; the remaining simple roots are the imaginary simple roots $\alpha \in \Pi ^ { \operatorname {im} }$. The Weyl group (cf. also Weyl group) $W$ of $\frak g$ is the group generated by the reflections $r_i : \mathfrak{h}^ { e ^ { * } } \rightarrow \mathfrak{h} ^ { e ^ { * } }$ for each $\alpha _ { i } \in \Pi ^ { \text{re} }$: $r ( \lambda ) = \lambda - \lambda ( h _ { i } ) \alpha _ { i }$. It will be a (crystallographic) Coxeter group. The real roots of $\frak g$ are defined to be those in $W ( \Pi ^ { re } )$; all other roots are called imaginary. For all real roots, $\operatorname { lim } \mathfrak { g } ^ { \alpha } = 1$ and $( \alpha | \alpha ) > 0$.

$V$ is called an integrable module if

\begin{equation*} V = \bigoplus _ { \lambda \in \mathfrak { h } ^ { e * } } V ^ { \lambda }, \end{equation*}

where the weight space $V ^ { \lambda } : = \{ v \in V : h . v = \lambda ( h ) v \}$, with $\operatorname{dim}V^\lambda < \infty$, and for each $i$ with $a _ { i i } = 2$ both $e _ { i }$ and $f_i$ are locally nilpotent: i.e. for all $v \in V$ and all sufficiently large $k$, $( e _ { i } ) ^ { k } . v = 0 = ( f _ { i } ) ^ { k } . v$. By the character one means the formal sum

\begin{equation*} \operatorname { ch } _ { V } : = \sum _ { \lambda \in \mathfrak{h} ^ {e* } } ( \operatorname { dim } V ^ { \lambda } ) e ^ { \lambda }. \end{equation*}

Let $P _ { + }$ be the set of all weights $\Lambda \in \mathfrak { h } ^ { * }$ obeying $\Lambda ( h _ { i } ) \in {\bf Z}_{ \geq 0}$ whenever $a _ { i i } = 2$, and $\Lambda ( h _ { i } ) \geq 0$ for all $i$. Define the highest-weight $\frak g$-module $L ( \Lambda )$ in the usual way as the quotient of the Verma module (cf. also Representation of a Lie algebra) by the unique proper graded submodule. Then one obtains the Weyl–Kac–Borcherds character formula: Choose $\rho \in \mathfrak { h } ^ { * }$ to satisfy

\begin{equation*} ( \rho \mid \alpha _ { i } ) = \frac { 1 } { 2 } ( \alpha _ { i } \mid \alpha _ { i } ) \end{equation*}

for all $i$, and define $S _ { \Lambda } = e ^ { \Lambda + \rho } \sum _ { s } \epsilon ( s ) e ^ { s }$, where $s$ runs over all sums of $\alpha _ { i } \in \Pi ^ { \text{im} }$ and $\epsilon ( s ) = ( - 1 ) ^ { m }$ if $s$ is the sum of $m$ distinct mutually orthogonal imaginary simple roots, each of which is orthogonal to $\Lambda$, otherwise $\epsilon ( s ) = 0$. Then \[ \operatorname{ch}_{L(\Lambda)} = \frac{\sum_{w \in W} \epsilon(w) w(S_\Lambda)}{e^\rho \prod_{\alpha \in \Delta_+} (1-e^{-\alpha})^{\operatorname{mult} \alpha}} \]

where $\operatorname{mult}\alpha = \dim \mathfrak{g}^\alpha$. $S _ { \Lambda }$ is the correction factor due to imaginary simple roots, much as the "extra" terms in the Macdonald identities are due to the imaginary affine roots. Putting $\Lambda = 0$ gives the denominator identity, as usual.

Thus, Borcherds algebras strongly resemble Kac–Moody algebras and constitute a natural and non-trivial generalization. The main differences are that they can be generated by copies of the Heisenberg algebra as well as $\operatorname{sl} _ { 2 } ( {\bf R} )$, and that there can be imaginary simple roots.

Interesting examples of Borcherds algebras are the Monster Lie algebra [a4], whose (twisted) denominator identity supplied the relations needed to complete the proof of the Monstrous Moonshine conjectures, and the fake Monster [a2]. A Borcherds algebra can be associated to any even Lorentzian lattice. The denominator identities of Borcherds algebras are often automorphic forms on the automorphism group $O _ { s + 2,2} (\bf R )$ of the even self-dual lattice $\text{II} _ { s + 2,2 }$ [a5]. They can serve as "automorphic corrections" to Lorentzian Kac–Moody algebras (see, for instance, [a6]). The space of BPS states in string theory carries a natural structure of a Borcherds-like algebra [a8].

References

[a1]	R.E. Borcherds, "Generalized Kac–Moody algebras" J. Algebra , 115 (1988) pp. 501–512
[a2]	R.E. Borcherds, "The monster Lie algebra" Adv. Math. , 83 (1990) pp. 30–47
[a3]	R.E. Borcherds, "Central extensions of generalized Kac–Moody algebras" J. Algebra , 140 (1991) pp. 330–335
[a4]	R.E. Borcherds, "Monstrous moonshine and monstrous Lie superalgebras" Invent. Math. , 109 (1992) pp. 405–444
[a5]	R.E. Borcherds, "Automorphic forms on $O _ { s + 2,2} (\bf R )$ and infinite products" Invent. Math. , 120 (1995) pp. 161–213
[a6]	V.A. Gritsenko, V.V. Nikulin, "Siegel automorphic form corrections of some Lorentzian Kac–Moody Lie algebras" Amer. J. Math. , 119 (1997) pp. 181–224
[a7]	K. Harada, M. Miyamoto, H. Yamada, "A generalization of Kac–Moody algebras" , Groups, Difference Sets, and the Monster , de Gruyter (1996)
[a8]	J.A. Harvey, G. Moore, "On the algebras of BPS states" Commun. Math. Phys. , 197 (1998) pp. 489–519
[a9]	E. Jurisich, "An exposition of generalized Kac–Moody algebras" Contemp. Math. , 194 (1996) pp. 121–159
[a10]	V.G. Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv. , 2 (1968) pp. 1271–1311
[a11]	V.G. Kac, "Infinite dimensional Lie algebras" , Cambridge Univ. Press (1990) (Edition: Third)
[a12]	R.V. Moody, "A new class of Lie algebras" J. Algebra , 10 (1968) pp. 211–230