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 $G$ be a group. A central extension of $G$ is an exact sequence

the center of $E$. For each section $s:G\to E$ of $p$ the map

is a 2-cocycle on $G$ and different choices of $s$ give cohomologous cocycles $\psi$. Conversely, given a 2-cocycle $\psi$ on $G$ with values in $C$ (with trivial $G$-action) the group

defines a central extension of $G$ by $C$. The associative condition on $E$ is equivalent to the 2-cocycle condition for $\psi$ since

$\left(g_1\cdot g_2\right)\cdot g_3=\left(g_1{g}_2\right)\psi (g_1,g_2)\cdot g_3=g_1{g}_2{g}_3\psi (g_1{g}_2,g_3)\psi (g_1,g_2)$

$g_1\cdot \left(g_2\cdot g_3\right)=g_1\cdot g_2{g}_3\psi (g_2,g_3)=g_1{g}_2{g}_3\psi (g_2,g_3)\psi (g_1,g_2{g}_3)$

Thus

The universal central extension of $G$ is the central extension $E$ such that every projective representation of $G$ is a linear representation of $E$. The Schur multiplier of $G$ is the $C$ for the universal central extension $E$. In a fairly general setting, the Schur multiplier of $G$ is C = $H^2(G,{ℂ}^{\times })$.

Central extensions for Lie Algebras

Let $𝔤$ be a Lie algebra. A central extension of $𝔤$ is an exact sequence

the center of $𝔢$. For each section $s:𝔤\to 𝔢$ of $p$ the map

is a 2-cocycle on $𝔤$ and different choices of $s$ give cohomologous cocycles $\psi$. Conversely, given a 2-cocycle $\psi$ on $𝔤$ with values in $𝔠$ (with trivial $𝔤$-action) the Lie algebra

defines a central extension of $𝔤$ by $𝔠$. The Jacobi identity on $𝔢$ is equivalent to the 2-cocycle condition for $\psi$ since

$\stackrel{\sim }{[}\stackrel{\sim }{[}x,y\stackrel{\sim }{]},z\stackrel{\sim }{]}=\stackrel{\sim }{[}[x,y]+\psi (x,y)c,z\stackrel{\sim }{]}=\stackrel{\sim }{[}[x,y],z\stackrel{\sim }{]}=\left[\right[x,y],z]+\psi \left(\right[x,y],z)c$,

so that

$\stackrel{\sim }{[}\stackrel{\sim }{[}x,y\stackrel{\sim }{]},z\stackrel{\sim }{]}+\stackrel{\sim }{[}\stackrel{\sim }{[}z,x\stackrel{\sim }{]},y\stackrel{\sim }{]}+\stackrel{\sim }{[}\stackrel{\sim }{[}y,z\stackrel{\sim }{]},x\stackrel{\sim }{]}=\left[\right[x,y],z]+\left[\right[z,x],y]+\left[\right[y,z],x]+\psi \left(\right[x,y],z)c+\psi \left(\right[z,x],y)c+\psi \left(\right[y,z],x)c$.

Thus

The universal central extension of $𝔤$ is the central extension $𝔢$ such that every projective representation of $𝔤$ is a linear representation of $𝔢$. The Schur multiplier of $𝔤$ is the $𝔤$ for the universal central extension $𝔢$. In a fairly general setting, the Schur multiplier of $𝔤$ is 𝔠 = $H^2(𝔤,ℂ)$.

Central extensions of the loop Lie algebra

Let $𝔤$ be a finite dimensional complex semisimple Lie algebra with Killing form $\mathrm{langle},\mathrm{rangle}:𝔤\times 𝔤\to ℂ$. Viewing elements of the loop Lie algebra

$𝔤[t,t^{–1}]=𝔤\otimes ℂ[t,t^{–1}]$ with bracket $[x{t}^m,y{t}^n]=[x,y]t^{m+n}$

as functions in $t$ identifies $𝔤[t,t^{-1}]$ with the space of maps $S^1\to 𝔤$, where $S^1=\{e^{i\theta }\mid 0\le \theta <2\pi \}$. Since the space $ℂc$ dual to the space of $𝔤$-invariant bilinear forms on $𝔤$ is one dimensional the universal central extension of $𝔤[t,t^{–1}]$ is

$𝔤[t,t^{-1}]\oplus ℂc$ with $[x{t}^m,y{t}^n]=[x,y]t^{m+n}+m{\delta }_{m,–n}\mathrm{langle}x,y\mathrm{rangle}c$,

where $c$ is central. The derivation $d:𝔤[t,t^{–1}]\to 𝔤[t,t^{-1}]$ given by $d=t\frac{d\phantom{t}}{dt}$ extends to $𝔤[t,t^{–1}]\oplus ℂc$ by $d\left(c\right)=0$ and adjoining this derivation gives the Lie algebra

$\stackrel{\sim }{𝔤}=𝔤[t,t^{-1}]\oplus ℂc\oplus ℂd$ with $[d,x{t}^m]=d\left(x{t}^m\right)=mx{t}^{m-1},[d,c]=0$.

Let $\varphi$ be the highest root of $𝔤$, normalise the scalar product on $𝔤$ so that $\mathrm{langle}\varphi ,\varphi \mathrm{rangle}=2????$ and let $x_{\varphi }\in 𝔤,x_{–\varphi }\in {𝔤}_{–\varphi }$ such that $\mathrm{langle}{x}_{\varphi },x_{–\varphi }\mathrm{rangle}=1$. Let

$e_0=–x_{–\varphi }t,f_0=x_{\varphi }{t}^{–1}$ and $h_0=[e_0,f_0]=[t{x}_{–\varphi },t^{–1}{x}_{\varphi }]=-h_{{\varphi }^{\vee }}+c$.

Let

$\stackrel{\sim }{𝔥}=𝔥\oplus ℂc\oplus ℂd$, and let $\stackrel{\sim }{{𝔥}^{*}}={𝔥}^{*}\oplus ℂ\gamma \oplus ℂ\delta$,

with $\alpha \left(c\right)=\alpha \left(d\right)$, for $\alpha \in {𝔥}^{*}$, and $\delta (𝔥\oplus ℂc)=0$, $\delta \left(d\right)=1$.

Let

${\alpha }_0=\delta –\varphi$, ${\alpha }_0^{\vee }=c–{\varphi }^{\vee }$, and $A^{\left(1\right)}=(\mathrm{langle}{\alpha }_i,{\alpha }_j^{\vee }\mathrm{rangle}{)}_{0<i,j<\ell }$.

Then $e_0,\dots ,e_n,h_0,\dots ,h_n,f_1,\dots ,f_n$ are generators of $\stackrel{\sim }{𝔤}$ as an affine Kac-Moody Lie algebra with Cartan matrix $A^{\left(1\right)}$ and minimal realization

$(\stackrel{\sim }{𝔥},({\alpha }_0^{\vee },\dots ,{\alpha }_n^{\vee }),({\alpha }_0,\dots ,{\alpha }_n\left)\right)$.

Note that $\dim \left(\stackrel{\sim }{𝔥}\right)=n+2=2(n+1)-\mathrm{rank}\left(A\right)$ since $\mathrm{rank}\left(A\right)=n$. The center of $\stackrel{\sim }{𝔤}$ is $ℂc$, the Cartan subalgebra of $\stackrel{\sim }{𝔤}$is

$\stackrel{\sim }{𝔥}=𝔥\oplus ℂc\oplus ℂd$ and $𝔤[t,t^{–1}]\simeq \frac{{\stackrel{\sim }{𝔤}}^{\prime }}{ℂc}$, where ${\stackrel{\sim }{𝔤}}^{\prime }=𝔤[t,t^{-1}]\oplus ℂc$

is the derived algebra of $\stackrel{\sim }{𝔤}$. Define $\delta \in {\stackrel{\sim }{𝔥}}^{*}$ by

$\delta \left(d\right)=1$, $\delta \left(c\right)=0$, $\delta \left(h\right)=0$, for $h\in 𝔥$.

The real and imaginary roots of $\stackrel{\sim }{𝔤}$ are

${\stackrel{\sim }{R}}_{\mathrm{re}}=\{\alpha +r\delta \mid \alpha \in R,r\in \mathbb{Z}\},{\stackrel{\sim }{R}}_{\mathrm{im}}=\{r\delta \mid r\in \mathbb{Z},r\ne 0\}$,

respectively, where each imaginary root has multiplicity $n=\mathrm{rank}\left(𝔤\right)$.

Kac-Moody Lie algebras

A symmetrizable matrix is a matrix $A=\left(a_{ij}\right)$ such that there exists a diagonal matrix $D$ with $DA$ symmetric. A generalised Cartan matrix $A=\left(a_{ij}\right)$ be a matrix with rows and columns indexed by a set $I$ such that

$a_{ij}\in \mathbb{Z},a_{ii}=2,a_{ij}\le 0\text{if}i\ne j,a_{ij}\ne 0\text{if andonly if}{a}_{ji}\ne 0$.

Let $A=\left(a_{ij}\right)$ is a square matrix with rows and columns indexed by a set $I$. A minimal realisation of $A$ is a triple $(𝔥,B,B^{\vee })$ where

• $𝔥$ is a finite dimensional vector space of dimension $\mathrm{Card}\left(I\right)+\mathrm{corank}\left(A\right)$,
• $B=\{{\alpha }_i\mid i\in I\}$ is a linearly independent set in ${𝔥}^{*}$,
• $B^{\vee }=\{{\alpha }_i^{\vee }\mid i\in I\}$ is a linearly independent set in $𝔥$.

The Lie algebra $\stackrel{\sim }{𝔤}\left(A\right)$ is given by generators $𝔥,e_i,f_i$ with relations

$[h,h^{\prime }]=0,[e_i,f_j]={\delta }_{ij}{h}_i,[h,e_i]={\alpha }_i\left(h\right)e_i,[h,f_i]=–{\alpha }_i\left(h\right)f_i,$

for $h,h^{\prime }\in 𝔥$. The Borcherds-Kac-Moody Lie algebra of $A$ is

$𝔤=\frac{\stackrel{\sim }{𝔤}\left(A\right)}{𝔯}$, where $𝔯$ is the largest ideal of $𝔤\left(A\right)$ such that $𝔯\cap 𝔥=0$. If $A$ is a generalised Cartan matrix the elements

$\left(\mathrm{ad}{e}_i{)}^{1–a_{ij}}\right(e_j)$ and $\left(\mathrm{ad}{f}_i{)}^{1–a_{ij}}\right(f_j)$ are in $𝔯$

and it is known that if $A$ is symmetrizable these elements generate $𝔯$.

The Lie algebra $\stackrel{\sim }{𝔤}\left(A\right)$ is graded by

$Q={\sum }_{i\in I}{\alpha }_i$ by setting $\mathrm{deg}\left(e_i\right)={\alpha }_i,\mathrm{deg}\left(f_i\right)=–{\alpha }_i,\mathrm{deg}\left(h\right)=0$,

for $h\in 𝔥$. Any ideal of $\stackrel{\sim }{𝔤}\left(A\right)$ is $Q$-graded and so $𝔤$ is $Q$-graded. An element $\alpha \in Q$ is

a root if ${𝔤}_{\alpha }\ne 0$ and $\dim \left({𝔤}_{\alpha }\right)$ is the multiplicity of $\alpha$.

Kac-Moody groups

An $\stackrel{\sim }{𝔤}$-module is integrable if elements of $\stackrel{\sim }{𝔤}{\stackrel{\sim }{𝔤}}_{\alpha }$, $\alpha \in R_{\mathrm{re}}$, act locally nilpotently. The Kac-Moody group is the group $G$ generate by $x_{\alpha }\left(c\right)$, $\alpha \in R_{\mathrm{re}}$, $c\in ℂ$, with relations

$x_{\alpha }\left(c_1\right)x_{\alpha }\left(c_2\right)=x_{\alpha }(c_1+c_2)$

and the additional relations coming from forcing an element to be $1$ if it acts by $1$ on every integrable ${\stackrel{\sim }{𝔤}}^{\prime }$-module, where the action of $x_{\alpha }\left(c\right)$ on an integrable $\stackrel{r}{g}{\stackrel{\sim }{𝔤}}^{\prime }$-module is given by

$x_{\alpha }\left(c\right)=e^{c{X}_{\alpha }}$, where $X_{\alpha }\in {\stackrel{\sim }{𝔤}}_{\alpha }^{\prime }$.