View source for Borcherds Lie algebra

''Borcherds algebra''

While a [[Kac–Moody algebra|Kac–Moody algebra]] is generated in a fairly simple way from copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302001.png" />, a Borcherds or generalized Kac–Moody algebra [[#References|[a1]]], [[#References|[a7]]], [[#References|[a9]]], [[#References|[a11]]] can also involve copies of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302002.png" />-dimensional Heisenberg algebra. Nevertheless, it inherits many of the Kac–Moody properties. Borcherds algebras played a key role in the proof of the [[Moonshine conjectures|Monstrous Moonshine conjectures]] [[#References|[a4]]], and also led to the development of a theory of automorphic products [[#References|[a5]]].

First recall the definition of a Kac–Moody algebra. By a (symmetrizable) Cartan matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302003.png" /> one means an integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302004.png" />-matrix obeying

C1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302006.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302007.png" />; and

C2) there is a diagonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302008.png" /> with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b1302009.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020010.png" /> is symmetric. A (symmetrizable) Kac–Moody algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020011.png" /> [[#References|[a10]]], [[#References|[a12]]] is the [[Lie algebra|Lie algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020012.png" /> generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020013.png" />, obeying the relations:

R1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020017.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020018.png" />; and

R2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020020.png" />.

A Borcherds algebra is defined similarly. By a generalized Cartan matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020021.png" /> one means a (possibly infinite) matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020023.png" />, obeying

GC1) either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020025.png" />;

GC2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020027.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020028.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020029.png" />; and

GC3) there is a diagonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020030.png" /> with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020032.png" /> is symmetric. By the (symmetrizable) universal Borcherds algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020033.png" /> one means the [[Lie algebra|Lie algebra]] (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020034.png" /> say) with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020035.png" />, subject to the relations [[#References|[a3]]]:

GR1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020038.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020039.png" />;

GR2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020040.png" />, whenever both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020042.png" />; and

GR3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020043.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020044.png" />.

Note that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020046.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020047.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020048.png" />, and to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020049.png" />-dimensional Heisenberg algebra when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020050.png" />. Immediate consequences of the definition are that:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020051.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020052.png" /> unless the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020053.png" />th and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020054.png" />th column of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020055.png" /> are identical;

iii) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020057.png" /> lie in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020058.png" />. Setting all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020059.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020060.png" /> gives the definition of the (symmetrizable) Borcherds algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020061.png" /> [[#References|[a1]]]. This central extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020063.png" /> is introduced for its role in the characterization of Borcherds algebras below. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020064.png" /> has no zero columns, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020065.png" /> equals its own universal central extension [[#References|[a3]]]. An important technical point is that both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020067.png" /> have trivial radical.

The basic structure theorem [[#References|[a1]]] is that of Kac–Moody algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020068.png" /> be a symmetrizable Borcherds algebra. Then:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020069.png" /> has triangular decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020071.png" /> is the subalgebra generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020073.png" /> is generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020074.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020075.png" /> is the [[Cartan subalgebra|Cartan subalgebra]]. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020077.png" />.

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020078.png" /> has a root space decomposition: formally calling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020079.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020081.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020082.png" />, and defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020083.png" /> to be the subspace of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020084.png" />, one gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020086.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020088.png" />;

c) there is an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020089.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020090.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020092.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020093.png" />;

d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020095.png" />;

e) there is an invariant symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020096.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020097.png" /> such that for each root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020098.png" />, the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b13020099.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200100.png" /> is non-degenerate, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200101.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200102.png" />;

f) there is a linear assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200103.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200105.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200106.png" />.

The condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200107.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200108.png" /> will lie in a dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200109.png" />. In particular, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200110.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200111.png" />; then each linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200112.png" /> is a derivation, and adjoining these to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200113.png" /> defines an Abelian algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200114.png" />. The simple root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200115.png" /> can be interpreted as the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200116.png" /> obeying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200118.png" />. Construct the induced bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200119.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200120.png" />, obeying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200121.png" />.

The properties a)–f) characterize Borcherds algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200122.png" /> be a Lie algebra (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200123.png" />) satisfying the following conditions:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200124.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200125.png" />-grading <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200126.png" /> (cf. also [[Lie algebra, graded|Lie algebra, graded]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200127.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200128.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200129.png" /> has an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200130.png" /> sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200131.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200132.png" /> and acting as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200133.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200134.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200135.png" /> has an invariant bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200136.png" /> invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200137.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200138.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200139.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200140.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200141.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200142.png" />. Then there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200143.png" /> from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200144.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200145.png" /> whose kernel is contained in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200146.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200147.png" /> is the semi-direct product of the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200148.png" /> with a subalgebra of the Abelian subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200149.png" />. That is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200150.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200151.png" /> by modding out some of the centre and adding some commuting derivations. See e.g. [[#References|[a4]]] for details.

Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200152.png" /> to be the set of all real simple roots, i.e. all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200153.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200154.png" />; the remaining simple roots are the imaginary simple roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200155.png" />. The Weyl group (cf. also [[Weyl group|Weyl group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200156.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200157.png" /> is the group generated by the reflections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200158.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200159.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200160.png" />. It will be a (crystallographic) [[Coxeter group|Coxeter group]]. The real roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200161.png" /> are defined to be those in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200162.png" />; all other roots are called imaginary. For all real roots, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200164.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200165.png" /> is called an integrable module if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200166.png" /></td> </tr></table>

where the weight space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200167.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200168.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200169.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200170.png" /> both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200171.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200172.png" /> are locally nilpotent: i.e. for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200173.png" /> and all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200174.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200175.png" />. By the character one means the formal sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200176.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200177.png" /> be the set of all weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200178.png" /> obeying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200179.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200180.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200181.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200182.png" />. Define the highest-weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200184.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200185.png" /> in the usual way as the quotient of the Verma module (cf. also [[Representation of a Lie algebra|Representation of a Lie algebra]]) by the unique proper graded submodule. Then one obtains the Weyl–Kac–Borcherds character formula: Choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200186.png" /> to satisfy

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200187.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200188.png" />, and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200189.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200190.png" /> runs over all sums of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200191.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200192.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200193.png" /> is the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200194.png" /> distinct mutually orthogonal imaginary simple roots, each of which is orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200195.png" />, otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200196.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200197.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200198.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200199.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200200.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200201.png" />, and that there can be imaginary simple roots.

Interesting examples of Borcherds algebras are the Monster Lie algebra [[#References|[a4]]], whose (twisted) denominator identity supplied the relations needed to complete the proof of the Monstrous Moonshine conjectures, and the fake Monster [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200202.png" /> of the even self-dual lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200203.png" /> [[#References|[a5]]]. They can serve as  "automorphic corrections"  to Lorentzian Kac–Moody algebras (see, for instance, [[#References|[a6]]]). The space of BPS states in string theory carries a natural structure of a Borcherds-like algebra [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.E. Borcherds,   "Generalized Kac–Moody algebras"  ''J. Algebra'' , '''115'''  (1988)  pp. 501–512</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.E. Borcherds,   "The monster Lie algebra"  ''Adv. Math.'' , '''83'''  (1990)  pp. 30–47</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.E. Borcherds,   "Central extensions of generalized Kac–Moody algebras"  ''J. Algebra'' , '''140'''  (1991)  pp. 330–335</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Borcherds,   "Monstrous moonshine and monstrous Lie superalgebras"  ''Invent. Math.'' , '''109'''  (1992)  pp. 405–444</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.E. Borcherds,   "Automorphic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130200/b130200204.png" /> and infinite products"  ''Invent. Math.'' , '''120'''  (1995)  pp. 161–213</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Harada,   M. Miyamoto,   H. Yamada,   "A generalization of Kac–Moody algebras" , ''Groups, Difference Sets, and the Monster'' , de Gruyter  (1996)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.A. Harvey,   G. Moore,   "On the algebras of BPS states"  ''Commun. Math. Phys.'' , '''197'''  (1998)  pp. 489–519</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E. Jurisich,   "An exposition of generalized Kac–Moody algebras"  ''Contemp. Math.'' , '''194'''  (1996)  pp. 121–159</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  V.G. Kac,   "Simple irreducible graded Lie algebras of finite growth"  ''Math. USSR Izv.'' , '''2'''  (1968)  pp. 1271–1311</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V.G. Kac,   "Infinite dimensional Lie algebras" , Cambridge Univ. Press  (1990)  (Edition: Third)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  R.V. Moody,   "A new class of Lie algebras"  ''J. Algebra'' , '''10'''  (1968)  pp. 211–230</TD></TR></table>