A flow in the formulation and proof of Lemma 2.7 of E. Felińska, Z. Jaskólski, and M. M. Kosztołowicz, J. Math. Phys. 53, 033504 (2012) is fixed in Sec. I of this Erratum. This has no consequences for the rest of the paper. An essential error was made in Theorems 3.5, 3.6, and Corollary 3.7 of Sec. III of E. Felińska, Z. Jaskólski, and M. M. Kosztołowicz, J. Math. Phys. 53, 033504 (2012). As it was pointed out by V. Mazorchuk and K. Zhao [“Simple Virasoro modules which are locally finite over positive part,” e-print arXiv:1205.5937v2 [math.RT]] they are valid only in the very limited case of n = 3. In order to introduce necessary correction we have rewritten a part of Sec. III after Theorem 3.3. This is presented in Sec. II of this Erratum.

Topics
Lie algebras

Lemma 2.7. Let |w⟩ be the universal Whittaker vector of a type ψr and a rank s and λ-a pseudo partition of order r. If k > 0 is the smallest number for which λ(−k) ≠ 0, then

\begin{eqnarray*}[L_{k+s},L_{\lambda _{-}}]L_{\lambda _{+}}\left|w\right\rangle &=&\\&& \hspace{-70.0pt} \lambda (-k) \psi _r(L_{s})(2k+s) L_{-l}^{\lambda (-l)} \hdots L_{-k-1}^{\lambda (-k-1)} L_{-k}^{\lambda (-k)-1} L_{\lambda _{+}}\left|w\right\rangle \; + \;\left|v\right\rangle \; + \textstyle \;\left|v^{\prime }\right\rangle ,\end{eqnarray*}
[Lk+s,Lλ−]Lλ+w=λ(−k)ψr(Ls)(2k+s)L−lλ(−l)⋯L−k−1λ(−k−1)L−kλ(−k)−1Lλ+w+v+v′,

where max |v⟩ < |λ| − kand

${\rm max}\,\left|v^{\prime }\right\rangle \leqslant |\lambda | - k,\; {\rm max}^\#\left|v^{\prime }\right\rangle < \#\lambda$
max v′⩽|λ|−k, max #v′<#λ⁠.

Proof: Let

$L_{\lambda _{-}^{{\prime }}} = L_{-n}^{\lambda (-n)} \hdots L_{-k-1}^{\lambda (-k-1)}$
Lλ−′=L−nλ(−n)⋯L−k−1λ(−k−1)⁠, so that
$L_{\lambda _{-}} = L_{\lambda _{-}^{{\prime }}}L_{-k}^{\lambda (-k)}$
Lλ−=Lλ−′L−kλ(−k)⁠. Then

(1)

Repeating the reasoning from the proof of Lemma 2.6, one can write

\begin{eqnarray*}[L_{k+s},L_{\lambda _{-}^{{\prime }}}]L_{-k}^{\lambda (-k)}L_{\lambda _{+}}\left|w\right\rangle &=& \hspace{-20.0pt} \underset{ \scriptstyle \# \gamma = \,0}{\sum _{ |\gamma |\,=\,|\lambda ^{\prime }|-k-s }} \hspace{-10.0pt}p_{\gamma } \,L_{\gamma }\, L_{-k}^{\lambda (-k)}L_{\lambda _{+}}\left|w\right\rangle \\&& \hspace{-50.0pt} + \sum \limits _{l=\,k+1}^{k+s} \underset{ \scriptstyle \# \gamma = \,0}{\sum _{ |\gamma |\,=\,|\lambda ^{\prime }|-l}} \hspace{-10.0pt}p_{\gamma } \,L_{\gamma }\,L_{k+s-l} L_{-k}^{\lambda (-k)}L_{\lambda _{+}}\left|w\right\rangle ,\end{eqnarray*}
[Lk+s,Lλ−′]L−kλ(−k)Lλ+w=∑|γ|=|λ′|−k−s#γ=0pγLγL−kλ(−k)Lλ+w+∑l=k+1k+s∑|γ|=|λ′|−l#γ=0pγLγLk+s−lL−kλ(−k)Lλ+w,

where the range in the sum over l follows from the assumption that

$L_{\lambda _{-}^{{\prime }}}$
Lλ−′ does not contain generators L−i with i ⩽ k. An element of the maximal degree in the second sum takes the form

$$q_{\gamma } L_{\gamma } L_{-k}^{\lambda (-k)}L_{s-1}L_{\lambda _{+}}\left|w\right\rangle ,$$
qγLγL−kλ(−k)Ls−1Lλ+w,

where |γ| = |λ′| − k − 1. Hence,

$${\rm max}\,[L_{k+s},L_{\lambda _{-}^{{\prime }}}]L_{-k}^{\lambda (-k)}L_{\lambda _{+}}\left|w\right\rangle < |\lambda |-k.$$
max [Lk+s,Lλ−′]L−kλ(−k)Lλ+w<|λ|−k.

Let us now turn to the second term in (1). A simple algebra yields

\begin{eqnarray*}L_{\lambda _{-}^{{\prime }}}[L_{k+s},L_{-k}^{\lambda (-k)}]L_{\lambda _{+}}\left|w\right\rangle &=& (2k+s)L_{\lambda _{-}^{{\prime }}} \sum _{j=1}^{\lambda (-k)}L_{-k}^{j-1} L_{s}L_{-k}^{\lambda (-k)-j}L_{\lambda _{+}}\left|w\right\rangle \\&=& \lambda (-k)\psi _r(L_s) (2k+s) L_{\lambda _{-}^{{\prime }}} L_{-k}^{\lambda (-k)-1}L_{\lambda _{+}}\left|w\right\rangle \\&+& (2k+s)L_{\lambda _{-}^{{\prime }}} \sum _{j=1}^{\lambda (-k)}L_{-k}^{j-1} [L_{s},L_{-k}^{\lambda (-k)-j}]L_{\lambda _{+}}\left|w\right\rangle \\&+& \lambda (-k)(2k+s) L_{\lambda _{-}^{{\prime }}} L_{-k}^{\lambda (-k)-1}[L_s,L_{\lambda _{+}}]\left|w\right\rangle .\end{eqnarray*}
Lλ−′[Lk+s,L−kλ(−k)]Lλ+w=(2k+s)Lλ−′∑j=1λ(−k)L−kj−1LsL−kλ(−k)−jLλ+w=λ(−k)ψr(Ls)(2k+s)Lλ−′L−kλ(−k)−1Lλ+w+(2k+s)Lλ−′∑j=1λ(−k)L−kj−1[Ls,L−kλ(−k)−j]Lλ+w+λ(−k)(2k+s)Lλ−′L−kλ(−k)−1[Ls,Lλ+]w.

By Lemma 2.6 maximal level of the second term on the rhs is strictly smaller than |λ| − k. If λ+(0) ≠ 0, the last term does not vanish. Since

\begin{equation*}[L_s, L_{\lambda _+}]|w\rangle = [L_s, L_0^k]L_{\lambda ^{\prime }_+}|w\rangle = \psi _{r}(L_s)\sum _{l=1}^{k}\binom{k}{l}s^l L_{0}^{k-l}L_{\lambda ^{\prime }_+}|w\rangle ,\end{equation*}
[Ls,Lλ+]|w⟩=[Ls,L0k]Lλ+′|w⟩=ψr(Ls)∑l=1kklslL0k−lLλ+′|w⟩,

its length is strictly smaller than

$\# \lambda _+$
#λ+⁠. □

The counterpart of Lemma 2.5 takes the form

Lemma 3.4: Let |w⟩ be the universal Whittaker vector of a type ψ1,n. Then for any pseudo partition 

$\lambda \in {\cal P}^{1,n}$
λ∈P1,n

  1. $[L_m,L_{\lambda _{+}}]\left|w\right\rangle = 0$
    [Lm,Lλ+]w=0  for m > n,

  2. ${\rm max}^{\#}[L_n,L_{\lambda _{+}}]\left|w\right\rangle <\#\lambda$
    max #[Ln,Lλ+]w<#λ⁠,

  3. there exists a positive integer mλ such that

    ${\rm max}^\#\,\underset{m_\lambda }{\underbrace{[L_1,[L_1,\hdots [L_1}},L_{\lambda _{+}}]\hdots ]\left|w\right\rangle <\#\lambda$
    max #[L1,[L1,⋯[L1︸mλ,Lλ+]⋯]w<#λ⁠.

By straightforward extensions of Lemmas 2.6 and 2.7, one gets

Lemma 3.5: Let 

$W_{\psi _{1,n}}$
Wψ1,n be the universal Whittaker module of type ψ1,n and let 
$\left|u\right\rangle \in W_{\psi _{1,n}}$
u∈Wψ1,n be an arbitrary vector. If max |u⟩ > 0, then |u⟩ is not a Whittaker vector of any type.

Lemma 3.6: Let 

$W_{\psi _{1,n}}$
Wψ1,n be the universal Whittaker module 
$W_{\psi _{1,n}}$
Wψ1,n of type ψ1,n and let

$$\left|u\right\rangle =\sum p_{\lambda } L_{\lambda }\left|w\right\rangle$$
u=∑pλLλw

be an arbitrary vector in

$W_{\psi _{1,n}}$
Wψ1,n⁠. If pλ ≠ 0 for pseudo-partitions
$\lambda \in {\cal P}^{1,n}$
λ∈P1,nwith λ(0) ≠ 0, then |u⟩ is not a Whittaker vector of any type.

Proof: By Lemma 3.5 it is enough to consider vectors of the form

$$\left|u\right\rangle =\sum p_{\lambda } L_{\lambda }\left|w\right\rangle$$
u=∑pλLλw

where |λ| = 0 for all λ in the sum. Let Λ0 be the set of all partitions such that pλ ≠ 0 and λ(0) ≠ 0. Let

$$\lambda ^0_{\rm max}= {\rm max} \lbrace \lambda (0)\,:\, \lambda \in \Lambda ^0\rbrace .$$
λ max 0= max {λ(0):λ∈Λ0}.

One can write

\begin{eqnarray*}(L_n - \psi _{1,n}(L_n))\left|u\right\rangle &=& \underset{\lambda (0)=\lambda ^0_{\rm max}}{\sum _{\lambda \in \Lambda ^0 }} p_{\lambda } [L_n, L_{\lambda }]\left|w\right\rangle + \underset{\lambda (0)<\lambda ^0_{\rm max}}{\sum _{\lambda \in \Lambda ^0 }} p_{\lambda } [L_n, L_{\lambda }]\left|w\right\rangle \\&=& n\lambda ^0_{\rm max}\, \psi _{1,n}(L_n)\underset{\lambda (0)=\lambda ^0_{\rm max}}{\sum _{\lambda \in \Lambda ^0 }} p_{\lambda } L_0^{\lambda (0)-1}L_{1}^{\lambda (1)}\dots L_{n-1}^{\lambda (n-1)} \left|w\right\rangle \\&+& {\sum _{\lambda ^{\prime }(0)<\lambda ^0_{\rm max}-1}} p^{\prime }_{\lambda ^{\prime }} L_{\lambda ^{\prime }}\left|w\right\rangle .\end{eqnarray*}
(Ln−ψ1,n(Ln))u=∑λ∈Λ0λ(0)=λ max 0pλ[Ln,Lλ]w+∑λ∈Λ0λ(0)<λ max 0pλ[Ln,Lλ]w=nλ max 0ψ1,n(Ln)∑λ∈Λ0λ(0)=λ max 0pλL0λ(0)−1L1λ(1)⋯Ln−1λ(n−1)w+∑λ′(0)<λ max 0−1pλ′′Lλ′w.

All terms of the first sum with the whole second sum form a set of linearly independent vectors. Hence, as each term in the first sum does not vanish neither the whole sum does. □

Let us now turn to the analysis of Whittaker vectors in the universal Whittaker modules

$W_{\psi _{1,n}}$
Wψ1,n of a type ψ1, n. By Lemmas 3.5 and 3.6 the only possible Whittaker vectors in
$W_{\psi _{1,n}}$
Wψ1,n are of the form

$$\left|u\right\rangle = \underset{\lambda (0)=\lambda (1)=0}{\sum _{|\lambda |=0}} p_{\lambda } L_{\lambda }\left|w\right\rangle .$$
u=∑|λ|=0λ(0)=λ(1)=0pλLλw.

All vectors of this form satisfy

$$L_{m} \left|u\right\rangle = \psi _{1,n}(L_{n})\delta _{n,m} \left|u\right\rangle ,\;\;\;m\geqslant n$$
Lmu=ψ1,n(Ln)δn,mu,m⩾n

and are therefore Whittaker vectors of the type

$$\psi _n=\lbrace \psi _{1,n}(L_n),0,\dots ,0\rbrace .$$
ψn={ψ1,n(Ln),0,⋯,0}.

We shall call them trivial.

Lemma 3.7: Let 

$W_{\psi _{1,n}}$
Wψ1,n be the universal Whittaker module 
$W_{\psi _{1,n}}$
Wψ1,n of a type ψ1, n. The only possible nontrivial Whittaker vectors 
$\left|u\right\rangle \in W_{\psi _{1,n}}$
u∈Wψ1,n are of the type ψ1, n.

Proof: We first show that a vector of the form

(2)

is not an eigenvector of any Lk, k = 2, …, Ln−1. To this end let us introduce the lexicographic order in the set of partitions of the form λ = (λ(2), …, λ(n − 1))

$$\lambda <\lambda ^{\prime } \Leftrightarrow \begin{array}{llllllll}\lambda (2)<\lambda ^{\prime }(2)\;\vee \; \\(\lambda (2)=\lambda ^{\prime }(2)\;\wedge \;\lambda (3)<\lambda ^{\prime }(3))\;\vee \; \\\vdots \\(\lambda (2)=\lambda ^{\prime }(2)\;\wedge \dots \wedge \; \lambda (n-2)=\lambda ^{\prime }(n-2)\; \wedge \;\lambda (n-1)<\lambda ^{\prime }(n-1)). \end{array}$$
λ<λ′⇔λ(2)<λ′(2)∨(λ(2)=λ′(2)∧λ(3)<λ′(3))∨⋮(λ(2)=λ′(2)∧⋯∧λ(n−2)=λ′(n−2)∧λ(n−1)<λ′(n−1)).

For vectors (2), we define

$$\lambda _{\max }(\left|u\right\rangle ) = \max \lbrace \lambda \,:\,p_\lambda \ne 0\rbrace .$$
λmax(u)=max{λ:pλ≠0}.

One easily checks that for k = 2, …, n − 1

$$\lambda _{\max }(L_k \left|u\right\rangle )>\lambda _{\max }(\left|u\right\rangle ),$$
λmax(Lku)>λmax(u),

hence, |u⟩ is not an eigenvector of Lk.

Let

$$\lambda _{\min }(\left|u\right\rangle ) = \min \lbrace \lambda \,:\,p_\lambda \ne 0\rbrace .$$
λmin(u)=min{λ:pλ≠0}.

For each partition λ = (λ(2), …, λ(n − 1))

$$\lambda _{\min }(\left[ L_1, L_{\lambda } \right]\left|w\right\rangle )<\lambda ,$$
λmin(L1,Lλw)<λ,

hence

$$\lambda _{\min }\left(L_1 \left|u\right\rangle -\psi _{1,n}(L_1)\left|u\right\rangle \right) \leqslant \lambda _{\min }(\left[ L_1, L_{\lambda _{\min }(\left|u\right\rangle )} \right]\left|w\right\rangle ) <\lambda _{\min }(\left|u\right\rangle ).$$
λminL1u−ψ1,n(L1)u⩽λmin(L1,Lλmin(u)w)<λmin(u).

It follows that if L1|u⟩ − ψ1, n(L1)|u⟩ ≠ 0, it is not proportional to |u⟩. Thus, if L1|u⟩ = λ|u⟩, then λ = ψ1, n(L1). □

Theorem 3.8: Let |w⟩ be a universal Whittaker vector of a type ψ1, n.

  1. For n = 3 there are no nontrivial Whittaker vectors in 

    $W_{\psi _{1,n}}$
    Wψ1,n of any type.

  2. For n = 4 the subspace of all nontrivial Whittaker vectors of the type ψ1, n in 
    $W_{\psi _{1,n}}$
    Wψ1,n     is span by the family of vectors
    (3)
    There are no other nontrivial Whittaker vectors in 
    $W_{\psi _{1,n}}$
    Wψ1,n     of any type.

Proof: By Lemma 3.7 it is enough to look for nontrivial solutions of the equation

$$L_{1} \left|u\right\rangle = \psi _{1,3}(L_1) \left|u\right\rangle .$$
L1u=ψ1,3(L1)u.

For n = 3, the only possibility is

$\left|u\right\rangle = \sum p_n L_{2}^n\left|w\right\rangle$
u=∑pnL2nw for which

$$(L_1-\psi _{1,3}(L_1))\left|u\right\rangle = - \sum n \,p_n \psi _{1,3}(L_3)L_2^{n-1}\left|w\right\rangle .$$
(L1−ψ1,3(L1))u=−∑npnψ1,3(L3)L2n−1w.

As the sum on the rhs is finite and non-vanishing there are no nontrivial Whittaker vectors in

$W_{\psi _{1,3}}$
Wψ1,3 of any type.

For n = 4 one checks by explicit calculations that vectors (3) are Whittaker vectors of the type ψ1, 4. We shall show that for any Whittaker vector of the type ψ1, 4 the decomposition

$$\left|u\right\rangle = {\sum _{\lambda =(\lambda (2),\lambda (3))}} p_{\lambda } L_{\lambda }\left|w\right\rangle$$
u=∑λ=(λ(2),λ(3))pλLλw

contains a nonzero term with λ = (k, 0). By assumption

(4)

For each term one has

$$[L_1,L_2^{\lambda (2)}L_3^{\lambda (3)}]\left|w\right\rangle = \left(-\lambda (2)L_2^{\lambda (2)-1}L_3^{\lambda (3)+1}-2\lambda (3)\psi _{1,n}(L_4)L_2^{\lambda (2)}L_3^{\lambda (3)-1} \right)\left|w\right\rangle .$$
[L1,L2λ(2)L3λ(3)]w=−λ(2)L2λ(2)−1L3λ(3)+1−2λ(3)ψ1,n(L4)L2λ(2)L3λ(3)−1w.

In order to achieve cancellation of terms on the rhs of (4), if the sum contains nonzero term with λ = (k, l) it has to contain non-vanishing terms with λ′ = (k − 1, l + 2) and λ″ = (k + 1, l − 2). It follows that the following terms must have nonzero coefficients:

$$L_2^{k^{\prime }}\left|w\right\rangle ,\;\;\; L_2^{k^{\prime \prime }}L_3\left|w\right\rangle .$$
L2k′w,L2k′′L3w.

The second case, however, cannot be realized as a non-vanishing term in the decomposition of a Whittaker vector of the type ψ1, 4 since

$$[L_1,L_2^{k^{\prime \prime }}L_3]\left|w\right\rangle =- k^{\prime \prime }L_2^{k^{\prime \prime }-1}L_3^2\left|w\right\rangle -2\psi _{1,4}(L_4)L_2^{k^{\prime \prime }}\left|w\right\rangle$$
[L1,L2k′′L3]w=−k′′L2k′′−1L32w−2ψ1,4(L4)L2k′′w

and the second term cannot be canceled on the rhs of (4). (This implies in particular that in decomposition (4) λ(3) assumes only even values.) It follows that |u⟩ contains at least one term with λ = (k, 0). Let λmin  = (kmin , 0) be the partition of this type with smallest k. Then

$$\left|u^{\prime }\right\rangle =\left|u\right\rangle -p_{\lambda _{\min }}\left|w^{k_{\min}}_2\right\rangle$$
u′=u−pλminw2kmin

is a new Whittaker vector of the type ψ1, n with

$k^{\prime }_{\min }>k_{\min }$
kmin′>kmin⁠. Repeating the subtraction above a finite number of times one gets a zero vector. Thus, |u⟩ is a linear combination of vectors (3). □

Construction (3) of Whittaker vectors can be generalized for n > 4 as follows:

$$\begin{array}{rlll}\left|w^l_{2;n}\right\rangle &=& \displaystyle \sum \limits _{k=0}^l \alpha _k L_{n-2}^{l-k}L_{n-1}^{2k} \left|w\right\rangle ,\;\;\;l\in \mathbb {N}, \\[15pt] \alpha _0&\ne & 0, \\\alpha _{k+1} &=& - \displaystyle \frac{(n-3)(l-k)}{2(n-2) (k+1)\,\psi _{1,n}(L_n)}\,\alpha _{k}. \end{array}$$
w2;nl=∑k=0lαkLn−2l−kLn−12kw,l∈N,α0≠0,αk+1=−(n−3)(l−k)2(n−2)(k+1)ψ1,n(Ln)αk.

Another generalization is given by

$$\begin{array}{rlll}\left|w^1_{l;n}\right\rangle &=& \displaystyle \sum \limits _{k=\,l}^{n-1} \alpha _k L_{k} L_{n-1}^{k-l} \left|w\right\rangle ,\;\;\;2\leqslant l\leqslant n-2, \\[15pt] \alpha _l &\ne & 0, \\\alpha _{k+1} &=& -\displaystyle \frac{k-1}{(n-2) (k+1-l)\,\psi _{1,n}(L_n)}\,\alpha _{k},\;\;\; l\leqslant k\leqslant n-3, \\\alpha _{n-1}&=& \displaystyle - \frac{n-3}{(n-2)(n-l)\,\psi _{1,n}(L_n)}\,\alpha _{n-2}. \end{array}$$
wl;n1=∑k=ln−1αkLkLn−1k−lw,2⩽l⩽n−2,αl≠0,αk+1=−k−1(n−2)(k+1−l)ψ1,n(Ln)αk,l⩽k⩽n−3,αn−1=−n−3(n−2)(n−l)ψ1,n(Ln)αn−2.

The constructions above do not exhaust all possibilities. As an illustration we give two more examples for n = 5

\begin{eqnarray*}\left|w^{1,1}_{2,3;5}\right\rangle &=& \left( L_2L_3 -\frac{1}{3\mu }L_2L_4^2 -\frac{1}{3\mu } L_3^2L_4 +\frac{5}{3^3 \mu ^2} L_3L_4^3 -\frac{2}{3^4\mu ^3}L^5_4 \right)\left|w\right\rangle , \\\left|w^2_{2;5}\right\rangle &=& \left( L_2^2 -\frac{2}{3\mu }L_2L_3L_4 + \frac{2^2}{3^3 \mu ^2} L_2L_4^3+ \frac{1}{3^2\mu ^2 }L_3^2L_4^2 -\frac{4}{3^4\mu ^3}L_3L_4^4 -\frac{1}{3} L_4+ \frac{4}{3^6\mu ^4}L_4^6 \right)\left|w\right\rangle ,\end{eqnarray*}
w2,3;51,1=L2L3−13μL2L42−13μL32L4+533μ2L3L43−234μ3L45w,w2;52=L22−23μL2L3L4+2233μ2L2L43+132μ2L32L42−434μ3L3L44−13L4+436μ4L46w,

where μ = ψ1, 5(L5). The dimension of the subspace of Whittaker vectors of the type ψ1, n grows very fast with n. A general discussion is rather involved and goes beyond the scope of this paper.

We close this section with the theorem characterizing

${\cal V}_{1,n}$
V1,n submodules. Its derivation parallels that of Theorem 2.13.

Theorem 3.9: Any submodule of a Whittaker module of a type ψ1, n contains a Whittaker vector of the same type.

We owe special thanks to Volodymyr Mazorchuk and Kaiming Zhao for sending us their work1 and in particular for pointing out that Theorems 3.5, 3.6, and Corollary 3.7 of Sec. III of our paper are valid only in the case of n = 3. This work was partially financed the NCN Grant No. DEC2011/01/B/ST1/01302.

1.
V.
 
Mazorchuk
 and
K.
 
Zhao
, “
Simple Virasoro modules which are locally finite over positive part
,” e-print arXiv:1205.5937v2 [math.RT].
View Original Article
© 2012 American Institute of Physics.
2012
American Institute of Physics