## The Virasoro algebra

Last update: 20 April 2012

## The Virasoro algebra

Let $A$ be an algebra over $ℂ.$ A derivation of $A$ is a linear map $d:A\to A$ such that The vector space of derivations on $A$ is a Lie algebra with bracket $[d1,d2] = d1d2 - d2d1.$

Let $𝔤$ be a Lie algebra. A central extension of $𝔤$ is a short exact sequence of Lie algebras $0→𝔠→ 𝔤1 →φ1 𝔤→0 such that 𝔠⊆Z(𝔤1),$ the center of $\stackrel{˜}{𝔤}.$ A morphism of central extensions is a Lie algebra homomorphism $\psi :{𝔤}_{1}\to {𝔤}_{2}$ such that ${\phi }_{2}\psi ={\phi }_{1}.$ A universal central extension is a central extension $\stackrel{˜}{𝔤}$ such that there is a unique morphism from $\stackrel{˜}{𝔤}$ to every other central extension of $𝔤.$ It classifies the projective representations of $𝔤$ (at least this is right for GROUPS, see Steinberg). Isomorphism classes of one-dimensional central extensions are in bijection with elements $c\in {H}^{2}\left(𝔤;𝔽\right)$ via the formula where $\mathbf{c}$ is a basis element of $Z\left(\stackrel{˜}{𝔤}\right).$

The Witt algebra is the Lie algebra of derivations of $ℂ\left[t,{t}^{-1}\right].$ If $d:ℂ\left[t,{t}^{-1}\right]\to ℂ\left[t,{t}^{-1}\right]$ is a derivation then and hence $d$ is determined by the value $d\left(t\right).$ Thus $W has basis {dj | j∈ℤ}, where dj = -tj+1 ddt,$ and $[dn,dm] = (n-m) dn+m.$ Note that $ℂ\left[t,{t}^{-1}\right]$ is the complexification of the ring of smooth functions on the circle ${S}^{1}.$

The Virasoro algebra is the universal central extension of $W.$ It has basis ${c,di | i∈ℤ} with [c,di] = 0, [dn,dm] = (n-m)dn+m + δn,-m n3-n12 c.$ To try to prove this note that if $[dn,dm] = (n-m)dn+m + c(n,m)z,$ then $[dn,dm] = -[dm,dn] forces c(n,m) = -c(m,n),$ and the Jacobi identity forces $c(n+m,l) + c(l+n,m) + c(m+l,n) = 0.$

The Virasoro algebra has triangular structure and skew linear $\left(\theta \left(\xi x\right)=\stackrel{_}{\xi }\theta \left(x\right),\phantom{\rule{.5em}{0ex}}for\phantom{\rule{.5em}{0ex}}\xi \in ℂ\phantom{\rule{.5em}{0ex}}and\phantom{\rule{.5em}{0ex}}x\in \mathrm{Vir}\right)$ Cartan involution given by $Vir<0 = span{di | i∈ℤ<0}, Vir0 = span{c,d0}, Vir>0 = span{di | i∈ℤ>0}, with θ: Vir → Vir dn ↦ d-n c ↦ c$

Let $U$ be the universal enveloping algebra of $\mathrm{Vir}.$ The action of $𝔥={\mathrm{Vir}}_{0}$ on ${U}_{<0}$ given ${U}_{<0}$ a ${ℤ}_{<0}$ grading such that $U-n has basis {d-λ | λ is a partition of n} where dPICTURE = d-λ = d-λ1⋯d-λl,$ if $\lambda =\left({\lambda }_{1},...,{\lambda }_{l}\right).$ This is the Poincaré-Birkhoff-Witt basis of ${U}_{<0}.$

### The action of admissible $\stackrel{^}{𝔤}$ modules

Because the Witt algebra is the space of derivations of $ℂ\left[t,{t}^{-1}\right]$ the Witt algebra acts on the loop algebra $𝔤\otimes ℂ\left[t,{t}^{-1}\right],$ and the Virasoro algebra also acts on $𝔤\otimes ℂ\left[t,{t}^{-1}\right]$ by $[ d˜k, tn⊗x ] = tk+1 ddt (tn⊗x) = ntn+k ⊗ x$ and $c$ acting by $0??$ We can "extend" this action to an action on admissible $\stackrel{^}{𝔤}$ modules.

Let $h$ be the Coxeter number of $𝔤$ and let $Tk = 12 ∑j∈ℤ∑i : ui(-j) ui(j+k) :$ where the normal ordering is

If $V$ is a restricted $\stackrel{^}{𝔤}-$module of level $l$ and $l\ne -h$ then $dk ↦ 1l+h Tk and z ↦ ll+h dim(𝔤)$ define an action of $\mathrm{Vir}$ on $V.$

Let $𝔤={\mathrm{𝔰𝔩}}_{2}$ and use the imbedding $ι: 𝔰𝔩2 → 𝔰𝔩2 ⊕ 𝔰𝔩2 x ↦ (x,x)$ to define an action of $\mathrm{Vir}$ on $L\left(\xi \right)\otimes L\left(m\xi +\frac{n}{2}\alpha \right)$ by $dk ↦ 1l+h (Tk⊗1 + 1⊗Tk) - 1l+h ι(Tk).$ This action of $\mathrm{Vir}$ commutes with the action of ${\stackrel{^}{\mathrm{𝔰𝔩}}\prime }_{2}.$ By a character computation where $I = {k∈ℤ | n2 - m+12 ≤ k ≤ n2},$ and $char(Um,n,k) = ∑j∈ℤ≥0 dim(Um,n,kj) qj = (fm,n,k - fm,n,n+1-k) ∏j∈ℤ≥0 11-qj,$ with $fm,n,k = ∑j∈ℤ q (m+2) (m+3) j2 + (n+1+2k(m+2))j +k2 .$ Then $z$ acts on ${U}_{m,n,k}$ by the constant $n(n+2) 4(m+2) - (n-2k) (n-2k+2) 4(m+3) +j$ and the minimum value of $j$ for which ${U}_{m,n,k}^{j}\ne 0$ is where

### The Shapovalov determinant

The highest power of $h$ in $\mathrm{det}\left(M{\left(h,c\right)}^{\left(h+n,c\right)}\right)$ is $∑λ⊢n l(λ), with coefficient ∏λ⊢n z2λ,$ where, for a partition $\lambda$ of $n,$ $\frac{n!}{{z}_{\lambda }}$ is the cardinality of the conjugacy class of the symmetric group ${S}_{n}$ labeled by $\lambda .$

 Proof. Let us first analyze the entries $⟨{d}_{-\mu }{v}^{+},{d}_{\lambda }{v}^{+}⟩$ in the matrix. Then $⟨ d-μ v+, d-λ v+ ⟩ = ⟨ v+, dmu d-λ v+ ⟩ = p0,0 (h,c),$ where ${p}_{0,0}\left({d}_{0},z\right)$ is the polynomial in ${d}_{0}$ and $z$ in the PBW basis expansion $dμ d-λ = ∑ν,τ d-ν pν,τ (d0,z) dτ.$ This expansion is obtained by using the relations to put the ${d}_{i}$ in increasing order. The first relation "combines" $j$ and $k$ into $j+k.$ If ${d}_{-\nu }{p}_{\nu ,\tau }\left({d}_{0},z\right){d}_{\tau }$ is a term in the PBW expansion then the parts of $-\nu$ and $\tau$ are combinations of parts of $\mu$ and $-\lambda$ and the degree in ${d}_{0}$ of the polynomial ${p}_{\nu ,\tau }\left({d}_{0},z\right)$ is the maximal number of 0 parts that can be obtained by combinations of the remaining parts of $\mu$ and $-\lambda$ (those that do not contribute to $\nu$ and $-\tau$). Thus the degree (in ${d}_{0}$) of ${p}_{0,0}\left({d}_{0},z\right)$ is the maximal number of 0 parts that can be obtained by combinations of the parts of $\mu$ and $-\lambda$ and is at most $l\left(\mu \right)$ and at most $l\left(\lambda \right).$ Since both $\lambda$ and $\mu$ are partitions of $n,$ a term of degree $l\left(\lambda \right)$ is produced only when $\lambda =\mu$ and each part of $\lambda$ is combined with a single part of $-\lambda .$ Thus the maximal degree term in row $\lambda$ of $A{\left(h,c\right)}^{\left(h+n,c\right)}$ appears in column $\lambda ,$ i.e. on the diagonal. The identity $drd-rs = d-rsdr + d-rs-1 ( 2rsd0 + 2r2 (s2) + s ( r3-r 12 )z ),$ is verified by induction on $s,$ the induction step being $drd-rs = d-rdrd-rs-1 + ( 2rd0 + ( r3-r 12 )z ) d-rs-1 = d-r ( d-rs-1dr + d-rs-2 ( 2r(s-1)sd0 + 2r2 ( s-12 ) + (s-1)s ( r3-r 12 )z ) ) + d-rs-1 ( 2r ( d0 + r(s-1) ) + ( r3-r 12 )z ) = d-rsdr + d-rs-1 ( 2rsd0 + 2r2 (s2) + s( r3-r 12 )z ).$ Suppose that $drk d-rs = d-rs drk + d-rs-1 p1k,s drk-1 + d-r2 p2k,s drk-2 + ⋯ + d-rs-k pkk,s,$ where ${p}_{i}^{k,s}$ are polynomials in ${d}_{0}$ and $z.$ Then $drk+1 d-rs = dr ∑j=0k d-rs-j pjk,s(d0,z) drk-j = ∑j=0k d-rs-j dr pjk,s(d0,z) drk-j + d-rs-j-1 ( 2r(s-j)d0 + 2r2 (s-j2) + (s-j) ( r3-r 12 )z ) pjk,s(d0,z) drk-j = ∑j=0k d-rs-j pjk,s(d0-r,z) drk-j+1 + d-rs-j-1 ( 2r(s-j)d0 + 2r2 (s-j2) + (s-j) ( r3-r 12 )z ) pjk,s(d0,z) drk-j,$ from which it follows that $plk+1,s(d0,z) = plk,s(d0-r,z) + ( 2r(s-l+1)d0 + 2r2 (s-l+12) + (s-j) ( r3-r 12 )z ) pl+1k,s(d0,z).$ (I'm not quite sure if this calculation is exactly right, I need to do some checks for $s=2$ and $s=3$ to make sure.) In particular, $\square$

There is a bijection ${(λ,i) | λ⊢n, 1≤i≤l(λ)} ↔ {(μ,(rs)) | rs=∅, |μ| + rs = n} (λ,i) → ( λ-(λisi), (λisi) ) (μ∪(rs),j) ← (μ,(rs)) PICTURE$ where $\lambda -\left({\lambda }_{i}^{{s}_{i}}\right)$ is the partition obtained by removing all rows of length ${\lambda }_{i}$ which are in rows with number $\ge i,$ ${s}_{i}$ is the number of $j\ge i$ such that ${\lambda }_{j}={\lambda }_{i}$ and $j-1$ is the row number of the largest part $\le r$ in the partition $\mu .$

This bijection proves the identity $∑λ⊢n l(λ) = ∑ (rs)≠∅ n-rs≥0 p(n-rs),$ where $p\left(k\right)$ is the number of partitions with $k$ boxes.

If $k and ${d}_{0}-h$ divides the determinant $\mathrm{det}\left({M}_{+k}\right)$ then ${\left({d}_{0}-h\right)}^{p\left(n-k\right)}$ divides the determinant $\mathrm{det}\left({M}_{+n}\right).$

${C}_{rs}\left(h,c\right)$ divides the determinant $\mathrm{det}\left({M}_{+rs}\right).$

 Proof. First proof: Second proof: Define a Vir action on the space of semi-infinite forms $ℋ(α,β) = span{ ⋯∧fik∧⋯∧fi1 | with i1 by setting $dn(fj) = ( j+β-(1-n)α ) fj-n.$ Then, for appropriate choice of $\alpha$ and $\beta ,$ the Vir module $ℋ\left(\alpha ,\beta \right)$ becomes a highest weight module of highest weight $\left(h,c\right).$ One can construct a number of highest weight vectors in $ℋ\left(\alpha ,\beta \right),$ see ???. $\square$

### Blocks

Given $\left(h,c\right)$ the equation $μ+1μ = 13-c6 determines {μ,1μ},$ and for each choice of $\mu$ in this set, $y2 = 4μ( 1-c24-h ) determines {y,-y}$ giving 4 lines $s = μr+y, s = μr-y, s = 1μr - 1μy, s = 1μr + 1μy.$ Conversely, given $\left(\mu ,y\right)$ then $13-c6 = μ+1μ determines c,$ and $h = -y2 4μ + 1-c24 determines h.$ Define $Crs (h,c) = 142 ( s-μr+y ) ( s-μr-y ) ( s-1μr-1μy ) ( s-1μr+1μy ) = 142 ( (s-μr)2-y2 ) ( (s-1μr)2 - y2μ2 ) = ( (s-μr) (1μs-r) 4 - y2 4μ ) ( (s-1μr) (μs-r)4 - y2 4μ ) = ( μr2 - 2rs + 1μs2 4 - y2 4μ ) ( μr2 - 2rs + μs2 4 - y2 4μ ) = ( 14 ( μr2 + 1μs2 ) - rs2 + h - 1-c24 ) ( 14 ( 1μ r2 + μs2 ) - rs2 +h- 1-c24 ).$ If $x = 12 25-c1-c$ then $(x-12) (x+12) = x2-14 = 14 (25-c1-c) - 14 = 14 ( 25-c-1+c 1-c ) = 14 (241-c) = 61-c$ so that $1-c6 = 1 (x-12)(x+12) and 13-c6 = 2+1-c6 = 2+ 1 (x-12)(x+12) .$ Then the solutions to $\mu +\frac{1}{\mu }=\frac{13-c}{6}$ are $μ = x+12 x-12 and 1μ = x-12 x+12 ,$ since $x+12 x-12 + x-12 x+12 = x2-x+14+x2+x+14 (x-12)(x+12) = 2x2+12 x2-14 = 2x2-12+1 x2-14 = 2+ 1 (x-12)(x+12) .$ Then $1-c24 - 14 ( μr2 + 1μs2 ) + rs2 = 1-c24 - 14 ( x+12 x-12 r2 + x-12 x+12 s2 ) + rs2 = 1-c24 - 14 ( (x2+x+14) r2 + (x2-x+14) s2 (x-12)(x+12) ) + rs2 = 1-c24 - 14 ( 2x2+12 (x-12)(x+12) (r2+s2) + x (x-12)(x+12) (r2-s2) ) + rs2 = 1-c24 - 14 ( (13-c6) (r2+s2) + 12 25-c1-c (1-c6) (r2-s2) ) + rs2 = 1-c24 - 14 ( (13-c6) (r2+s2) + 112 (25-c) (1-c) (r2-s2) ) + rs2$ and $1-c24 - 14 ( μr2 + 1μ s2 ) + rs2 = 1-c24 - 14 (s-μr)2 μ = 1-c24 - 14 ( s - x+12 x-12 r )2 x-12 x+12 = 14 ( 1 (x-12)(x+12) - ( (x-12)s - (x+12)r )2 (x-12)(x+12) ) = ( (x+12)r - (x-12)s )2-1 -4(x-12)(x+12) .$ Now put $m+\frac{5}{2}=x$ so that $x+\frac{1}{2}=m+3$ and $x-\frac{1}{2}=m+2.$

$det(A-n) = ∏1≤r≤s≤n ( (2r)s s! ) p(n-rs) - p(n-r(s+1)) ∏ r,s∈ℤ≥0 rs≤n ( h-hrs ) p(n-rs) ,$ where $hrs = 148 ( (13-c) (r2+s2) + (c-1) (c-25) (r2-s2) - 24rs - 2+2c ).$ Then $Cr,s (h,c) = { (h-hrs) (h-hsr), if r≠s, h-hrr, if r=s. }$

## Notes and References

These notes are from lecture notes of Arun Ram from 2005.

## References

[Cu1] C. Curtis, "Representations of Hecke algebras." Astérisque, 9 (1988), 13-60.

[Li1] P. Littelmann, Paths and root operators in representation theory, Ann. Math. 142 (1995), 499-525.

[Li2] P. Littelmann, Bases for representations, LS-paths and Verma flags, Volume in honor of the 70th birthday of C.S. Seshadri, ?????.