Twisted and Non-Twisted Deformed Virasoro Algebra via Vertex Operators of $U_q(\widehat{\mathfrak{sl}}_2)$
Mikhail Bershtein Roman Gonin
math.QA math-ph math.MP math.RT

	
	Effect of random anisotropy in stabilization of topological chiral textures
    Subhankar Bedanta
    March 2020
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		The work is devoted to a probably new connection between deformed Virasoro algebra and quantum . We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of . The same is done for a twisted version of deformed Virasoro algebra.   
	
	
	

§ INTRODUCTION

	The work is devoted to a probably new connection between deformed Virasoro algebra and quantum . More specifically, we use integrable representations  V(Λ_0) and V(Λ_1) of quantum  on level 1, they possess realization in terms of Heisenberg algebra. Also there are vertex operators
	
    Φ(z)    V(Λ_i) → V(Λ_1-i) ⊗ V_z,    Ψ(z)     V(Λ_i) → V_z⊗ V(Λ_1-i).

	for evaluation representation V_z of . The main result is realization of deformed Virasoro algebra in terms of these vertex operators see Theorem <ref> and Theorem <ref>. Let us remark that deformed Virasoro algebra depend on parameters q_1, q_2, q_3 such that q_1 q_2 q_3 =1. It turns out, that deformed Virasoro algebra is connected with  for q =q_3^1/2.
	
	To be more precise, in Theorem <ref> we have a realization of twisted deformed Virasoro algebra. This algebra was defined in <cit.> but its bosonization was unknown.
	
	In Theorem <ref> we have constructed realization of ordinary (non-twisted)  deformed Virasoro algebra defined in <cit.>. A bosonization of this algebra is known since <cit.>, but our bosonization is a different one. The bosonization from <cit.> can be also realized by the same formula as in Theorem <ref> but using another vertex operators <cit.> defined by (<ref>) with respect to Drinfeld coproduct.
	
	The existence of two realizations (our Theorems <ref> and <ref>) is similar to existence of two choices of twist in XXZ model, see e.g. <cit.>.
	
	

  
Further development Deformed Virasoro algebra is a particular case of W_q_1,q_2(𝔰𝔩_n) for n=2. Twisted deformed Virasoro is a particular case of twisted W-algebras W_q_1,q_2(𝔰𝔩_n, n_tw) for n=2 and n_tw=1. Generally n_tw is a parameter of twist, i.e. for n_tw=0 we obtain non-twisted W-algebra. We expect that one can construct bosonization of W_q_1,q_2(𝔰𝔩_n, n_tw) algebra from vertex operators of quantum 𝔰𝔩_n on the level 1 (see <cit.>).
	
	Also we expect that tensor product of W_q_1,q_2(𝔰𝔩_n, n_tw) with Heisenberg algebra 𝖧𝖾𝗂𝗌 are certain quotients of toroidal algebra U_q_1,q_2,q_3(𝔤̈𝔩̈_1); for non-twisted case such relation is known <cit.>, <cit.>, <cit.>. Hence a representation of W_q_1,q_2(𝔰𝔩_n, n_tw) ⊗𝖧𝖾𝗂𝗌 becomes a representation of  U_q_1,q_2,q_3(𝔤̈𝔩̈_1) automatically. We expect that for (n,n_tw)=1, the discussed above bosonized representation of W_q_1,q_2(𝔰𝔩_n, n_tw) will lead to Fock modules of U_q_1,q_2,q_3(𝔤̈𝔩̈_1) with slope n_tw/n. For the case q_3=1 all this was done in <cit.>. Let us remark W_q_1,q_2(𝔰𝔩_n, n_tw) ⊗𝖧𝖾𝗂𝗌 acts on integrable level 1 representations of U_q (𝔤𝔩_n) = U_q (𝔰𝔩_n) ⊗𝖧𝖾𝗂𝗌 if this holds without 𝖧𝖾𝗂𝗌 factors.
	
	One of our motivations for this project comes from <cit.>. It was conjectured in loc. cit. that there is an action (with certain properties) of U_q𝔤𝔩_n on the Fock module of toroidal algebra U_q_1,q_2,q_3(𝔤̈𝔩̈_1) with slope n'/n. As it was explained above, we also expect U_q𝔤𝔩_n acts on the Fock module of toroidal algebra U_q_1,q_2,q_3(𝔤̈𝔩̈_1). So we hope that both actions exist and coincide.
	
	

  
Our methods. The main technical tool of our paper is R-matrix relations (Theorems <ref> and <ref>). One can find these relations without delta-function term in <cit.>. In loc. cit. the parameters of vertex operator are numbers, but in our paper the parameters are formal variables. Therefore our formulas are close to formulas in loc. cit., but have a different meaning and probably are new. 
	
	Technically, we write down formulas (<ref>) and (<ref>) for current of deformed Virasoro algebra T(z) via vertex operators Φ(z) and Ψ^*(z), and then we check relations of deformed Virasoro algebra using interchanging relations for vertex operators. Delta-function term on RHS of deformed Virasoro relation appears from the delta-function term in the R-matrix relation.
	

  
Plan of the paper. The paper is organized as follows
	
	In Section <ref> we recall bosonization of  and its vertex operators following <cit.>.
	
	In Section <ref> we study relations for vertex operators: interchanging relations (in particular R-matrix relations), and `special point relations'.
	
	In Section <ref> we construct realizations of twisted and non-twisted Virasoro algebra via vertex operators of . A connection of obtained representations with Verma modules is studied.
	
	

  
Acknowledgments We are grateful to B. Feigin,  E. Gorsky, M. Jimbo, M. Lashkevich, A. Negut, Y. Pugai, J. Shiraishi for interest to our work and discussions. The work is supported by the Russian Foundation of Basic Research under grant mol_a_ved  18-31-2006. R.G. was also supported by the “Young Russian Mathematics" award.
	
	

§ BOSONIZATION OF  AND ITS VERTEX OPERATORS
 
	In this section we will recall the bosonization of the level 1 representations of  and its vertex operators. All this can be found in <cit.>. Our notation almost coincide with <cit.>; however there are differences in normalization of vertex operators.
	

  
Fock modules. Algebra  is generated by x_k^±, a_l for k ∈ℤ, l ∈ℤ_≠ 0, K^± 1 and central elements γ^± 1/2. These elements are called Drinfeld generators. The relations are <cit.>, although let us recall 
	
    [a_k, a_l]= δ_k+l,0[2k]/kγ^k - γ^-k/q - q^-1,

	here [n]=(q^n-q^-n)/(q-q^-1).
	
	Denote by Λ_0, Λ_1 fundamental weights of 𝔰𝔩_2 and by α root of 𝔰𝔩_2 ⊂𝔰𝔩_2
	The algebra  admits two basic representations V(Λ_0) and V(Λ_1). As vector spaces
	
    V(Λ_i) = ℂ[a_-1, a_-2, … ] ⊗( ⊕_nℂ e^Λ_i + n α).

	As representations of Heisenberg subalgebra a_k, these modules are infinite sums of Fock modules 
	
    V_j = ℂ[a_-1, a_-2, … ] ⊗ℂ e^Λ_i + ⌊j/2⌋α  for i ≡ j  2.

	Let us define operators e^±α and ∂ as follows
	
    e^±α( f ⊗ e^β) = f ⊗ e^β±α,     ∂( f ⊗ e^β) = (α, β) f ⊗ e^β.

	The action of other  generators is given by
	
    K =q^∂,   γ = q, 
    	
       X^±(z)= exp( ±∑_n=1^∞a_-n/[n] q^∓ n/2 z^n ) exp( ∓∑_n=1^∞a_n/[n] q^∓ n/2 z^-n) e^±α z^±∂,

	here X^±(z) = ∑ x_k^± z^-k-1. Obtained representations are irreducible highest weight representations with highest vectors |Λ_i ⟩ = 1 ⊗ e^Λ_i∈ V(Λ_i).
	

  
Vertex operators. Vertex operator of  are certain formal power series of operators 
	
    Φ^(1-i, i)_±(z)     V(Λ_i) → V(Λ_1-i),    Ψ^(1-i, i)_±(z)     V(Λ_i) → V(Λ_1-i).

	Below we will abbreviate Φ_±(z) = Φ^(1-i, i)_±(z), if a statement holds for both i=0,1.
	
	A conceptual definition of these operators via certain intertwiner relations is given in <cit.>. For us it is more convenient to give an ad hoc definition
	
    Φ_-(z) =   exp( ∑_n=1^∞a_-n/[2n] q^7n/2 z^n ) exp(- ∑_n=1^∞a_n/[2n] q^-5n/2 z^-n) e^α/2 (-q^3 z)^∂/2, 
    Ψ_+(z) =   exp( - ∑_n=1^∞a_-n/[2n] q^n/2 z^n ) exp( ∑_n=1^∞a_n/[2n] q^-3n/2 z^-n) e^-α/2 (-q z)^-∂/2.

	Operator Φ_+(z), Ψ_-(z) are determined by each of these formulas
	
    Φ_+ (z)    = [Φ_- (z) , x_0^- ]_q,      q^2 z Φ_+ (z) =    [ Φ_- (z), x_1^- ]_q^-1, 
    Ψ_-(z)    = [ Ψ_+(z),  x_0^+]_q,      (q^2 z)^-1Ψ_-(z) =    [Ψ_+(z), x_-1^+]_q^-1.

	here we use following notation [A,B]_p= AB-pBA.
	
	We will also need the dual operator Ψ_ε^*(z) = Ψ_-ε(q^2 z). Then
	
    Ψ^*_-(z) = exp( - ∑a_-n/[2n] q^5n/2 z^n ) exp( ∑a_n/[2n] q^-7n/2 z^-n) e^-α/2 (-q^3 z)^-∂/2.

	
	
	Denote
	
    (x) =   (q^4 x; q^4)_∞/(q^2 x; q^4)_∞,    (x) =   (q^2 x; q^4)_∞/(x; q^4)_∞,   (x) =   (q x; q^4)_∞/(q^3 x; q^4)_∞.

	It is straightforward to check that
	
    (-q^3 z)^-1/2(w/z) Φ_-(z) Φ_-(w) =    :Φ_-(z) Φ_-(w):,  
    
    	(-qz)^-1/2(w/z) Ψ_+(z) Ψ_+(w) =    :Ψ_+(z) Ψ_+(w):, 
    
    	(-q^3 z)^1/2(w/z) Φ_-(z) Ψ_-^*(w) =    :Φ_-(z) Ψ_-^*(w):, 
    
    	(-q^3 w)^1/2(z/w) Ψ_-^*(w) Φ_-(z) =    :Φ_-(z) Ψ_-^*(w):.

	here : … : stands for normal ordering in terms of Heisenberg algebra. Then 
	
    z^-1/2(w/z) Φ_-(z) Φ_-(w) =    w^-1/2(z/w) Φ_-(w) Φ_-(z), 
    
    	z^-1/2(w/z) Ψ^*_-(z) Ψ^*_-(w) =    w^-1/2(z/w) Ψ^*_-(w) Ψ^*_-(z),
    
    	z^1/2(w/z) Φ_-(z) Ψ_-^*(w) =    w^1/2(z/w) Ψ_-^*(w) Φ_-(z).

	

  
π-involution Recall, that  is also generated by e_i, f_i, t_i for i=0,1. These generators are called Chevalley generators. The connection with Drinfeld generators as follows
	
    t_1=   K,     x_0^+=   e_1,     x_0^-=   f_1, 
    
    	t_0 =   γ K^-1,     x_1^- =    e_0 t_1,     x_-1^+ =    t_1^-1 f_0.

	Let us consider an exterior automorphism π of  given by π(e_i)=e_1-i,  π(f_i)=f_1-i. Then π acts on the Drinfeld generators as follows
	
    π(K)= γ K^-1,       π(x_0^+) =  x_1^- K^-1,       π(x_0^-) =  K  x_-1^+, 
             π(x_1^-)=  γ x_0^+ K^-1,      π(x_-1^+) = γ^-1 K x_0^-.

	
    π(K)= γ K^-1,       π(x_0^+) = e_0= x_1^- K^-1,       π(x_0^-) = f_0 = K  x_-1^+, 
             π(x_1^-)= e_1 γ K^-1= γ x_0^+ K^-1,      π(x_-1^+) = t_0^-1 f_1 =γ^-1 K x_0^-.

	
		
		There exist an involution π̃ interchanging V(Λ_0) and V(Λ_1), such that π̃ X π̃ = π(X) for any X ∈.
	
	
		V(Λ_0) and V(Λ_1) are irreducible highest weights representations and π preserves triangular decomposition. To finish the proof we notice that action of π interchange the highest weights of the representations.
	
	To determine π̃ uniquely we require π̃(|Λ_i ⟩) = |Λ_1-i⟩.
	 
		Conjugation by involution π̃ is expressed as follows
		
    π̃(Φ_+^(1-i,i)(z) ) π̃ =    (-q^3)^1/2-i z^-1/2Φ^(i,1-i)_-(z),    π̃(Φ^(1-i,i)_-(z) ) π̃=    (-q^3)^1/2-i z^1/2Φ_+^(i,1-i)(z),
    π̃(Ψ_+^(1-i,i)(z) ) π̃=    (-q)^3(1/2-i ) z^-1/2Ψ^(i,1-i)_-(z),    π̃(Ψ_-^(1-i,i)(z) ) π̃=    (-q)^3(1/2-i ) z^1/2Ψ^(i,1-i)_+(z).

	
	
		One can prove the formulas up to a constant via the intertwining properties <cit.>
		
    π̃(Φ_+^(1-i,i)(z) ) π̃ =    c_1^(i)Φ^(i,1-i)_-(z),    π̃(Φ^(1-i,i)_-(z) ) π̃=    c_1^(i) z Φ_+^(i,1-i)(z),
    π̃(Ψ_+^(1-i,i)(z) ) π̃=    c_2^(i)Ψ^(i,1-i)_-(z),    π̃(Ψ_-^(1-i,i)(z) ) π̃=    c_2^(i) z Ψ^(i,1-i)_+(z),

		here c_1^(i) and c_2^(i) are some z-dependent scalars.
		Then one can find the constants by comparison with normalization <cit.>.
	
	  The following relations hold
		
    z^-1/2(w/z) Φ_+(z) Φ_+(w) =    w^-1/2(z/w) Φ_+(w) Φ_+(z), 
    
    		z^-1/2(w/z) Ψ^*_+(z) Ψ^*_+(w) =    w^-1/2(z/w) Ψ^*_+(w) Ψ^*_+(z), 
    
    		z^1/2(w/z) Φ_+(z) Ψ_+^*(w) =    w^1/2(z/w) Ψ_+^*(w) Φ_+(z).

	
	
		These relations are obtained from (<ref>)–(<ref>) after conjugation by π̃.
	
	

§ VERTEX OPERATORS RELATIONS REVISITED
 
	The main results of this section are R-matrix relations (Theorems <ref> and <ref>). One can find these relations without delta-function term in <cit.>. In loc. cit. parameters of vertex operator are numbers, but in our paper the parameters are formal variables. Although formulas below are close to formulas in loc. cit., they have a different meaning and can be considered as a new result.
	

 §.§ R-matrix relations

	
	R-matrix is an operator on ℂ^2 ⊗ℂ^2. Let v_+, v_- be a basis of each ℂ^2. The matrix of this operator with respect to basis v_+⊗ v_+, v_+⊗ v_-, v_-⊗ v_+, v_-⊗ v_- looks as follows
	
    R(x)=  [                1                0                0                0;                0   (1-x)q/1-q^2 x    1-q^2/1-q^2 x                0;                0 (1-q^2)x/1-q^2 x   (1-x)q/1-q^2 x                0;                0                0                0                1 ].

	This R-matrix is an important object in the representation theory of , though in this paper we will not use any information on R-matrix apart from (<ref>). Below we will see that R-matrix encodes certain interchanging relations for vertex operators.
	

  §.§.§ Interchanging relation on Φ-vertex operators

	Denote by δ(x,y)= ∑_k+l=-1 x^k y^l.
	 
		Following relations hold
		
    z^-1/2(w/z)  Φ_-(z) Φ_+ (w) - w^-1/2(z/w) ( q(1-z/w)/q^2-z/wΦ_+ (w) Φ_- (z) + (q^2-1)z/w/q^2-z/wΦ_- (w) Φ_+ (z) ) 
    =(-1)^∂ (-q^3)^1/2 q^-2δ(z,q^2 w),

		

		
    z^-1/2(w/z)  Φ_+(z) Φ_- (w) - w^-1/2(z/w) ( q(1-z/w)/q^2-z/wΦ_- (w) Φ_+ (z) + q^2-1/q^2-z/wΦ_+ (w) Φ_- (z) ) 
    = -(-1)^∂ (-q^3)^1/2 q^-3δ(z,q^2 w).
 
	
	
		Using (<ref>), we obtain
		
		
    [Φ_- (z) Φ_- (w), x_0^-]_q^2 = Φ_- (z) [Φ_- (w), x_0^- ]_q + q [Φ_-(z),    x_0^-]_q Φ_- (w) 
       =Φ_- (z) Φ_+(w) + q Φ_+ (z) Φ_- (w),
    		
    
    		[ Φ_- (z) Φ_- (w), x_1^- ]_q^-2 =  Φ_- (z) [Φ_- (w), x_1^- ]_q^-1 + q^-1 [   Φ_- (z), x_1^- ]_q^-1Φ_- (w)
       =q^2 w Φ_- (z) Φ_+(w) + q z Φ_+ (z) Φ_- (w).

		Solving the system of two linear equations, one can find
		
    Φ_-(z) Φ_+ (w)    =  z[ Φ_- (z) Φ_- (w), x_0^-]_q^2 - [ Φ_- (z) Φ_- (w), x_1^- ]_q^-2/z(1-q^2 w/z),
    Φ_+ (z) Φ_- (w)    = -qw[ Φ_- (z) Φ_- (w), x_0^-]_q^2 + q^-1 [ Φ_- (z) Φ_- (w), x_1^- ]_q^-2/z(1-q^2 w/z).

		Using (<ref>), we see that			
		
    (-q^3 z)^-1/2(w/z) Φ_-(z) Φ_+ (w)    =  z[ : Φ_-(z) Φ_-(w)  :  , x_0^-]_q^2 - [: Φ_-(z) Φ_-(w)  :  ,  x_1^- ]_q^-2/z(1-q^2 w/z), 
    
    		(-q^3 z)^-1/2(w/z) Φ_+ (z) Φ_- (w)     = -qw[ : Φ_-(z) Φ_-(w)  :  , x_0^-]_q^2 + q^-1 [ : Φ_-(z) Φ_-(w)  :  , x_1^- ]_q^-2/z(1-q^2 w/z).

		
		Then
		
    z^-1/2(w/z) Φ_-(z) Φ_+ (w) - w^-1/2(z/w) ( q(1-z/w)/q^2-z/wΦ_+ (w) Φ_- (z) + (q^2-1)z/w/q^2-z/wΦ_- (w) Φ_+ (z) ) 
    
    		=(-q^3)^1/2(  q^2 w[ : Φ_- (q^2 w) Φ_- (w)  :, x_0^-]_q^2 - [ : Φ_- (q^2 w) Φ_- (w)  :, x_1^- ]_q^-2)   δ(z,q^2 w),

		
    z^-1/2(w/z) Φ_+(z) Φ_- (w) - w^-1/2(z/w) ( q(1-z/w)/q^2-z/wΦ_- (w) Φ_+ (z) + q^2-1/q^2-z/wΦ_+ (w) Φ_- (z) ) 
    
    		=(-q^3 )^1/2(  -q w[ : Φ_- (q^2 w) Φ_- (w)  :, x_0^-]_q^2 + q^-1 [ : Φ_- (q^2 w) Φ_- (w)  :, x_1^- ]_q^-2)   δ(z,q^2 w).

		
		
		 
			Following relation holds
			
    q^2 w[ : Φ_- (q^2 w) Φ_- (w)  :, x_0^-]_q^2 - [ : Φ_- (q^2 w) Φ_- (w)  :, x_1^- ]_q^-2 = (-1)^∂ q^-2.

		
		
		
			We will proof the lemma assuming w to be a number, not a formal variable; formal variable version follows. Let us consider two contours of integration C_+ = {y | |y|=R_+ ≫ |w|  }, C_- = {y | |y|=R_- ≪ |w|  }
			
			Denote Ω(w) =  :Φ_- (q^2 w) Φ_- (w):. Note that
			
    [ Ω(w) , x_0^-]_q^2 =   ∫_C_-Ω(w) X^-(y) dy - q^2 ∫_C_+ X^-(y) Ω(w)dy,
    
    			[ Ω(w) , x_1^-]_q^-2 =   ∫_C_- y Ω(w) X^-(y) dy - q^-2∫_C_+ y X^-(y) Ω(w)dy.

			Hence
			
    q^2 w[ Ω(w), x_0^-]_q^2 - [ Ω(w), x_1^- ]_q^-2 =
    			∫_C_-( q^2 w -  y ) Ω(w) X^-(y) dy -∫_C_+( q^4 w - q^-2 y ) X^-(y) Ω(w)dy
    
    			=∫_C_-( q^2 w -  y )/q^2 (q^4 w-y)(q^2 w - y) :Ω(w) X^-(y):dy -∫_C_+ q^4 w - q^-2 y /(y -q^4 w)(y-q^6 w) :X^-(y) Ω(w):dy
    
    			=∫_C_+ q^-2/y -q^4 w :X^-(y) Ω(w):dy - ∫_C_-q^-2/y-q^4 w :Ω(w) X^-(y):dy   
    			
     =_y=q^4 w q^-2/y -q^4 w :  X^-(y) Ω(w):dy.
    			= (-1)^∂ q^-2.

			Here we used  :X^-(q^4 w) Ω(w): =(-1)^∂.
		
		To finish the proof of Proposition <ref> we apply Lemma <ref> to (<ref>) and (<ref>).
	
	
	

  
Matrix notation Denote Φ(z)=Φ_+ (z)⊗ v_+ + Φ_-(z) ⊗ v_- ∈(V(Λ_i), V(Λ_1-i) ) ⊗ℂ^2. Denote products 
	
    Φ^(1)(z) Φ^(2) (w) = ∑_ϵ_1, ϵ_2 = ±Φ_ϵ_1(z) Φ_ϵ_2(w) ⊗ v_ϵ_1⊗ v_ϵ_2,   Φ^(2)(w) Φ^(1) (z) = ∑_ϵ_1, ϵ_2 = ±Φ_ϵ_2(w) Φ_ϵ_1(z) ⊗ v_ϵ_1⊗ v_ϵ_2.
 Finally denote by R^-1(z/w) Φ^(2) (w) Φ^(1)(z) the result of the action of R^-1(z/w) on ℂ^2 ⊗ℂ^2 tensor multiple of  (V(Λ_i), V(Λ_i) ) ⊗ℂ^2⊗ℂ^2.
	
	 
		The following relation holds
		
    z^-1/2(w/z)  Φ^(1)(z) Φ^(2) (w) = 
     w^-1/2(z/w)  R^-1(z/w) Φ^(2) (w) Φ^(1)(z) + (-1)^∂  (-q)^1/2 (q^-1 v_- ⊗ v_+ - q^-2 v_+ ⊗ v_-) δ(z,q^2 w).

	
	
		The theorem is just a reformulation of (<ref>), (<ref>) and Proposition <ref>.
	
	

  §.§.§ Interchanging relation for Ψ-vertex operators

	 
		The following relations hold
		
    z^-1/2(w/z)  Ψ_+(z) Ψ_-(w) - 
    		w^-1/2(z/w)  ( q(1-z/w)/1 - q^2 z/wΨ_-(w) Ψ_+(z) + 1-q^2/1-q^2 z/wΨ_+(w) Ψ_-(z) )
    =(-1)^∂ (-q)^1/2 q^2 δ(q^2z, w),

		
    z^-1/2(w/z) Ψ_-(z) Ψ_+(w) - w^-1/2(z/w) ( q(1-z/w)/1-q^2 z/wΨ_+(w) Ψ_-(z) +(1-q^2)z/w/1-q^2 z/wΨ_-(w) Ψ_+(z) )
    = -(-1)^∂  (-q)^1/2 q δ(q^2z, w).

	
	
		Using (<ref>) we obtain
		
    [Ψ_+(z) Ψ_+(w), x_0^+]_q^2 = Ψ_+(z) [Ψ_+(w), x_0^+]_q + q[Ψ_+(z), x_0^+]_q Ψ_+(w) = Ψ_+(z) Ψ_-(w) + q Ψ_-(z) Ψ_+(w),

		

		
    [Ψ_+(z) Ψ_+(w), x_-1^+]_q^-2 = Ψ_+(z) [Ψ_+(w), x_-1^+]_q^-1 + q^-1[Ψ_+(z), x_-1^+]_q^-1Ψ_+(w) 
     =(q^2 w)^-1Ψ_+(z) Ψ_-(w)+ (q^3 z)^-1Ψ_-(z) Ψ_+(w).

		Solving the system of linear equations, we obtain
		
    Ψ_+(z) Ψ_-(w) = w/z (- [Ψ_+(z) Ψ_+(w), x_0^+]_q^2 + q^4 z [:Ψ_+(z) Ψ_+(w):, x_-1^+]_q^-2)/q^2 -w/z,
    Ψ_-(z) Ψ_+(w) = q[Ψ_+(z) Ψ_+(w), x_0^+]_q^2  -  q^3 w [:Ψ_+(z) Ψ_+(w), x_-1^+]_q^-2/q^2 - w/z.

		Using (<ref>), we see that
		
    (-qz)^-1/2(w/z) Ψ_+(z) Ψ_-(w) =    w (- [:Ψ_+(z) Ψ_+(w):, x_0^+]_q^2 + q^4 z [:Ψ_+(z) Ψ_+(w):, x_-1^+]_q^-2)/q^2 z(1 -  w/q^2 z), 
    
    		(-qz)^-1/2(w/z) Ψ_-(z) Ψ_+(w) =   q[:Ψ_+(z) Ψ_+(w):, x_0^+]_q^2  -  q^3 w [:Ψ_+(z) Ψ_+(w):, x_-1^+]_q^-2/q^2 (1 - w/q^2 z).

		Then
		
    z^-1/2(w/z)  Ψ_+(z) Ψ_-(w) - w^-1/2(z/w) ( 1-q^2/1-q^2 z/wΨ_+(w) Ψ_-(z) +q(1-z/w)/1 - q^2 z/wΨ_-(w) Ψ_+(z) ) 
     = (-q)^1/2 q^2 z (-[:Ψ_+(z) Ψ_+(q^2 z):, x_0^+]_q^2 + q^4 z [:Ψ_+(z) Ψ_+(q^2 z):, x_-1^+]_q^-2)  δ(q^2 z, w).

		
    z^-1/2(w/z) Ψ_-(z) Ψ_+(w) -  w^-1/2(z/w) ( q(1-z/w)/1-q^2 z/wΨ_+(w) Ψ_-(z) +(1-q^2)z/w/1-q^2 z/wΨ_-(w) Ψ_+(z) ) 
     =(-q)^1/2 q z( [:Ψ_+(z) Ψ_+(q^2 z):, x_0^+]_q^2  -  q^4 z [:Ψ_+(z) Ψ_+(q^2 z):, x_-1^+]_q^-2) δ(q^2 z, w).

		 
			Following relation holds
			
    [:Ψ_+(z) Ψ_+(q^2 z):, x_0^+]_q^2  -  q^4 z [:Ψ_+(z) Ψ_+(q^2 z):, x_-1^+]_q^-2 = - (-1)^∂ z^-1.

		
		
			Denote by
			
    Υ(z) = :Ψ_+(z) Ψ_+(q^2 z):.
 
			Note that
			
    [Υ(z), x_0^+]_q^2   = ∫_C_-Υ(z) X^+(y)dy- q^2∫_C_+  X^+(y)Υ(z)dy,
    
    			[Υ(z), x_-1^+]_q^-2   = ∫_C_- y^-1Υ(z) X^+(y)dy - q^-2∫_C_+ y^-1 X^+(y)Υ(z)dy.

			Hence
			
    [Υ(z), x_0^+]_q^2  -  q^4 z [Υ(z), x_-1^+]_q^-2 = 
    			∫_C_- (1-q^4 z/y)Υ(z) X^+(y) dy - q^2 ∫_C_+  (1 - z/y)X^+(y)Υ(z)dy
    =
    			∫_C_-q^2(1-q^4 z/y)/(q^2 z-y)(q^4z-y) :Υ(z) X^+(y):dy - q^2 ∫_C_+(1 - z/y)/(y-z)(y-q^2 z) :X^+(y)Υ(z):dy 
    =-∫_C_-q^2/y(q^2 z-y) :Υ(z) X^+(y):dy + ∫_C_+q^2/y(q^2 z-y) :X^+(y)Υ(z):dy 
    			
    
    			=_y=q^2zq^2/y(q^2 z-y) :X^+(y)Υ(z):dy = - (-1)^∂ z^-1.

			Here we used :X^+(q^2 z)Υ(z):  =(-1)^∂. 
	
	In terms of the operators Ψ^*, Proposition <ref> can be rewritten as follows.
	 
		The following relation holds
		
    z^-1/2(w/z)  Ψ^*_-(z) Ψ^*_+(w)  
    		-w^-1/2(z/w)  ( q(1-z/w)/1 - q^2 z/wΨ^*_+(w) Ψ^*_-(z) + 1-q^2/1-q^2 z/wΨ^*_-(w) Ψ^*_+(z) ) 
    = (-1)^∂(-q)^1/2 q δ(q^2z, w),

		
    z^-1/2(w/z) Ψ^*_+(z) Ψ^*_-(w) - w^-1/2(z/w) ( q(1-z/w)/1-q^2 z/wΨ^*_-(w) Ψ^*_+(z) +(1-q^2)z/w/1-q^2 z/wΨ^*_+(w) Ψ^*_-(z) )
     = - (-1)^∂ (-q)^1/2δ(q^2z, w).
		
	
	
	

  
Matrix notation Denote Ψ^*(z)=Ψ^*_+ (z)⊗ v^*_+ + Ψ^*_-(z) ⊗ v^*_- ∈(V(Λ_i), V(Λ_1-i) ) ⊗( ℂ^2 )^*. Let us emphasise that ( ℂ^2 )^* is the dual space to ℂ^2, considered in the definition of Φ(z).
	
	Denote products 
	
    Ψ^*, (1)(z) Ψ^*, (2) (w) = ∑_ϵ_1, ϵ_2 = ± 1Ψ^*_ϵ_1(z) Ψ^*_ϵ_2(w) ⊗ v^*_ϵ_1⊗ v^*_ϵ_2, 
    Ψ^*, (2)(w) Ψ^*, (1) (z) = ∑_ϵ_1, ϵ_2 = ± 1Ψ^*_ϵ_2(w) Ψ^*_ϵ_1(z) ⊗ v^*_ϵ_1⊗ v^*_ϵ_2.

	Finally denote by Ψ^*,(2) (w) Ψ^*,(1)(z) R(z/w) the result of the dual action of R(z/w) on ( ℂ^2 ⊗ℂ^2 )^* tensor multiple of  (V(Λ_i), V(Λ_i) ) ⊗( ℂ^2  ⊗ℂ^2 )^*. In other words, we multiply the operator-valued row vector on the matrix.
	 
		The following relation holds
		
    z^-1/2(w/z)  Ψ^*,(1)(z) Ψ^*,(2) (w) 
     = w^-1/2(z/w)  Ψ^,*(2) (w) Ψ^*,(1)(z) R(z/w) + (-1)^∂ (-q)^1/2 (q v^*_- ⊗ v^*_+ -  v^*_+ ⊗ v^*_-) δ(q^2 z, w).

	
	
		The theorem is just a reformulation of (<ref>), (<ref>) and Corollary <ref>.
	
	
	

 §.§ Special point relation

	

  §.§.§ Special point for Φ

	 
		We have the following identity
		
    . (-q z)^-1/2(w/z) ( q Φ_-(z) Φ_+ (w)  - Φ_+ (z) Φ_- (w) ) |_w=q^2z =  (-1)^∂/z q^2 (1-q^2).

	
	
		Note, that a priori LHS of (<ref>) is not well defined since a coefficient of any power of z is an infinite sum of operators. So we have to prove that the result of substitution exists as well as to find the result. Also note, that we substitute w ↦ q^2 z to the whole expression, not to the individual multiples; the result of substitution to the individual multiples does not have to exist.
	
	
		Substituting w ↦ q^2z to (<ref>) and (<ref>) we obtain
		
    . (-q^3 z)^-1/2(w/z) ( q Φ_-(z) Φ_+ (w)  - Φ_+ (z) Φ_- (w) ) |_w=q^2 z =  
      q^2 z[ : Φ_-(z) Φ_-(q^2 z)  :  , x_0^-]_q^2 - [: Φ_-(z) Φ_-(q^2 z)  :  ,  x_1^- ]_q^-2/qz(1-q^2).

		To finish the proof we apply Lemma <ref>.
	
	

  §.§.§ Special point for Ψ

	 We have the following identity
		
    . (-qz)^-1/2(w/z)  ( Ψ_-(z) Ψ_+ (w) - q Ψ_+(z) Ψ_- (w) ) |_z=q^2w =  q w^-1/1-q^2 (-1)^∂.

	
	
		Let us substitute z ↦ q^2 w to (<ref>) and (<ref>) 
		
    . (-qz)^-1/2(w/z)  ( Ψ_-(z) Ψ_+ (w) - q Ψ_+(z) Ψ_- (w) ) |_z=q^2 w =  
    q [  : Ψ_+ (q^2 w) Ψ_+ (w)  : , x_0^+]_q^2 - q^5 w [ : Ψ_+ (q^2 w) Ψ_+ (w)  : , x_-1^+ ]_q^-2/q^2-1 .

		To finish the proof we applying Lemma <ref>.
	
	
	 
		For any q_1 ∈ℂ\{0} we have the following identity
		
    . (-q_1/z)^1/2(w/z)  ( Ψ^*_+(q q_1 z) Ψ^*_- (q q_1 w) - q Ψ^*_-(q q_1 z) Ψ^*_+ (q q_1 w) ) |_z=q^2w = -   w^-1/1-q^2 (-1)^∂.

	
	
	

 §.§ Interchanging relation on Φ and Ψ

	 
		The following relation holds
		
    z^1/2(w/z) Φ_ϵ_1 (z) Ψ^*_ϵ_2 (w) = w^1/2(z/w) Ψ^*_ϵ_2 (w)  Φ_ϵ_1 (z).

	
	
		We have already seen the cases  ϵ_1= ϵ_2= ±, see (<ref>) and (<ref>). To prove remaining cases, let us use a relation from <cit.> and relation <cit.>
		
    Ψ_-^*(z) x_0^- -  x_0^- Ψ_-^*(z) =   0,    Φ_-(z) x_0^+ -  x_0^+ Φ_-(z) =   0.

		To be combined with (<ref>) and (<ref>), the relations yield
		
    [Φ_-(z) Ψ_-^*(w), x^-_0 ]_q = [Φ_-(z), x^-_0 ]_q Ψ_-^*(w) =Φ_+(z) Ψ_-^*(w),
    
    		[Φ_-(z) Ψ_-^*(w), x^+_0 ]_q= Φ_-(z) [Ψ_-^*(w), x^+_0 ]_q  = Φ_-(z) Ψ_+^*(w).

		Considering  q-commutator of (<ref>) with x_0^- and x_0^+, we obtain cases ϵ_1=+, ϵ_2=- and ϵ_1=-, ϵ_2=+ correspondingly. 
	
	
	

§ REALIZATION OF (TWISTED) DEFORMED VIRASORO ALGEBRA
 
	In this section, we will consider two algebras: deformed Virasoro algebra and twisted deformed Virasoro algebra. Deformed Virasoro algebra is extensively studied. Twisted Virasoro was defined in <cit.>, though this algebra is considerably less famous.
	
	The algebras depend on two parameters q_1, q_2. It is also convenient to consider q_3 such that q_1 q_2 q_3 = 1. In this section we study connection between the algebras and  for q^2 = q_3.
	
	To define (twisted) deformed Virasoro algebra, we need the following notation
	
    ∑_l=0^∞ f_l x^l = f(x) = exp( ∑_n=1^∞1/n(1-q_1^n)(1-q_2^n)/1+q_3^-n x^n).

	Note that
	
    f(x) = 1/1-x(q_1 q x ) ( q_1^-1 q^-1 x ).

	

 §.§ Deformed Virasoro algebra

	 Deformed Virasoro algebra  is generated by T_n for n ∈ℤ. The defining relation is
		
    ∑_l=0^∞f_l T_n-l T_m+l - ∑_l=0^∞f_l T_n-l T_m+l = -(1-q_1)(1-q_2)/1-q_3^-1( q_3^-n - q_3^n) δ_n+m,0.

		Denote T(z)= ∑_n∈ℤ T_n z^-n, δ(x) = ∑_k ∈ℤ x^k.
		Relation (<ref>) is equivalent to
		
    f(w/z) T(z) T(w) - f(z/w) T(w) T(z) = -(1-q_1)(1-q_2)/1-q_3^-1( δ(w/q_3 z) -  δ(q_3 w/z) ).
	
	
	

  
Representation Recall that V_j were defined by (<ref>).
	 
		The formula below determines an action of  on V_j for all j ∈ℤ.
		
    T(z) =  z^1/2q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)(u Ψ_+^* (q q_1 z)  Φ_+ (z)  + u^-1Ψ_-^* (q q_1 z) Φ_- ( z) ).

	
	Denote the obtained representation by .
	
	 
		A bosonization of deformed Virasoro algebra is known since <cit.>, but our bosonization is a different one. In both cases current T(z) is presented as a sum two summands. Surprisingly, the first summands in both cases are `the same normally ordered exponent of Heisenberg a_k'; however, the second ones are different. In <cit.> the second summand is also an exponent of the same Heisenberg, but this is not true for our bosonization. Note that  Ψ_-^* (q q_1 z) Φ_- ( z) is an exponent of Heisenberg π(a_k), but not of a_k.
	
	
	
		The proof is basically verification of (<ref>). Let us rewrite (<ref>) in the matrix form
		
    T(z) = q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)Ψ^*(q q_1  z)  ε_z Φ(z),  for ε_z = z^1/2[    u    0;    0 u^-1 ].

		Using (<ref>), Proposition <ref> and (<ref>) we obtain
		
    f(w/z)  ( Ψ^*(q q_1  z)  ε_z Φ(z) ) ( Ψ^*(q q_1  w) ε_w Φ(w) ) 
    = 
    		1/1-w/z(q_1 q w/z) (w/q_1 q z) ( Ψ^*(q q_1 z)  ε_z Φ(z) ) ( Ψ^*(q q_1 w) ε_w Φ(w) ) 
    = 
    		1/1-w/z(q q_1 w/z)^1/2(z/q_1 q w) (w/q_1 q z) ( Ψ^*, (1)(q q_1  z)  Ψ^*,(2)(q q_1 w) ) ε_z ⊗ε_w (Φ^(1)(z)  Φ^(2)(w) )
    =
    		(z/q_1 q w) (w/q_1 q z) (q q_1 z w)^1/2
    ×(z^-1/2(w/z) Ψ^*, (1)(q q_1  z)  Ψ^*,(2)(q q_1 w) )  ε_z ⊗ε_w ( z^-1/2(w/z) Φ^(1)(z)  Φ^(2)(w) ).

		To continue the calculation, we apply Theorems <ref> and <ref>. The RHS of (<ref>) can be 
		presented as sum of three summands. The first summand is
		
    (z/q_1 q w) (w/q_1 q z) (q q_1 z w)^1/2
    ×(w^-1/2(z/w)  Ψ^*,(2)(q q_1 w) Ψ^*, (1)(q q_1  z) ) R(z/w) ε_z ⊗ε_w R^-1(z/w) ( w^-1/2(z/w) Φ^(2)(w) Φ^(1)(z) )
    =
    		1/1-z/w(q q_1 z /w)^1/2(z/q_1 q w) (w/q_1 q z)   ( Ψ^*,(2)(q q_1 w) Ψ^*, (1)(q q_1  z)  ) ε_z ⊗ε_w  (Φ^(2)(w) Φ^(1)(z) )
    =
    		1/1-z/w(z/q_1 q w) (q_1 q z/w)   ( Ψ^*(q q_1 w) ε_w Φ(w)  ) ( Ψ^*(q q_1  z) ε_z Φ(z) )
    =
    		f(z/w)  ( Ψ^*(q q_1 w) ε_w Φ(w)  ) ( Ψ^*(q q_1  z) ε_z Φ(z) ).

		Here we have used  Proposition <ref> and an important property 
		
    R(z/w) ε_z ⊗ε_w R^-1(z/w) = ε_z ⊗ε_w.

		The second summand without factor (z/q_1 q w) (w/q_1 q z) is
		
    (q q_1 z w)^1/2( z^-1/2(w/z) Ψ^*,(1)(q q_1  z)  Ψ^*,(2)(q q_1 w) )ε_z⊗ε_w( (-q)^1/2 (q^-1 v_-⊗v_+-q^-2 v_+⊗v_-) δ(z,q^2 w))(-1)^∂
    
    		=(q q_1)^1/2 zw ( z^-1/2(w/z) Ψ^*, (1)(q q_1  z)  Ψ^*,(2)(q q_1 w) )  ( (-q)^1/2 (q^-1 v_-⊗v_+ - q^-2 v_+⊗v_-) δ(z,q^2 w) )(-1)^∂
    =
    		q^-1  zw  (-q_1/z)^1/2(w/z)  ( q Ψ^*_-(q q_1  z)  Ψ^*_+(q q_1 w) - Ψ^*_+(q q_1  z)  Ψ^*_-(q q_1 w)  ) (-1)^∂δ(z,q^2 w)
    =
    		q^-1  zw   w^-1/1-q^2δ(z,q^2 w)=  1 /q(1-q^2)δ(z/q^2 w).

		Here we used Corollary <ref>.
		
		The third summand  without factor (z/q_1 q w) (w/q_1 q z) is
		
    q q_1 (zw)^1/2( (-q)^1/2 (q v^*_- ⊗ v^*_+ -  v^*_+ ⊗ v^*_-) δ(q^3 q_1 z, q q_1w)  ) ε_z ⊗ε_w ( z^-1/2(w/z) Φ^(1)(z)  Φ^(2)(w)  ) (-1)^∂
    
    		=q q_1 zw ( (-q)^1/2 (q v^*_- ⊗ v^*_+ -  v^*_+ ⊗ v^*_-) δ(q^3 q_1 z, q q_1w)  )  ( z^-1/2(w/z) Φ^(1)(z)  Φ^(2)(w)  ) (-1)^∂
    =
    		- q^2 q_1  zw  (-qz)^-1/2(w/z)   ( q Φ_-(z) Φ_+(w) -  Φ_+(z)  Φ_-(w)   ) (-1)^∂δ(q^3 q_1 z, q q_1w) 
    =
    		-q^2 q_1  zw   1/z q^2 (1-q^2)δ(q^3 q_1 z, q q_1w)=- 1/q (1-q^2)δ(q^2 z/w).

		Here we used Proposition <ref>
		
		When we calculated the second and the third summands, we have omitted the multiple
		
    (q^2/q_1 q ) (q^-2/q_1 q )   =1-q^-2 q_1^-1/1- q_1^-1( (q/q_1)  )^2 = 1-q_2/1- q_1^-1( (q/q_1)  )^2.

		So the delta-function coefficient is
		
    1-q_2/1- q_1^-1( (q/q_1)  )^2 ×1/q(1-q^2)= (1-q_1)(1-q_2)/(1-q_3^-1)( (q/q_1)/q^3/2(q_1^1/2-q_1^-1/2))^2.

		So we have proven
		
    f(w/z)  ( Ψ^*(q q_1  z)  ε_z Φ(z) ) ( Ψ^*(q q_1  w) ε_w Φ(w) ) - f(z/w)  ( Ψ^*(q q_1 w) ε_w Φ(w)  ) ( Ψ^*(q q_1  z) ε_z Φ(z) ) 
    =
    		(1-q_1)(1-q_2)/(1-q_3^-1)( (q/q_1)/q^3/2(q_1^1/2-q_1^-1/2))^2  ( δ( z/q^2 w)  -  δ( q^2 z/w)  ).

		Evidently, this is equivalent to the theorem.
	
	

  
Connection with Verma module. Highest weight vector | λ⟩ for  with highest weight λ∈ℂ in a -module is defined by the following properties
	
    T_0 | λ⟩   = λ | λ⟩,     T_n |λ⟩   =0   for n>0.

	Denote by = 1 ⊗ℂ e^Λ_i + ⌊j/2⌋α∈ V_j the highest weight vector with respect to Heisenberg algebra.
	 
		Vector ∈ is a highest weight vector for  with the highest weight 
		
    λ_u,j = (-q)^1/2 (q q_1)^j/2 u + ( (-q)^1/2 (q q_1)^j/2 u )^-1.

	
	
		Consider a grading on  given by a_-k = k and =0. One can check that  is a graded -module with respect to grading T_-n=n. Grading consideration implies that  is a highest weight vector. It remains to calculate the highest weight. 
		Using (<ref>), we obtain
		
    q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1) z^1/2Ψ_-^* (q q_1 z) Φ_- ( z) = q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)(-q^3 × q q_1 z)^-1/2/(1/q q_1) z^1/2:Ψ_-^* (q q_1 z) Φ_- ( z): 
    =
    		q^3/2(q_1^1/2- q_1^-1/2)(-q^3 × q q_1 )^-1/2/(1-1/q_1)  :Ψ_-^* (q q_1 z) Φ_- ( z): = (-q)^-1/2  :Ψ_-^* (q q_1 z) Φ_- ( z):.

		Using (<ref>) and formulas for explicit bosonization (<ref>) and (<ref>), we obtain 
		
    z^1/2q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)Ψ_-^* (q q_1 z)  Φ_- (z)  = (-q)^-1/2 (qq_1)^-j/2 + O(z).

		here O(z) is a formal power series, which vanishes at z=0, i.e. ∑_n>0α_n z^n.
		 
			Vector π̃ coincides up to a scale with vector | 1 - j ⟩.
		
		
			Let us consider two grading on V(Λ_0) ⊕ V(Λ_1)
			
    = j(j-1)/4        _K v = j   iff Kv=q^jv
        a_-k =  k

			One can check that
			
    _K( π(v) )    = 1 - _K v   ( π(v) )    = _pr v

			Up to a scale, vector  is the only vector with _K=j and = j(j-1)/4.
		
		Let us apply π-involution to (<ref>); Proposition <ref> and Lemma <ref> imply
		
    z^1/2q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1) (-q)^3 ( 1/2 - i )(q_1 q^3 z)^-1/2Ψ_+^(i, 1-i),*(q q_1 z)  (-q^3)^i-1/2 z^1/2Φ^(1-i,i)_+ (z) | 1 - j ⟩
    = (-q)^-1/2 (qq_1)^-j/2 |1-j ⟩ + O(z).
 
		Replacing of j ↦ 1-j, we obtain 
		
    z^1/2q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)Ψ_+^* (q q_1 z)  Φ_+ (z)  = (-q)^1/2 (qq_1)^j/2 + O(z).

		Comparison of (<ref>) and (<ref>) with (<ref>) finishes the proof. 
	
	
	
	Verma module M(λ) is a module with cyclic highest weight vector |λ⟩ and without any other relations apart from (<ref>).  Verma module M(λ) enjoys a universal property: it maps to any module with a highest weight vector of weight λ.
	According to Proposition <ref>, there is a natural map ϕ_u,j M(λ_u,j) →. We will say that λ is generic if λ≠± (q_1^r/2q_2^s/2+q_1^-r/2q_2^-s/2) for r,s ∈ℤ_≥ 1.
	
	
		For generic λ the Verma module M(λ) is irreducible. Dimension of nth graded component is p(n), i.e. the number of partitions of n elements.
	
	
	This proposition follows from the fact that determinant of the Shapovalov form for such λ is nonzero, this fact was proven in <cit.>, using <cit.>. One can also deduce this from the irreducibility of tensor product of Fock modules of toroidal algebra U_q_1,q_2,q_3(𝔤̈𝔩̈_1) <cit.> and relations to W-algebras <cit.> <cit.>.
	
	We will say that pair u,j is generic if the corresponding highest weight λ_u,j is generic.
	
	
		For generic values of u,j the module  is irreducible. The natural map ϕ_u,j M(λ_u,j) → is an isomorphism.
	
	
		Note that dimensions of graded components of both M(λ) and   equals to p(n), in particular they coincide. If M(λ_u,j) is irreducible, then the map ϕ_u,j M(λ_u,j) → is an isomorphism.
	
	
		As it was mentioned in Remark <ref>, another bosonization of  was constructed in <cit.>. Moreover, their formula for the highest weight essentially coincides with our formula (<ref>). Namely, in notation of <cit.> the highest weight of the representation on the Fock space π_μ equals to λ_u,j if q^μ in notation of loc. cit. equals to (-q^3)^1/2 (qq_1)^ju in notation of this paper (note that parameters q,p in loc. cit. correspond to q_1, q_3^-1 in this paper). For generic u,j these modules are isomorphic since they both are isomorphic to irreducible Verma module.
		
		
		
		
		
		
	
	

 §.§ Twisted Deformed Virasoro algebra

	 Twisted deformed Virasoro algebra is generated by T_r for r ∈ 1/2+ℤ. The defining relation is
		
    ∑_l=0^∞f_l T_r-l T_s+l - ∑_l=0^∞f_l T_s-l T_r+l = -(1-q_1)(1-q_2)/1-q_3^-1( q_3^-r - q_3^r) δ_r+s,0.

	
	Denote T(z)= ∑_r∈ 1/2 + ℤ T_r z^-r, (x) = ∑_r∈ 1/2 + ℤ x^r
	Relation (<ref>) is equivalent to
	
    f(w/z) T(z) T(w) - f(z/w) T(w) T(z) = -(1-q_1)(1-q_2)/1-q_3^-1( (w/q_3 z) -  (q_3 w/z) ).
	
	 
		Formulas below determines an action of Twisted Deformed Virasoro algebra on V(Λ_i) for i=0, 1
		
    T(z) = (-1)^1/2q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)( z Ψ_-^* (q q_1 z)  Φ_+ (z)  +  Ψ_+^* (q q_1 z) Φ_- ( z) ).

	
	Denote the obtained representation by .
	
		Let us rewrite (<ref>) in the matrix form
		
    T(z) = (-1)^1/2q^3/2(q_1^1/2- q_1^-1/2)/(q/q_1)Ψ^*(q q_1  z)  ε_z Φ(z),  for ε_z =  [ 0 1; z 0 ].

		The proof is very similar to the proof of Theorem <ref>. A crucial point is that (<ref>) holds for the new ε_z. Hence RHS of (<ref>) still can be presented as sum of three summands. The first summand is still given by (<ref>). The second summand without factor (z/q_1 q w) (w/q_1 q z)
		
    (q q_1 wz )^1/2( z^-1/2(w/z) Ψ^*, (1)(q q_1  z)  Ψ^*,(2)(q q_1 w) )ε_z⊗ε_w ( (-q)^1/2 (q^-1 v_-⊗v_+-q^-2 v_+⊗v_-) δ(z,q^2 w)) (-1)^∂
    =
    		(- q_1  )^1/2( z^-1/2(w/z) Ψ^*, (1)(q q_1  z)  Ψ^*,(2)(q q_1 w) ) ε_z ⊗ε_w (  q^-1 v_-⊗v_+ - q^-2 v_+⊗ v_- ) (z/q^2 w) (-1)^∂
    
    		=(-q_1/z )^1/2(w/z) Ψ^*, (1)(q q_1  z)  Ψ^*,(2)(q q_1 w)   ( q^-1 w  v_+⊗v_- - w v_-⊗v_+ ) (z/q^2 w) (-1)^∂
    
    		=q^-1w ×(-q_1/z )^1/2(w/z)  ( Ψ^*_+(q q_1 z) Ψ^*_-(q q_1w) - q Ψ_-(q q_1 z) Ψ^*_+(q q_1w) )  (z/q^2 w) (-1)^∂
    =
    		-q^-1w w^-1/1-q^2(z/q^2 w) =  -1/q(1-q^2)(z/q^2 w).

		The third summand without factor (z/q_1 q w) (w/q_1 q z)
		
    q q_1 (wz )^1/2( (-q)^1/2 (q v^*_- ⊗ v^*_+ -  v^*_+ ⊗ v^*_-) δ(q^3 q_1 z, q q_1 w)  ) ε_z ⊗ε_w ( z^-1/2(w/z) Φ^(1)(z)  Φ^(2)(w) ) (-1)^∂
    =
    		- (- q)^-1/2(  (q v^*_- ⊗ v^*_+ -  v^*_+ ⊗ v^*_-)  (q^2 z/w)  ) ε_z ⊗ε_w ( z^-1/2(w/z) Φ^(1)(z)  Φ^(2)(w) ) (-1)^∂
    =
    		-(- q)^-1/2(  qz v^*_+ ⊗ v^*_- - w v^*_- ⊗ v^*_+  )  ( z^-1/2(w/z) Φ^(1)(z)  Φ^(2)(w) )  (q^2 z/w) (-1)^∂
    =
    		qz (-qz)^-1/2(w/z) ( q  Φ_-(z) Φ_+(w) - Φ_+(z) Φ_-(w) )    (q^2 z/w) (-1)^∂
    =
    		qz 1/z q^2 (1-q^2) (q^2 z/w)= 1/q (1-q^2) (q^2 z/w).

		The end of the proof is almost the same as in non-twisted case.
	
	
	

  
Connection with Verma module.
	Highest weight vector  in a representation of twisted deformed Virasoro algebra is defined by the following properties
	
    T_r  =0   for r>0.

	
		The vectors | Λ_i ⟩∈ are highest weight vectors.
		
	
		Recall the grading  on V(Λ_i) defined by (<ref>) and (<ref>).  One can check that  is a graded -module with respect to grading T_-r=r. To finish the proof one has to note that |0 ⟩ =  |1 ⟩=0 and >0 for j ≠ 0,1.
	
	
	Verma module  of twisted Virasoro algebra is a cyclic module with cyclic vector  and without any other relations apart from (<ref>). Verma module enjoys a universal property: it maps to any module with a highest weight vector. Hence there exist a natural map ϕ_i → such that ↦ | Λ_i ⟩.
	
	 
		Verma module  is spanned by 
		
    T_-r_m… T_-r_1  for 0 <r_1 ≤ r_2 ≤…≤ r_m.

	
	
	
		One can prove that any element T_s_1… T_s_k can be presented as a linear combination of vectors (<ref>) using (<ref>) by induction.
	
	
	
		For generic q_3 the Verma module  is irreducible. Natural maps ϕ_i → are isomorphisms.
	
	
		The representation  for q_3=1 was considered in <cit.>; it follows from <cit.> that the representations are irreducible. Hence  is irreducible for generic q_3. Then maps ϕ_i  M^tw→ are surjective. Now recall the Gauss identity
		
    ∏_r=1/2+ℤ_≥ 01/1- 𝔮^r =∑_j∈ i+2ℤ𝔮^j(j-1)/4/∏_n=1^∞ (1-𝔮^n)  for i=0,1.

		According to Lemma <ref>, dimensions of graded components of  do not exceed corresponding coefficient of LHS of (<ref>). On the other hand, coefficients of RHS of (<ref>) are equal to dimensions of graded components of . Hence, it follows from surjectivity of ϕ_i that ϕ_i is an isomorphism.
		
	
		For generic q_3 vectors (<ref>) form a basis of .
	
	

§ Π-INVOLUTION

	Let us consider an involution π given by π(e_i)=e_1-i,  π(f_i)=f_1-i. Recall that Drinfeld generators are expressed by following formulas
	
    t_1=   K     x_0^+=   e_1     x_0^-=   f_1 
    
    	t_0 =   γ K^-1    x_1^- =    e_0 t_1     x_-1^+ =    t_1^-1 f_0

	Then
	
    π(K)=   γ K^-1   π(x_0^+) =    e_0= x_1^- K^-1   π(x_0^-) =    f_0 = K  x_-1^+ 
          π(x_1^-)=    e_1 γ K^-1= γ x_0^+ K^-1   π(x_-1^+) =   γ^-1 K x_0^-

	
	
		Up to an i-dependent constant
		π(Φ_+(z) )= z^-1/2Φ_-(z), π(Φ_-(z))= z^1/2Φ_+(z)
	
	
		The vertex operators are defined by following relations (up to normalization)
		
    Φ_+ (z) x_0^- - q^-1 x_0^- Φ_+ (z) = 0 
    Φ_+(z) = Φ_-(z) x_0^- - q x_0^- Φ_-(z)
    
    		K Φ_- (z) = Φ_+ (z) x_0^+ - x_0^+ Φ_+ (z) 
    Φ_- (z) x_0^+ - x_0^+ Φ_- (z) = 0
    
    		(qzK)^-1Φ_-(z) = Φ_+ (z) x_-1^+ - x_-1^+ Φ_+(z)
    Φ_-(z) x_-1^+ - x_-1^+ Φ_-(z)=0
    Φ_+(z) x_1^- - q x_1^-Φ_+ (z) = 0
    
    		q^2 z Φ_+ (z) = Φ_-(z) x_1^- - q^-1 x_1^-Φ_-(z)

		After the involution these relations give
		
    z^-1Φ_- (z) K  x_-1^+ - q^-1 K  x_-1^+ z^-1Φ_- (z) = 0 
    
    		z^-1Φ_-(z) = Φ_+(z) K  x_-1^+  - q K  x_-1^+  Φ_+(z)
    γ K^-1Φ_+ (z) = z^-1Φ_- (z) x_1^- K^-1 - z^-1 x_1^- K^-1Φ_- (z) 
    Φ_+ (z) x_1^- K^-1 - x_1^- K^-1Φ_+ (z) = 0
    
    		(qz)^-1 (γ K^-1)^-1Φ_+(z) = z^-1Φ_- (z) γ^-1 K x_0^- -γ^-1 K x_0^- z^-1Φ_-(z)
    Φ_+(z)γ^-1 K x_0^- - γ^-1 K x_0^-Φ_+(z)=0
    Φ_-(z) γ x_0^+ K^-1 - q γ x_0^+ K^-1Φ_- (z) = 0
    
    		q^2  Φ_- (z) = Φ_+(z)  γ x_0^+ K^-1 - q^-1γ x_0^+ K^-1Φ_+(z)
 
		
		Recall that
		
		
    K X^±(z) K^-1 = q^± 2 X^±(z) 
    
    		K Φ_±(z) K^-1 = q^∓ 1Φ_±(z) 
    
    		K Ψ_±(z) K^-1 = q^∓ 1Ψ_±(z)

		
		
		
		
		
		
		
		
		Hence
		
    Φ_- (z)   x_-1^+ -  x_-1^+ Φ_- (z) = 0 
    
    		(qKz)^-1Φ_-(z) =  Φ_+(z)  x_-1^+  -  x_-1^+  Φ_+(z)
    
    		q^2 z Φ_+ (z) =  Φ_- (z) x_1^-  - q^-1 x_1^-  Φ_+ (z)
    Φ_+ (z) x_1^-  - q x_1^-  Φ_+ (z) = 0
    Φ_+(z) = Φ_- (z)  x_0^- - q x_0^-Φ_-(z)
    Φ_+(z) x_0^- -  q^-1x_0^-Φ_+(z)=0
    Φ_-(z)  x_0^+ -   x_0^+Φ_-(z)=0
    Φ_-(z) x_0^+  -   x_0^+  Φ_- (z) = 0
    
    		K Φ_- (z) = Φ_+(z)  x_0^+  - x_0^+  Φ_+(z)

	
	
		π(Ψ_+(z) )= z^-1Ψ_-(z), π( Ψ_-(z) )=Ψ_+(z)
	
	
	
		
    Ψ_+ (z) x_0^- -  x_0^- Ψ_+ (z) = 0 
    
    		K^-1Ψ_+ (z) = Ψ_- (z) x_0^- - x_0^- Ψ_- (z)
    Ψ_- (z) = Ψ_+ (z) x_0^+ - q x_0^+  Ψ_+ (z) 
    Ψ_- (z) x_0^+ - q^-1 x_0^+  Ψ_- (z) = 0
    
    		(qz)^-1Ψ_- (z) = q Ψ_+ (z)  x_-1^+ -  x_-1^+  Ψ_+ (z)
    
    		q^-1Ψ_- (z) x_-1^+ -  x_-1^+  Ψ_- (z)=0
    Ψ_+ (z) x_1^-  -  x_1^-Ψ_+ (z) = 0
    
    		q z K Ψ_+ (z) =  Ψ_- (z) x_1^-  -   x_1^-Ψ_- (z)

		After the involution
		
    Ψ_- (z) K  x_-1^+  -  K  x_-1^+  Ψ_- (z) = 0 
    γ^-1 z^-1 K Ψ_- (z) = Ψ_+ (z) K  x_-1^+ - K  x_-1^+ Ψ_+ (z)
    Ψ_+ (z) = z^-1Ψ_- (z) x_1^- K^-1 - q x_1^- K^-1 z^-1Ψ_- (z) 
    Ψ_+ (z) x_1^- K^-1 - q^-1 x_1^- K^-1Ψ_+ (z) = 0
     
    		(qz)^-1Ψ_+ (z) = q z^-1Ψ_- (z)  γ^-1 K x_0^- -  γ^-1 K x_0^-  z^-1Ψ_- (z)
    
    		q^-1Ψ_+ (z) γ^-1 K x_0^- -  γ^-1 K x_0^-Ψ_+ (z)=0
    Ψ_- (z) γ x_0^+ K^-1  - γ x_0^+ K^-1Ψ_- (z) = 0
     
    		q γ K^-1Ψ_- (z) =  Ψ_+ (z) γ x_0^+ K^-1  -  γ x_0^+ K^-1Ψ_+ (z)
		
		Hence
		
    q^-1Ψ_- (z)   x_-1^+  -  x_-1^+  Ψ_- (z) = 0
    
    		(qz)^-1Ψ_- (z) = q Ψ_+ (z)   x_-1^+ -  x_-1^+ Ψ_+ (z)
    
    		q z K Ψ_+ (z) =  Ψ_- (z) x_1^-  - x_1^-  Ψ_- (z) 
    Ψ_+ (z) x_1^- -  x_1^-  Ψ_+ (z) = 0
     
    		K^-1Ψ_+ (z) = Ψ_- (z)   x_0^- -   x_0^-Ψ_- (z)
    Ψ_+ (z) x_0^- -  x_0^-Ψ_+ (z)=0
    Ψ_- (z)  x_0^+   -  q^-1 x_0^+  Ψ_- (z) = 0
    Ψ_- (z) =  Ψ_+ (z)  x_0^+  -  q x_0^+  Ψ_+ (z)

	
	
	
	
	
	
	
	
	alpha
	
	
	Landau Institute for Theoretical Physics, Chernogolovka, Russia,

		Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia,

		National Research University Higher School of Economics, Moscow, Russia,

		Institute for Information Transmission Problems, Moscow, Russia,

		Independent University of Moscow, Moscow, Russia
	
	E-mail:  mbersht@gmail.com

	
	
	Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia

		National Research University Higher School of Economics, Moscow, Russia
	
	E-mail:  roma-gonin@yandex.ru	
	
	
