Jet schemes, Quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces
aa r X i v : . [ m a t h . QA ] O c t JET SCHEMES, QUANTUM DILOGARITHM ANDFEIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES
HAO LI AND ANTUN MILAS
Abstract.
We analyze the structure of the infinite jet algebra, or arc algebra,associated to level one Feigin-Stoyanovsky’s principal subspaces. For A -series,we show that their Hilbert series can be computed either using the quantumdilogarithm or as certain generating functions over finite-dimensional repre-sentations of A -type quivers. In particular, we obtain new fermionic characterformulas for level one A -type principal subspaces, which implies that they areclassically free. We also analyze the infinite jet algebra of the principal sub-space of the affine vertex algebra L ( so (5)) and of L ( so (8)). We give newconjectural expressions for their characters and provide strong evidence thatthey are also classically free. Introduction
Associated varieties and associated schemes are important geometric object at-tached to a vertex algebras V . They have been investigated in [AM18, AL19, AM,AM16, vEH18] and also in physics due to their relevance to four-dimensional N = 2superconformal field theory [BLL + ψ from the jet algebra J ∞ ( R V ) of V to theassociated graded algebra gr( V ) [Ara12], where R V denotes Zhu’s Poisson algebraof V .Following the terminology of [vEH18] we say that V is classically free if ψ isinjective. Another way of expressing injectivity is by saying that V is ”chiralizationof its Poisson algebra” [Li20]. Vertex algebras with this property are commonand we expect that they include many (but not all) rational vertex algebras andof course large families of irrational vertex algebras. It is relatively easy to findcounterexamples to injectivity of ψ [AL19], [vEH18]. However, a full description ofthe kernel, if non-trivial, of the ψ -map is an interesting and difficult problem. Veryrecently, Andrews, Van Ekeren and Heluani [AvEH20] found a remarkable q -seriesidentity that allowed them to describe the kernel of ψ for the c = Ising Virasorovertex algebra.As in [Li20] (see also [JSM20]), here we are interested in associated schemesand infinite jet algebras of Feigin-Stoyanovsky’s principal subspaces [FS93, MP12].Although our understanding of principal subspaces and their characters has greatlyimproved with the help of vertex algebra tools (cf. [Geo96, CLM06, CLM10, Pri05]),we found that known character formulas are not sufficient to determine whetherthe ψ map is injective. The main reason is that in most approaches to characterformulas only simple roots are used instead of all positive roots (for some recentresult on this subject see for instance [PS18], [But20]). In this paper, [Li20] anda sequel, we try to fill the gap between two approaches to principal subspaces by studying character formulas coming from all positive roots. Here are q -seriesidentities needed in this paper. Theorem 1.1.
Let A be the Cartan matrix ( h α i , α j i ) ≤ i,j ≤ n − of type A n − , n ≥ ,that is h α i , α j i = i = j − | i − j | = 10 otherwise and m = ( m , , ...., m n − ,n ) = ( m i,j ) ≤ i
Example 1.2.
For sl , we have λ = m , + m , , λ = m , + m , . Letting newvariables m = n , n = n and n = m , , n = m , and n = m , , both equations(1) and (2) give the following identity [FS93](3) X m,n ≥ q m + n − mn x m y n ( q ) m ( q ) n = X n ,n ,n ≥ x n + n y n + n q n + n + n + n n + n n ( q ) n ( q ) n ( q ) n . Notice that both the LHS and RHS of our identities are ”stable” with respectto the rank n . In other words, if we let m · ,n = 0 in the theorem, then all identitiesare valid for sl n − (i.e. type A n − ).In this paper we are concerned with q -series identities of the form(4) X n ∈ Z k ≥ q n ⊤ A n x n ( q ) n · · · ( q ) n k = X m ∈ Z ℓ ≥ q B ( m ) x U m ( q ) m · · · ( q ) m ℓ , where x n = x n · · · x n k k , A is a positive definite integral k × k matrix (e.g. Cartanmatrix), B ( m ) is a quadratic form on ℓ -dimensional space (normally ℓ is biggerthan k ) and U is a k × ℓ matrix. We say that the q -series identity (4) is equivalent to(5) X n ∈ Z k ≥ q n ⊤ ˜ A n x n ( q ) n · · · ( q ) n k = X m ∈ Z ℓ ≥ q ˜ B ( m ) x U m ( q ) m · · · ( q ) m ℓ , if for every n = ( n , ..., n k ), there is a solution of U m = n such that B ( m ) − n ⊤ A n = ˜ B ( m ) − n ⊤ ˜ A n . Under this condition, clearly, (4) implies (5) and vice-versa as all x coefficients areequal.The main idea in the proof of Theorem 1.1 is to show that identities (1) and (2)are in fact equivalent to a pair of identities derived on one hand from the quantumdilogarithm and on the other hand from quiver representations. Since the right-hand side of identities (1) and (2) is the known (charged) character formula of theprincipal subspace W (Λ ) of the affine vertex algebra of type A n − , as a corollaryof Theorem 1.1, we also get two new charged character formulas for the principalsubspace. Moreover, following the idea in [Fei09] on PBW filtration, we carefullyconstruct a series of filtrations on certain infinite jet algebras of quotient polynomialalgebras. From that and Theorem 1.1, we obtain the Hilbert series formulas forcertain jet algebras. This in particular, clarifies an argument in [Li20, Theorem5.7], where it was claimed that W (Λ ) is classically free. Our result also providesa new combinatorial basis of W (Λ ).This paper is organized as follows. In Section 2 we recall standard definitions ofthe infinite jet algebra, arc space, affine vertex algebra and the Feigin-Stoyanovsky’sprincipal subspace. Here, for simplicity, we only consider principal subspaces forthe vacuum representation. In Section 3, by using filtration we get two inequalitiesfor Hilbert series of jet algebras. This is then used to prove that W (Λ ) is classicallyfree. In the main part, Section 4 and Section 5, we show equivalences of identitiesamong (1), (2), and two identities derived from the quantum dilogarithm and quiverrepresentations. In sections 6 and 7, we apply the level one spinor representationsfor orthogonal series to investigate the C -algebras of the principal subspace for HAO LI AND ANTUN MILAS L ( so (5)) and of L ( so (8)). We present conjectural formulas for their characters,which if true would imply that these principal subspaces are also classically free.2. preliminary Affine jet algebras.
We first define affine jet algebras. As usual, let C [ x , x , . . . , x n ] be the polynomial algebra in x i , 1 ≤ i ≤ n . Let f , f , . . . , f n be elements in the polynomial algebra. We will define the jet algebra of the quotientalgebra: R = C [ x , x , . . . , x n ]( f , f , . . . , f n ) . Firstly, let us introduce new variables x j, ( − − i ) for i = 0 , . . . , m . We define aderivation T on C [ x j, ( − − i ) | ≤ i ≤ m, ≤ j ≤ n ] , as T ( x j, ( − − i ) ) = ( ( − − i ) x j, ( − i − for i ≤ m −
10 for i = m. Here we identify x j with x j, ( − . Set R m = C [ x j, ( − − i ) | ≤ i ≤ m, ≤ j ≤ n ]( T j f i | i = 1 , . . . n, j ∈ N ) , the algebra of m -jets of R . The infinite jet algebra of R is J ∞ ( R ) = lim → m R m = C [ x j, ( − − i ) | ≤ i, ≤ j ≤ n ]( T j f i | i = 1 , . . . n, j ∈ N ) . The scheme X ∞ = lim ← m X m , where X m = Spec( R m ), is called the infinite jet scheme,or arc space, of X = Spec( R ).We often omit ”infinite” and call it jet algebra for brevity. The jet algebra isa differential commutative algebra. Moreover, this construction is functorial and J ∞ ( · ) is right exact. If R is graded, then J ∞ ( R ) is also graded, and we can definethe Hilbert-(Poincare) series of J ∞ ( R ) as: HS q ( J ∞ ( R )) = X m ∈ Z dim( J ∞ ( R ) ( m ) ) q m . For later use, we denote the defining ideal of J ∞ ( R ) by( f , . . . , f n ) ∂ := ( T j f i ; i = 1 , . . . , n, j ≥ . Principal subspaces.
Let V be a vector space. A field is a formal series ofthe form a ( z ) = P n ∈ Z a ( n ) z − n − , where a ( n ) ∈ End( V ) and for each v ∈ V onehas a ( n ) v = 0for n ≫
0. A vertex algebra is a vector space of states V , together with a distin-guished vector called vacuum ∈ V, a linear derivation map T ∈ End( V ), and thestate-field correspondence map a Y ( a, z ) = X n ∈ Z a ( n ) z − n − , EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 5 satisfying the usual axioms of creation, T -derivative properties, and Jacobi identity(for details see for instance [LL12]). A vertex algebra V is called commutative if a ( n ) = 0 for n ≥ g be a simple finite dimensional complex Lie algebra. Following [LL12], Wecan define an affine Lie algebra b g and its vacuum representation of level k = − h ∨ , V k ( g ), which has a natural conformal vertex algebra structure. We denote by L k ( g )the irreducible quotient of V k ( g ) which is also a vertex algebra. Next we define theprincipal subspace in L k ( g ) . We choose simple roots { α , . . . , α n } , and denote theset of positive roots by ∆ + . Let ( · , · ) be a rescaled Killing form on g such that( α, α ) = 2 for every long root α . (as usual we identify h and h ∗ via the Killingform).Let n + := a α ∈ ∆ + C x α , where x α is the corresponding root vector, and c n + = n + ⊗ C [ t, t − ] be its affinization. For any affine vertex algebra L k ( g ), isomorphicto L ( k Λ ) as a ˆ g -module, we define the (FS)-principal subspace of simple b g -module L k ( g ) as W ( k Λ ) := U ( b n + ) · , where is the vacuum vector. It is easy to see that this is a vertex algebra (withoutconformal vector). For k = 1 and g is of ADE type we have W L ∼ = W (Λ ) where W L ⊂ V L , where L is the root lattice, and W L is the principal space inside thelattice vertex algebra V L [MP12]. Principal subspaces were introduced in [FS93]and since then have been studied in many papers some by the authors.Given a vertex algebra V, we define its C -algebra as: R V := V / h a ( − b | a ∈ V, b ∈ V i , where the multiplicative structure is given by a · b = a ( − b and the Lie bracket by { a, b } = a (0) b. According to [Zhu96], the algebra R V is a Poisson algebra. Then X V := Spec( R V )is called the associated scheme of V . Definition 2.1.
A commutative associated unital algebra V is called a vertexPoisson algebra if it is equipped with a linear operation, V → Hom(
V, z − V [ z − ]) , a → Y − ( a, z ) = X n ≥ a ( n ) z − n − , such that • ( T a ) n = − na ( n − , • a ( n ) b = P j ≥ ( − n + j +1 1 j ! T j ( b ( n + j ) a ) , • [ a ( m ) , b ( n ) ] = P j ≥ (cid:0) mj (cid:1) ( a ( j ) b ) ( m + n − j ) , • a ( n ) ( b · c ) = ( a ( n ) b ) · c + b · ( a ( n ) c ) , for a, b, c ∈ V and n, m ≥ { F n ( V ) } ofthe algebra V , where for n ∈ Z , F n ( V ) is linearly spanned by the vectors u (1)( − − k ) . . . u ( r )( − − k r ) HAO LI AND ANTUN MILAS for r ≥ u (1) , . . . , u ( r ) ∈ V, k , . . . , k r ≥ k + . . . + k r ≥ n. Then V = F ( V ) ⊃ F ( V ) ⊃ . . . such that u ( n ) v ∈ F r + s − n − ( V ) for u ∈ F r ( V ) , v ∈ F s ( V ) , r, s ∈ N , n ∈ Z ,u ( n ) v ∈ F r + s − n ( V ) for u ∈ F r ( V ) , v ∈ F r ( V ) , r, s, n ∈ N . The corresponding associated graded algebra gr F ( V ) = ` n ≥ F n ( V ) /F n +1 ( V ) is avertex Poisson algebra.According to [Li05], we know that F n ( V ) = (cid:8) u ( − − i ) v | u ∈ V, i ≥ , v ∈ F n − i ( V ) (cid:9) . In particular, F ( V ) /F ( V ) = V /C ( V ) = R V ⊂ gr F ( V ). We can extend thisembedding to a surjective map ψ : J ∞ ( R V ) ։ gr F ( V ) . Following [Ara12], J ∞ ( R V ) has a unique Poisson vertex algebra structure such that u ( n ) v = ( { u, v } , if n = 00 , if n > u, v ∈ J ∞ ( R ). The map ψ is a Poisson vertex algebra epimorphism.We say that a vertex algebra is classically free if its corresponding map ψ is anisomorphism. 3. Principal subspace W (Λ ) of sl n This principal subspace was studied by several authors [FS93, Cal08, CLM10,Geo96, Sad15]. According to [FFL11] and [Li20], the C -algebra R W (Λ ) of theprincipal subspace of the affine vertex algebra L ( sl n ) is isomorphic to C [ E i,j | ≤ i < j ≤ n ] / h X σ ∈ S E i ,j σ E i ,j σ | j > i i , where E i,j can be identified with the roots vectors in n + , and 1 ≤ i ≤ i ≤ n ,1 ≤ j ≤ j ≤ n . We let A := R W (Λ ) for sake of brevity. Inside the definingquadratic binomial ideal of A we have the following three types of elements (it ishelpful to use graphical interpretation to visualize these elements) : Type I : E i ,j E i ,j = − E i ,j E i ,j , where 1 ≤ i < j ≤ n, ≤ i < j ≤ n, ≤ i < i ≤ n, ≤ j < j ≤ n, j > i , j > i + 1 .i j i j i + 1 Type II : E i ,j E i ,j , where 1 ≤ i < j ≤ n, ≤ i < j ≤ n, i = i , ≤ j ≤ j ≤ n, or 1 ≤ i < j ≤ n, ≤ i < j ≤ n, ≤ j = j ≤ n, ≤ i < i ≤ n.i j i j i j i j Type III : E i,i +1 E j,j ′ = − E i,j ′ E j,i +1 , where j < i < i + 1 < j ′ .i + 1 i j ′ j Remark 3.1.
Inside the jet algebra J ∞ ( A ), Type I relations generate furtherrelations coming from z -coefficients of E + i ,j ( z ) E + i ,j ( z ) + E + i ,j ( z ) E + i ,j ( z ) = 0 , where E + i,j ( z ) = P n ∈ Z < ( E i,j ) ( n ) z − n − denotes the holomorphic part of the fieldassociated to E i,j . For Type II and III relations, we use similar notation. In orderto simplify presentation, we often write A + ( z ) B + ( z ) = 0 to denote set of relationsobtained by taking all z -coefficients in the expansion.Let us consider another commutative algebra B = C [ E i,j | ≤ i < j ≤ n ] /I, where I is general all Type II relations and left-hand side of all relations of type Iand III. This way I is a monomial ideal. This algebra and its jet algebra are easierto analyze. We will show that HS q ( J ∞ ( A )) ≤ HS q ( J ∞ ( B )) , where inequality among q -series has the obvious meaning: all q -coefficients on theleft-hand side are less than or equal to the corresponding coefficients on the right-hand side.Following an idea of E. Feigin on PBW filtration in [Fei09], for n ≥
3, weintroduce a sequence of increasing filtrations G i,j ( n − ≥ j ≥ i ≥
1) on jetalgebras J ∞ ( B ), where B is a quotient polynomial algebra, C [ E a,b | ≤ a < b ≤ n ] / I by an ideal generated by homogeneous monomials and binomials of degree2. Although its definition is somewhat technical, the idea behind is very simple -it essentially replaces some binomial quotient relations of B with monomial onesinside the corresponding associated graded algebra. Eventually we will end up witha quotient polynomial by a quadratic monomial ideal; the jet algebra associated toit is easier to handle using vertex algebra methods, and its Hilbert series can becomputed explicitly [Li20].Fix i and j as above. We define the filtration G i,j on J ∞ ( B ) by letting G i,j begenerated with E i,i +2 , E i − ,i +2 , · · · , E j,i +2 . We let E i,j = { ( i, i + 2) , ( i − , i + 2) , · · · , ( j, i + 2) } , and G i,js = span n ( E u,u ′ ) ( i ) v | i ≤ − , ( u, u ′ ) / ∈ E i,j , v ∈ G i,js − o + G i,js − . Here the subscript s is the filtration parameter, and G i,j = span n ( E i,i +2 ) l ( k ) ( E i − ,i +2 ) l ( k ) · · · ( E j,i +2 ) l i − j +1 ( k i − j +1 ) | l r ∈ N , k s ∈ Z < o . HAO LI AND ANTUN MILAS
Then the associated graded algebra of J ∞ ( B ) is defined as gr i,j ( J ∞ ( B )) = G i,j ⊕ M s> G i,js /G i,js − . For better transparency of formulas we abuse notation and use ( E a,b ) − t , 1 ≤ a < b ≤ n , to denote its representative inside the associated graded algebra gr i,j ( J ∞ ( B )).Next, we introduce a sequence of ( i, j ) filtrations in a particular order and alsoconstruct a commutative algebra at each step of filtration. Step
1: Let B . := J ∞ ( A ). We can define an algebra homomorphism from P = C [ E i,j | ≤ i < j ≤ n ] to gr , ( J ∞ ( A )) by sending E i,j to its equivalenceclass. We denote the kernel of this map by I i,j , and define a new algebra B . := C [ E i,j | ≤ i < j ≤ n ] / I . . Step .
1: Define B . := C [ E i,j | ≤ i < j ≤ n ] / I . , where I . is the kernel of the algebra homomorphism from P to gr , ( J ∞ ( B . ))by sending E i,j to its equivalence class. Step .
2: Define B . := C [ E i,j | ≤ i < j ≤ n ] / I . , where I . is the kernel of the map defined in the same manner as abovefrom P to gr , ( J ∞ ( B . )).We list all generators of G i,j in each step at stage 2: Step . G , : E , E , Step . G , : E , . ... Step i.
1: Define B i. := C [ E i,j | ≤ i < j ≤ n ] / I i. analogously, where I i. is the kernel of the map from P to gr i. ( B i − .i − ).... Step i.i : Define B i.i := C [ E i,j | ≤ i < j ≤ n ] / I i.i similarly, where I i.i is the kernel of the map from P to gr i.i ( B i.i − ). EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 9
We list generators of G i,j as following: Step i. G i, : E i,i +2 E i − ,i +2 · · · E ,i +2 E ,i +2 Step i. G i, : E i,i +2 E i − ,i +2 · · · E ,i +2 ... Step i.i G i,i : E i,i +2 . Eventually, this procedure terminates at Step n − .n −
2, and we reached a desiredalgebra B n − .n − . Claim 3.2.
From above construction of B i.j , we have a surjective homomorphism ( J ∞ ( B i.j ) ։ gr i,j ( J ∞ ( B i.j − )) if j − ≥ J ∞ ( B i.j ) ։ gr i.j ( J ∞ ( B i − .i − )) if j = 1 , (6)at Step i.j . Proof.
It is enough to prove the Claim at Stage 1 . .
1, there is a natural differential algebra epi-morphism φ from J ∞ ( C [ E i,j | ≤ i < j ≤ n ]) to gr , ( J ∞ ( A )), and this leads to anisomorphism J ∞ ( C [ E i,j | ≤ i < j ≤ n ]) /J ∼ = gr , ( J ∞ ( A )) , where J is the kernel of map φ . Note J ∞ ( B . ) = J ∞ ( C [ E i,j | ≤ i < j ≤ n ]) / ( I . ) ∂ . We also have ( I . ) ∂ ⊂ J since J is closed under the differentiationin jet algebra and contains I . . Therefore, we have a surjective homomorphism: J ∞ ( C [ E i,j | ≤ i < j ≤ n ]) / ( I . ) ∂ ։ J ∞ ( C [ E i,j | ≤ i < j ≤ n ]) /J. We proved the Claim. (cid:3)
Observe that given any increasing filtration G i,j on an algebra Q , we alwayshave HS q ( gr G i,j ( Q )) = lim s →∞ HS q ( G i,js ) = HS q ( Q ) , which follows from gr G i,j ( Q ) = G i,j ⊕ L s> G i,js /G i,js − by taking Hilbert series on both sides. In particular, wehave ( HS q ( gr i,j ( J ∞ ( B i.j − ))) = HS q ( J ∞ ( B i.j − )) if j − ≥ HS q ( gr i.j ( J ∞ ( B i − .i − ))) = HS q ( J ∞ ( B i − .i − )) if j = 1 . (7)Hence, we have a chain of inequalities: HS q ( J ∞ ( B n − .n − )) ≥ HS q ( J ∞ ( B n − .n − )) ≥ · · · ≥ HS q ( J ∞ ( B n − . )) ≥ · · · ≥ HS q ( J ∞ ( A )) . (8) Example 3.3.
Let us consider the case of sl . We have A = C [ E , , E , , E , , E , , E , , E , ] /I, where I = h E , , E , , E , , E , , E , , E , , E , E , , E , E , , E , E , , E , E , ,E , E , + E , E , , E , E , , E , E , , E , E , , E , E , i . In this case, we have two steps of the filtration. We list the generators of G i,j foreach step. Step
1: We have
Step 1.1 G , : E , Here we write down the G , and G , explicitly. G , = span n ( E , ) l ( k ) | l ∈ N , k ∈ Z < o .G , = span { ( E , ) ( k ) ( E , ) l ( k ) , ( E , ) ( k ) ( E , ) l ( k ) , ( E , ) ( k ) ( E , ) l ( k ) , ( E , ) ( k ) ( E , ) l ( k ) , ( E , ) ( k ) ( E , ) l ( k ) , | l ∈ N , k , k ∈ Z < } + G , , and similarly we define G , s , s ≥ B . equals C [ E , , E , , E , , E , , E , , E , ] /I ′ , where I ′ = h E , , E , , E , , E , , E , , E , , E , E , , E , E , , E , E , , E , E , ,E , E , , E , E , , E , E , , E , E , , E , E , i . Indeed, all generating relations of I remain true in gr . ( J ∞ ( A )) except for E , E , = − E , E , . At Step 1 . E , E , ⊂ G , and E , E , ⊂ G , \ G , . Thus, the relation E , E , = − E , E , gives us E , E , = 0in gr , ( A ). Step
2: We have
Step . G , : E , E , Step . G , : E , . It is not hard to see that B ∼ = B . ∼ = B . ∼ = B . . Therefore, we have a surjective differential algebra homomorphism, φ , from J ∞ ( B ) to gr . ( J ∞ ( A )) by sending E i,j to its equivalent class in gr . ( J ∞ ( A )),which implies HS q ( J ∞ ( A )) ≤ HS q ( J ∞ ( B )) . Now we prove this for general sl n . Proposition 3.4.
We have HS q ( J ∞ ( A )) ≤ HS q ( J ∞ ( B )) . Proof.
According to (8), we know that HS q ( J ∞ ( B n − .n − )) ≥ HS q ( J ∞ ( A )) . In order to prove the statement, it is sufficient to show that the ideal of B , i.e., I ,belongs to I n − .n − . Indeed, if it is true, we would have HS q ( J ∞ ( B )) ≥ HS q ( J ∞ ( B n − .n − ))since both B and B n − .n − are quotient polynomial algebras with the same gener-ators. Next, let us show I ⊂ I n − .n − .For Type II relations, it is clear that they all belong to I n − .n − .Let us consider Type I relations in A , i.e., E i ,j E i ,j = − E i ,j E i ,j , where1 ≤ i < j ≤ n, ≤ i < j ≤ n, ≤ i ≤ i ≤ n, ≤ j ≤ j ≤ n, j > EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 11 i + 1 . Following above procedure, we see that these Type I relations belong to I i,j until the step ( j − . ( i + 1). Indeed, at that step, G j − ,i +10 is generated by E j − ,j , E j − ,j , · · · , E i +1 ,j . Note E i ,j E i ,j ⊂ G j − ,i +12 \ G j − ,i +11 , while E i ,j E i ,j ⊂ G j − ,i +11 . Therefore, we get E i ,j E i ,j = 0 in B j − .i +1 at the step ( j − . ( i + 1), andthese relations also remain true in B n − .n − . For Type III relations of A , i.e., E i,i +1 E j,j ′ = − E i,j ′ E j,i +1 where j < i < i + 1
1: Define H . := C [ E i,j | ≤ i < j ≤ n ] / T . , where T . is the kernel of the map from P to gr , ( J ∞ ( A )) by sending E i,j to its equivalence class. Step .
2: Define H . := C [ E i,j | ≤ i < j ≤ n ] / T . , where T . is the kernel of the map from P to gr , ( J ∞ ( H . )) by sending E i,j to its equivalence class. We list all generators of G i,j at each step : Step . G , : E , E , Step . G , : E , . Then we can define H i,j iteratively. So we only list the generators of G i,j at each step in the following: Step Step . G , : E , E , E , Step . G , : E , E , Step . G , : E , . Step i : Step i. G i, : E i +1 ,i +2 E i,i +2 · · · E ,i +2 E ,i +2 Step i. G i, : E i +1 ,i +2 E i,i +2 · · · E ,i +2 ... Step i.i + 1 G i,i +10 : E i +1 ,i +2 . Using similar arguments, we can prove the following result.
Proposition 3.5.
We have HS q ( J ∞ ( A )) ≤ HS q ( J ∞ ( H )) . We also have:
Proposition 3.6.
The Hilbert series of J ∞ ( B ) equals X m ∈ N n ( n − / ≥ q B ( m ) Y ≤ i First, we consider an integral lattice L of rank n ( n − with a basis: α , , · · · , α ,n , · · · , α n − ,n . The symmetric bilinear form of this lattice is defined as: ( α i ,j , α i ,j ) = 1 when • ≤ i < j ≤ n, ≤ i < j ≤ n, ≤ i < i ≤ n, ≤ j < j ≤ n, j >i + 1 , • ≤ i < j ≤ n, ≤ i < j ≤ n, ≤ i = i ≤ n, ≤ j < j ≤ n, • ≤ i < j ≤ n, ≤ i < j ≤ n, ≤ j = j ≤ n, ≤ i < i ≤ n, • i + 1 = j , ≤ i ≤ i < i + 1 < j ≤ n, ( α i,j , α i,j ) = 2 for all α i,j , and ( α i ,j , α i ,j ) = 0 otherwise. According to [Li20,Theorem 5.13], we have J ∞ ( B ) ∼ = gr F ( W L ) . It is known that the character of W L ⊂ V L equals (9) [MP12]. Thus, we have the first identity. The identity for J ∞ ( H ) follows along the same lines. (cid:3) EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 13 Quantum dilogarithm The quantum dilogarithm is defined as φ ( x ) := Q i ≥ (1 − q i x ). Let x and y benon-commutative variables such that xy = qyx. The quantum dilogarithm satisfies an important pentagon identity of Faddeev andKashaev [FK94],(11) φ ( y ) φ ( x ) = φ ( x ) φ ( − yx ) φ ( y ) . Using the binomial q -series identity we also have φ ( x ) = X n ≥ ( − n q n ( n − x n ( q ) n . We also record another useful form ( j ≥ φ ( − q j x ) = X n ≥ q n + j − n x n ( q ) n . Warm up: Proof of (3) using quantum dilogarithm. We first observethat two sides are equal if and only if Coeff x m y n agree. Comparing coefficients onboth sides leads to q m + n − mn ( q ) m ( q ) n = X n n mn n n q n + n + n + n n + n n ( q ) n ( q ) n ( q ) n , where, after letting n = m − n and n = n − n , the right hand-side can berewritten as(13) X n ≥ q ( m − n ) +( n − n ) + n +( m − n ) n +( n − n ) n ( q ) m − n ( q ) n − n ( q ) n . After simplifying exponents on both sides we end up with an equivalent identity(14) 1( q ) m ( q ) n = X n ≥ q ( n − n )( m − n ) ( q ) m − n ( q ) n − n ( q ) n . This famous identities appeared in a variety of situation and there are severaldifferent proofs in the literature [Zag07], [Lee12] (credited to Zwegers), [FK94], etc.Using quantum pentagon identity (15) in a slightly different form(15) φ ( − q / y ) φ ( − q / x ) = φ ( − q / x ) φ ( − qyx ) φ ( − q / y )together with Euler expansion we get X m,n ≥ q m + n y m x n ( q ) m ( q ) n = X n ,n ,n ≥ q n + n ( n +1)+ n x n ( yx ) n y n ( q ) n ( q ) n ( q ) n . Using identities y m x n = x n y m q − mn , ( yx ) n = x n y n q − n ( n +1) we can write X m,n ≥ q m + n − mn x n y m ( q ) m ( q ) n = X n ,n ,n ≥ q n + n x n + n y n + n ( q ) n ( q ) n ( q ) n . Extracting the term next to x n y m now gives (after letting n = n + n and m = n + n ) q m + n − mn ( q ) m ( q ) n = X n ≥ q ( n − n ) + ( m − n ) ( q ) n − n ( q ) m − n ( q ) n which is clearly equivalent to identity (13). (cid:3) In summary, the pentagonal identity for the quantum dilogarithm is equivalentto the q -series identity formula for the character of the level one principal subspaceof sl (3).4.2. General case. Let x i , 1 ≤ i ≤ n − 1, be formal variables. Assume that(16) x i x i +1 = qx i +1 x i , and other pairs commute. Then φ ( − q x n − ) · · · φ ( − q x ) = X k , ··· ,k n − ≥ q k / ··· + k n − / − k k −···− k n − k n − x k · · · x k n − n − ( q ) k · · · ( q ) k n − . (17)By definition we have two useful formulas: φ ( − q j − i x j − x j − · · · x i ) = X m ≥ ( − m q m ( m − ( − q j − i x j − · · · x i ) m ( q ) m = X m ≥ q m q m ( j − i )2 − m ( x j − · · · x i ) m ( q ) m , (18) ( x j − · · · x i ) m = q − ( j − i − m ( m +1)2 x mi x mi +1 · · · x mj − . (19)These two formulas combined together give us φ ( − q j − i x j − x j − · · · x i ) = X m ≥ q (2 − ( j − i )) m x mi x mi +1 · · · x mj − ( q ) m . (20)Now for φ ( − q x n − ) · · · φ ( − q x ) and formula (15) we commute factors suchthat φ ( − q x i ) is in front of φ ( − q x i +1 ) . Therefore, we get the quantum diloga-rithmic identity (in this particular order!): φ ( − q x n − ) · · · φ ( − q x )= φ ( − q x ) φ ( − qx x ) φ ( − q x )... φ ( − q n − x n − · · · x ) φ ( − q n − x n − · · · x ) · · · φ ( − q x n − ) φ ( − q n − x n − · · · x ) φ ( − q n − x n − · · · x ) · · · φ ( − qx n − x n − ) φ ( − q x n − ) . (21) EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 15 Next, we expand for each j > iφ ( − q j − i x j − x j − · · · x i ) = X m i,j ≥ ( − m i,j q mi,j ( mi,j − ( − q j − i x j − x j − · · · x i ) m i,j ( q ) m i,j . (22)This allows us to rewrite dilogarithmic identity (21) in the form (5). X m ∈ N n ( n − / ≥ q C ( m ) · F Y ≤ i Theorem 4.1. The identity (1) and quantum dilogarithmic identity (21) are equiv-alent.Proof. We are going to show that the identity (21) is equivalent to the q -seriesidentity (1) by induction. We already proved that this is true for A in the previoussection. Assume that quantum dilogarithmic identity (21) and identity (1) areequivalent for A n − .Now we prove the equivalence in the case of A n . In order to prove the equivalencebetween identity (1) and dilogarithmic identity, it suffices to show (with m i,i +1 asabove) that(24) C ( m ) + E ( m ) − ( k / · · · + k n / − k k − .. − k n − k n ) = B ( m ) − k ⊤ A k . According to our induction hypothesis, if we let m i,n +1 = 0 where 1 ≤ i ≤ n, theyare equal. Then we need show that both sides of (24) have the same terms whichinvolve m i,n +1 (1 ≤ i ≤ n ). After expanding both sides of (24), it is easy to see theterms of the form k i k j ( j > i + 1) are absent on both sides, and the coefficients ofthe terms of the form k i k i +1 equal 1 on both sides (so they cancel out). Similarly,terms of the form k i also cancel out. We are left to analyze 6 possible types of terms involving m i,n +1 : Type I: m i ,j +1 m i ,n +1 , where j + 1 ≤ i − i n + 1 i j + 1 i − Type II: m i ,j +1 m i ,n +1 , where i < i ≤ j + 1 < n + 1. i n + 1 i j + 1 Type III: m i ,j +1 m i ,n +1 , where i ≤ i < j + 1 < n + 1, or i < i < j + 1 ≤ n + 1. i n + 1 i j + 1 Type IV: k i m i ,n +1 , where i ≥ i + 1. i n + 1 i i − Type V: k i m i ,n +1 , where n ≥ i ≥ i . i n + 1 i Type VI: m i,n +1 , where n ≥ i .Now we compare terms of each type on the left- and right-hand side of (24). Weonly provide a few details for brevity. First, on the left-hand side of (24), straight-forward computations with powers of the q -series give the following coefficients: forterms of Type I and IV coefficients are zero as they are absent from the formula.Similarly, for all Type III terms coefficients are also zero. For Type V and VI thecoefficients are − a . ( j Y s = i x s ) m i ,j · · · ( n Y s = i x s ) m i ,n +1 = q − ( j − i ) m i ,n +1 m i ,j · · · , b. j Y s = i q m s,s +12 = q ( j − i +1) m i ,j m i ,n +1 · · · . Therefore, the overall coefficient of Type II terms is 1 . Now we compute coefficients on the right-hand side of (24) for the same six typesof terms. Again, Type I terms are absent so their coefficients are zero. We list allpossible ways, in which we can get Type II terms: EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 17 a. j X s = i m s,s +1 = 2( j − i + 1) m i ,j +1 m i ,n +1 + · · · , b. ( j X s = i m s,s +1 ) m i ,n +1 = − ( j − i + 1) m i ,j +1 m i ,n +1 + · · · , c. ( j X s = i m s,s +1 ) m i ,j +1 = − ( j − i + 1) m i ,j +1 m i ,n +1 + · · · , d. m i ,j +1 m i ,n +1 . Adding these up gives the coefficient of Type II to be 1 . Along the same lines TypeIII and IV coefficients are zero, and Type V and VI coefficients are − m i,n +1 (1 ≤ i ≤ n ). Therefore, two identities are equivalent by induction. (cid:3) Corollary 4.2. The Hilbert series of J ∞ ( R W (Λ ) ) is given by X m ∈ N n ( n − / ≥ q B ( m ) Q ≤ i Then we conclude [Li20]: Corollary 4.3. The principal subspace W (Λ ) of sl n is classically free. Next result is needed in Section 6 and is proven using similar arguments as above. Proposition 4.4. For the case of A n , the coefficients of six types of terms in B ′ ( m ) − k ⊤ A k are: Type Coefficient Condition I 0II j − i ) + 1 III j − i ) IV 0V -1 i = i , i = n V -2 i < i < n VI n-i q -identities from quiver representations In this part we discuss identity (2) from a perspective of quiver representationsfollowing [RWY18]. We will use formulas from [RWY18] and several basic factsabout quiver representations. Indecomposable representations of the quiver of type A n − , Q , are in one-to-one correspondence with positive roots of A n − , and can beenumerated by segments [ i, j ], 1 ≤ i ≤ j ≤ n − Theorem 5.1. [RWY18, Corollary 1.5] Let Q be a quiver of type A n − . Then forevery k = ( k , ..., k n − ) ∈ N n − ≥ we have an identity n − Y i =1 ( q ) k i = X η q codim( η ) Y ≤ i ≤ j ≤ n − ( q ) m [ i,j ] ( η ) (25) where summation is over all finite-dimensional representations η (up to equivalence)of Q such that dim( η ) = k , codim( η ) is the co-dimension of a certain orbit, and m [ i,j ] ( η ) indicates the multiplicity of [ i, j ] in η . Recall another result from [RWY18] for codimensions: Lemma 5.2. codim( η ) = X [ I,J ] ∈ ConditionStrands m I · m J , where ConditionStrands = { [ I, J ] : [ I, J ] satisfies conditions (1) , (2) or (3) } , where strand is a pair I = [ a, b ] , ≤ a ≤ b ≤ n − (corresponding to indecomposablerep of Q ) and(1) I = [ w, x − , J = [ x, z ] , w < x ≤ z, e.g., w x − x z (2) I = [ w, y ] , J = [ x, z ] , w < x ≤ y < z and the arrows a x − and a y point inthe same direction, e.g., w yx z (3) I = [ x, y ] , J = [ w, z ] , w < x ≤ y < z and the arrows a x − and a y point indifferent directions, e.g., EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 19 w yx z For our purpose, we rewrite (25) as a family of identities in the shape of (5): X k =( k ,...,k n − ) ∈ N n − ≥ x k Q n − i =1 ( q ) k i = X η q codim( η ) x dim ( η ) Y ≤ i ≤ j ≤ n − ( q ) m [ i,j ] ( η ) , (26)and the summation is over all finite-dimensional representation η of A n − (up toisomorphism). Example 5.3 ( sl ) . Representation η of the A quiver of type dim( η ) = ( m, n ) isgiven by C m ◦ → C n ◦ . Indecomposable representations are given by [1 , 1] := C ◦ → ◦ ,[2 , 2] := ◦ → C ◦ , and [1 , 2] = C ◦ → C ◦ . Condition (1) is [1 , , [2 , 2] and there are nopairs with conditions two and three. Therefore we havecodim( η ) = m [1 , ( η ) · m [2 , ( η ) , and we are summing over all representations η with dim ( η ) = k = ( m, n ). Thatmeans 1( q ) m ( q ) n = X m [1 , ,m [2 , ,m [1 , ≥ m [1 , m [1 , mm [2 , m [1 , n q m [1 , · m [2 , q ) m [1 , ( q ) m [2 , ( q ) m [1 , , which is precisely formula (8) (we only have to rewrite m − n = n and n − n = n ,and use that summation variables are m [1 , = n , m [1 , = n and m [2 , = n ).Then we can prove the following: Theorem 5.4. The identities (26) and (2) are equivalent for A n ( n ≥ . Proof. We have proved the equivalence for A . We assume the direction of A n quiver is always from left to right, i.e, . Because of the orien-tation, there is no pair with condition (3) in codim( η ). Now we prove that thesetwo identities are equivalent for A n ( n ≥ 2) using induction. Assume that twoidentities are equivalent for A n − . For the case of A n , by identifying m [ i,j ] with m i,j +1 , and letting m i,i +1 ( m [ i,i ] ) be k i − X ≤ s It was briefly mentioned in [RWY18] that Keller’s quantum dilog-arithm identity for type A quivers [Kel11] is closely related to Theorem 5.1. Thispart can be viewed as a precise clarification of that claim.6. Level one principal subspace for B In this and next section we consider principal subspaces of two level one repre-sentation for which we have a well-known spinor realization. For more about levelone spinor representations for B (1) ℓ and D (1) ℓ affine Kac-Moody Lie algebras we referthe reader to [FF85].6.1. Spinor representation of B (1) ℓ . Level one affine vertex algebra of type B (1) ℓ has a spinor representation via 2 ℓ + 1 fermions. We take F ℓ ( Z + 1 / 2) := Λ( a i ( − / , a i ( − / , · · · , a ∗ i ( − / , a ∗ i ( − / , ... ; 1 ≤ i ≤ ℓ )and F = Λ( e ( − / , e ( − / , ... ). Then the even part of F ℓ ( Z + 1 / ⊗ F is isomorphic to affine vertex algebras L ( so (2 ℓ + 1)). It is easy to see from therealization that the principal subspace W (Λ ) ⊂ L (Λ ) is strongly generated bythe following fields: a i a j : , ≤ i < j ≤ ℓ : a i a ∗ j : 1 ≤ i < j, : a i e : , ≤ i ≤ ℓ, This gives ℓ ( ℓ − / ℓ ( ℓ − / ℓ = ℓ root vectors.6.2. Principal subspace of L ( so (5)) . For B (1)2 = C (1)2 , the principal subspace W (Λ ) is generated by : a a : , a a ∗ : , : a e : , : a e :corresponding to positive roots ǫ + ǫ , ǫ − ǫ , ǫ , ǫ , respectively. Its C -algebra, R W (Λ ) , is generated by x ǫ + ǫ , x ǫ − ǫ , x ǫ , x ǫ , and we have the following relationsin R W (Λ ) : x ǫ + ǫ = 0 , x ǫ − ǫ = 0 , x ǫ − x ǫ + ǫ x ǫ − ǫ = 0 , x ǫ x ǫ + ǫ = 0 ,x ǫ x ǫ − ǫ = 0 , x ǫ x ǫ + ǫ = 0 , x ǫ = 0 , x ǫ x ǫ = 0 . We denote by A the commutative algebra C [ x ǫ + ǫ , x ǫ − ǫ , x ǫ , x ǫ ]( x ǫ + ǫ , x ǫ − ǫ , x ǫ − x ǫ + ǫ x ǫ − ǫ , x ǫ x ǫ + ǫ , x ǫ x ǫ − ǫ , x ǫ x ǫ + ǫ , x ǫ , x ǫ x ǫ ) . Character formula. It is known that the character of the principal subspaceof L ( so (5)) is (given by [But14]):ch[ W (Λ )] = X r ,r ,r ≥ q r +( r + r ) + r − r ( r +2 r ) ( q ) r ( q ) r ( q ) r . (28)This identity is more complicated because x α ( z ) = 0 but x α ( z ) = 0.Let us introduce a filtration, G , on J ∞ ( A ) by letting G be generated with x ǫ + ǫ , x ǫ − ǫ , and G s = span (cid:8) ( x u ) ( i ) v | u ∈ { ǫ , ǫ } , v ∈ G s − (cid:9) + G s − . Then we have the following lemma. Lemma 6.1. The Hilbert series of the jet algebra of B := C [ x ǫ + ǫ , x ǫ − ǫ , x ǫ , x ǫ ]( x ǫ + ǫ , x ǫ − ǫ , x ǫ , x ǫ x ǫ + ǫ , x ǫ x ǫ − ǫ , x ǫ x ǫ + ǫ , x ǫ , x ǫ x ǫ ) , (29) is greater than or equal to HS q ( gr G ( J ∞ ( A ))) .Proof. Since both J ∞ ( B ) and gr G ( J ∞ ( A )) are generated by x ǫ + ǫ , x ǫ − ǫ , x ǫ , x ǫ ,it is enough to show that the quotient relations of J ∞ ( B ) also hold in gr G ( J ∞ ( A )). Note ( x ǫ + ǫ , x ǫ − ǫ , x ǫ x ǫ + ǫ , x ǫ x ǫ − ǫ , x ǫ x ǫ + ǫ , x ǫ , x ǫ x ǫ ) ∂ are valid in gr G ( J ∞ ( A )). For the quotient relations of J ∞ ( A ), ( x ǫ − x ǫ + ǫ x ǫ − ǫ ) ∂ ,we have ( x ǫ ) ∂ ⊂ G \ G , while ( x ǫ + ǫ x ǫ − ǫ ) ∂ ⊂ G . Therefore, quotient relations, ( x ǫ − x ǫ + ǫ x ǫ − ǫ ) ∂ , in J ∞ ( A ) give us ( x ǫ ) ∂ in gr G ( J ∞ ( A )). The result follows. (cid:3) Proposition 6.2. The Hilbert series of J ∞ ( B ) is less than or equal to X n ,n ,n ,n ,n ≥ q n + n +( n + n ) + n + n +(2 n + n )( n )+ n ( n + n )+ n n ( q ) n ( q ) n ( q ) n ( q ) n ( q ) n , Proof. We will prove that J ∞ ( B ) is spanned by a set of monomials whose characteris given by the q -series in the statement. We first observe that we can filter J ∞ ( B )with the number of ”particles” of type x ǫ . In the zero component of the filtrationwe have the jet algebra with relations( x ǫ + ǫ , x ǫ − ǫ , x ǫ , x ǫ x ǫ − ǫ , x ǫ x ǫ + ǫ ) ∂ whose Hilbert series is given by [Li20] X n ,n ,n ≥ q n + n + n + n ( n + n ) ( q ) n ( q ) n ( q ) n . Now we include x ǫ generator and additional relations. It is know that the jetalgebra of C [ x ǫ ] / ( x ǫ ) is known to admit a combinatorial basis satisfying differencetwo at the distance two condition as in Rogers-Selberg identities. Therefore, itsHilbert series is given by X n ,n ≥ q ( n + n ) + n ( q ) n ( q ) n . EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 23 This formula can be also explained using jet schemes by letting a = x ǫ and b = x ǫ and considering ( ab, a − b , a ) ∂ . Since a is of degree two, we get contribution2 n + 2 n n + n in the exponent.To finish the proof we have to analyze two additional relations and their contri-bution: x ǫ x ǫ + ǫ = 0 , and x ǫ x ǫ = 0 . The first relation contributes with the boundary condition (2 n + n ) n in the q -exponent as (2 n + n ) counts the power of x ǫ in ( x ǫ ) n x n ǫ and we have relation x ǫ x ǫ + ǫ = 0 ( n summation variable corresponds to x ǫ + ǫ ). For the secondrelation, using x ǫ x ǫ = 0 and x ǫ x ǫ = 0, we get a spanning set of monomialsinvolving a = x ǫ and x ǫ to be (cid:8) a ( − n i ) · · · a ( − n ) ( x ǫ ) ( m j ) · · · ( x ǫ ) ( m ) | j + 1 ≤ n , n s − n s − ≥ , m s − m s − ≥ (cid:9) , where j + 1 ≤ n (boundary condition), n s − n s − ≥ , m s − m s − ≥ x ǫ x ǫ = 0, x ǫ = 0 and x ǫ = 0, respectively. Thenthe character of the space spanned by these monomials is at most X n ,n ≥ q n +2 n + n n ( q ) n ( q ) n . If we combining these arguments together, we obtain the result. (cid:3) Remark 6.3. It is possible to prove the equality in Proposition 6.2 but this involvesfurther analysis as in [CLM10] using ”simple currents” and ”intertwining maps” indisguise.The Hilbert series of J ∞ ( B ) is greater than or equal to ch[ W (Λ )]. Using Math-ematica, we checked that (6.2) and the character formula (28) agree up to O ( q ).Thus, we expect the following stronger result using charge variables: Conjecture 6.4. We have the identity X r ,r ,r ≥ y r y r + r q r +( r + r ) + r − r ( r +2 r ) ( q ) r ( q ) r ( q ) r = X n ,n ,n ,n ,n ≥ y n + n + n y n +2 n + n + n q n + n +( n + n ) + n + n +(2 n + n )( n )+ n ( n + n )+ n n ( q ) n ( q ) n ( q ) n ( q ) n ( q ) n Provided Conjecture 6.4 is true, then we have Corollary 6.5. The C -algebra of the principal subspace of L ( so (5)) , W (Λ ) , isisomorphic to A , and the principal subspace is classically free.Proof. We can use the same filtration G on J ∞ ( R W (Λ ) ). Then using the surjec-tivity of ψ and previous discussion we havech[ W (Λ )]( q ) ≤ HS q ( J ∞ ( R W (Λ ) )) = HS q ( gr G ( J ∞ ( R W (Λ ) ))) ≤ HS q ( gr G ( J ∞ ( A )) , Also from Lemma 6.1, HS q ( gr G ( J ∞ ( A )) ≤ HS q ( J ∞ ( B )) = ch[ W (Λ )] , where in the last equation we use Conjecture 6.4. Combining these inequalitiestogether, it follows that the character and all intermediate Hilbert series of jetalgebras are equal, therefore, ch[ W (Λ )]( q ) = HS q ( J ∞ ( R W (Λ ) )) and ψ is injective. (cid:3) We believe that the q -series identity (6.4) is new and does not come from a simpleapplication of the pentagon identity for the quantum dilogarithm. Perhaps it is aconsequence of a more complicated relation which involves both φ ( x ) and φ ( x ).For instance, the coefficients on the left-hand side of the identity can be extractedfrom the coefficients of φ ( x ) − φ ( − yq ) φ ( y ) − . After specialization y = y = 1, we expect the following elegant q -series identitywith a modular product side. Conjecture 6.6. X r ,r ,r ≥ q r +( r + r ) + r − r ( r +2 r ) ( q ) r ( q ) r ( q ) r = ( − q ; q ) ∞ ( − q ; q ) ∞ ( q, q ; q ) ∞ . Interestingly, this conjecture would imply(30) ch[ W B (Λ )]( τ ) = ch[ W A (Λ )]( τ ) · ch[ W A (Λ )]( τ ) , where the right hand side is the product of characters of the level one principalsubspace for A and A vertex algebras.7. Level one principal subspace of D Spinor representation of D (1) ℓ . Level one VOA of D (1) ℓ has a spinor repre-sentation via 2 ℓ fermions. We take F ℓ ( Z + 1 / 2) = Λ( a i ( − / , a i ( − / , · · · , a ∗ i ( − / , a ∗ i ( − / , ... ; 1 ≤ i ≤ ℓ )and F = Λ( e ( − / , e ( − / , ... ). Then the even part of F ℓ ( Z + 1 / L so (2 ℓ ) (Λ ). The principal subspace W (Λ ) ⊂ L (Λ ) is stronglygenerated by : a i a j : , ≤ i < j ≤ ℓ : a i a ∗ j : 1 ≤ i < j ≤ ℓ, all together ℓ ( ℓ − 1) = ℓ root vectors.For D (1)4 , the principal subspace is generated by: a i a j : a i a ∗ j : (1 ≤ i < j ≤ , corresponding to ǫ i + ǫ j , ǫ i − ǫ j , respectively. And its C -algebra, R W (Λ ) , is gener-ated by x ǫ i + ǫ j , x ǫ i − ǫ j (1 ≤ i < j ≤ W ij and V ij , respectively.And we have the following relations in the quotient of R W (Λ ) : EIGIN-STOYANOVSKY’S PRINCIPAL SUBSPACES 25 W W , W W , W W , W W , W W , W W , W W ,W W , W W , W W , W W , W W , W W , W W ,W W , − W W = W W = W W , W V , W V , W V ,W V , W V , W V , W V , W V , W V , W V ,W V , W V , W V = W V = W V , W W = W W , − W V = W V = W V , − W V = W V , V V , V V , V V ,V V , V V , V V , V V , V V , V V , V V , V V , − V V = V V , W V + W V = W V + W V . We denote by D the algebra C [ W ij , V ij | ≤ i < j ≤ /I , where I is generatedby above quadratic relations. We expect that the algebra D is the C -algebra of R W (Λ ) . Jet algebra and quantum dilogarithm. We first assume x x = qx x , x x = qx x , x x = qx x . Then by using properties of quantum dilogarithm we have φ ( − q x ) φ ( − q x ) φ ( − q x ) φ ( − q x )= φ ( − q x ) φ ( − qx x ) φ ( − q x x x ) φ ( − q x x x ) φ ( − q x ) φ ( − q x x x x x ) φ ( − q x x x x ) φ ( − qx x ) φ ( − qx x ) φ ( − q x x x ) φ ( − q x ) φ ( − q x ) . It is equivalent with X ( m , n ) ∈ N q B ( m , n ) x λ x λ x λ x λ Q ≤ i The following two statements are equivalent: i The Hilbert series of J ∞ ( R W (Λ )) equals X ( m , n ) ∈ N q B ( m , n ) Q ≤ i For algebra D , we can choose filtration G and G by letting G and G be generated with { V , V } and { V , W } , respectively. And let G s = span (cid:8) ( W ij ) ( n ) v, ( V ij ) ( n ) v | n ≤ , V ij = V , or V , v ∈ G s − (cid:9) + G s − , and G s = span (cid:8) ( W ij ) ( n ) v, ( V ij ) ( n ) v | n ≤ , V ij = V , W ij = W , v ∈ G s − (cid:9) + G s − . 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