Central extensions
Arun Ram
Department of Mathematics
University of Wisconsin
Madison, WI 53706 USA
ram@math.wisc.edu
This page is the result of joint work with James Parkinson and Christoph
Schwer.
Central extensions for groups
Let be a group. A central extension of is an exact sequence
such that ,
the center of . For each section of the map
given by ,
is a 2-cocycle on and different choices of give cohomologous cocycles . Conversely, given a 2-cocycle on with values in (with trivial -action) the group
with and central in ,
defines a central extension of by . The associative condition on is equivalent to the 2-cocycle condition for since
Thus
(central extensions of by ) .
The universal central extension of is the central extension such that every projective representation of is a linear representation of . The Schur multiplier of is the for the universal central extension . In a fairly general setting, the Schur multiplier of is C = .
Central extensions for Lie Algebras
Let be a Lie algebra. A central extension of is an exact sequence
such that ,
the center of . For each section of the map
given by ,
is a 2-cocycle on and different choices of give cohomologous cocycles . Conversely, given a 2-cocycle on with values in (with trivial -action) the Lie algebra
with and central in ,
defines a central extension of by . The Jacobi identity on is equivalent to the 2-cocycle condition for since
,
so that
.
Thus
(central extensions of by ) .
The universal central extension of is the central extension such that every projective representation of is a linear representation of . The Schur multiplier of is the for the universal central extension . In a fairly general setting, the Schur multiplier of is 𝔠 = .
Central extensions of the loop Lie algebra
Let be a finite dimensional complex semisimple Lie algebra with Killing
form . Viewing elements of the loop Lie algebra
with bracket
as functions in identifies with the space of maps , where . Since the space dual to the space of -invariant bilinear forms on is one dimensional the universal central extension of is
with ,
where is central. The derivation given by extends to by and adjoining this derivation gives the Lie algebra
with .
Let be the highest root of , normalise the scalar product on so that and let such that . Let
and .
Let
, and let ,
with , for , and , .
Let
, , and .
Then are generators of as an affine Kac-Moody Lie algebra with Cartan matrix and minimal realization
.
Note that since . The center of is , the Cartan subalgebra of is
and , where
is the derived algebra of . Define by
, , , for .
The real and imaginary roots of are
,
respectively, where each imaginary root has multiplicity .
Kac-Moody Lie algebras
A symmetrizable matrix is a matrix such that there exists a diagonal matrix with symmetric. A generalised Cartan matrix be a matrix with rows and columns indexed by a set such that
.
Let is a square matrix with rows and columns indexed by a set . A minimal realisation of is a triple where
- is a finite dimensional vector space of dimension ,
- is a linearly independent set in ,
- is a linearly independent set in .
The Lie algebra is given by generators with relations
for . The Borcherds-Kac-Moody Lie algebra of is
, where is the largest ideal of such that . If is a generalised Cartan matrix the elements
and are in
and it is known that if is symmetrizable these elements generate .
The Lie algebra is graded by
by setting ,
for . Any ideal of is -graded and so is -graded. An element is
a root if and is the multiplicity of .
Kac-Moody groups
An -module is integrable if elements of , , act locally nilpotently. The Kac-Moody group is the group generate by , , , with relations
and the additional relations coming from forcing an element to be if it acts by on every integrable -module, where the action of on an integrable -module is given by
, where .