A New Realization of Matrix Bialgebra Associated to A Hecke Operator

Phùng Hô` Ha’i

Hanoi Institute of Mathematics, P.O.Box 631, 10000 Boho, Hanoi and

Max-Planck Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany

                        Email: phung@ioit.ncst.ac.vn

0. Introduction

Let V be a vector space of finite dimension over a field of characteristic zero. There are two ways of interpreting the symmetric tensor algebra over V. The first one is to consider it as a factor algebra of the tensor algebra T (V) over V. Thus we have a relation between two tensors, like a ⊗ b = b ⊗ a. The second way is to consider it as a subalgebra of T (V) consisting of symmetrized tensors, e.g., for a, b ∈ V, the element a ⊗ b + b ⊗ a belongs to the symmetric tensor algebra on V. We are thus speaking of two realizations of the symmetric tensor algebras over V. The analogous procedure applies also for the exterior (anti-symmetric) tensor algebra.

For a quantum space associated to a Hecke operator, the two realizations for its (quantum) symmetric and anti-symmetric tensor algebras was first obtained by Gurevich [1]. While the first realization was taken as the definition, the second realizations follows from the general theory of Hecke algebras. The aim of this paper is to give the two realization for the (anti-) symmetric tensor algebras on the quantum semi-group of endormorphism (associated to a Hecke operator).

The main idea in giving the second realization is to construct a projector on the tensor products of V. For the classical case, it is the (anti-) symmetrizer operator, e.g., a ⊗ b ⟶ (a ⊗ b + b ⊗ a)/2. For the quantum space, it is the quantum (anti-) symmetrizer, constructed in terms of the trivial and signature representations of the Hecke algebras. The direct generalization of the latter operators to the case of quantum semi-groups fails, these operators are no more projectors. Moreover the eigenvalues of these operators a known only for small ranks.

By a simple observation (Equation (1)), we are able to find the eigenvalues of these operators and then we can modify them to get projectors.

The paper is organised as follows. In Section 1 we recall the definitions and the two realizations of the quantum (anti-) symmetric algebras associated to a Hecke operator. In Section 2 we introduce the two operators Φ and Ψ and find out they eigevalues. Then we define the projector Φ̄ and Ψ̄ and give the second realization for the quantum (anti-) symmetric algebras on the quantum semi-group of endomorphisms. In the last section we define the (anti-) symmetric algebras on the quantum space of homomorphism of two quantum spaces. The second realization of these algebras is also given. 0.1. Preliminaries Partitions A partitions λ of n ∈ ℕ is a sequence λ = (λ_{1}, λ_{2}, ..., λ_{r}, ...) of nondecreasing non-negative integers, whose sum is n, we write |λ| = n. The maximal number r, for which λ_{r} ≠ 0, is called the length of λ. The diagram [λ] associated to λ is a matrix, whose first row contains λ_{1} elements, second row contains λ_{2} elements, and so on. The conjugated to λ, denoted by λ′, is the one, whose diagram [λ′] is obtained from [λ] by rotating it 180^{∘} along its diagonal.

A standard λ-tableau is the diagram [λ], filled by numbers 1, 2, ..., |λ|, in such a way that they increase along rows and columns. The number of standard tableaux is

d_{λ} = ∏_{x∈[λ]} \frac{n!}{h_{λ}(x)}

where the product is taken over all nodes x of the diagram λ, h_{λ}(x) is the hooklength of the node x: if x is in the i-th row and j-th column of [λ], then h_{λ}(x) := λ_{i} + λ′_{j} − i − i + 1. Symmetric Groups The symmetric group 𝔖_{n} consists of permutations of the set 1, 2, …, n. It can be regarded as group generated by transposition v_{i} = (i, i + 1), 1 ≤ i ≤ n − 1, subject to the relations: v_{i}^{2} = 1, v_{i}v_{i+1}v_{i} = v_{i+1}v_{i}v_{i+1}. The length of l(w) of an element w is the minimal length of the words in v_{i} expressing w. It equal the number of pairs 1 ≤ i < j ≤ n for which wi > wj. Hecke Algebras The Hecke algebra H_{n} = H_{n,q} is the q-analogue of the group algebra 𝕂[𝔖_{n}]. It is generated over 𝕂 by 1 and the elements T_{i}, 1 ≤ i ≤ n − 1, subject to the relations T_{i}^{2} = (q − 1)T_{i} + q and T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}. When q = 1, H_{n,1} reduces to 𝕂[𝔖_{n}], T_{i} ⟶ v_{i}. H_{n} has a basis consisting of T_{w}, w ∈ 𝔖_{n}, T_{1} := 1, T_{v_{i}} := T_{i}, T_{w}T_{u} = T_{wu} if l(w) + l(u) = l(wu).

We shall always assume that q^{n} ≠ 1, ∀n > 1. In this case H_{n,q} is semisimple. Representations of the Hecke Algebras Representations of H_{n,q}, for q not being root of unity, can be parameterized by partitions of n. Let S_{λ} be the simple representation of H_{n}, corresponding to λ, then the dimension of S_{λ} is d_{λ}.

We shall also consider the algebra H_{n}^{op}. Its simple comodules are canonically identified with S_{λ}^{∗}: if ϕ_{λ} is the representation of H_{n} on S_{λ}, then the representation of H_{n}^{op} on S_{λ}^{∗} is given by ϕ̄_{λ}(w) := ϕ_{λ}(w)^{∗}.

Let A_{λ} be the block in H_{n}, that corresponds to λ. Let E_{λ}^{ij}, 1 ≤ i, j ≤ d_{λ} be a basis of A_{λ} such that E_{λ}^{ij}E_{λ}^{kl} = δ_{k}^{j}E_{λ}^{il}. Thus, E_{λ}^{ii} are mutually orthogonal primitive idempontents of H_{n}. Notice that for λ ≠ μ, E_{λ}^{ij}E_{μ}^{kl} = 0. A bilinear form on H_{n} There exists a non-degenerate, symmetric, associative bilinear form on H_{n}, defined as follows: (T_{u}, T_{w}) := q^{l(u)}δ_{w}^{u}. Thus, we see that {q^{−l(w)}T_{w^{−1}}, w ∈ 𝔖_{n}} is the dual basis to {T_{w}, w ∈ 𝔖_{n}} with respect to this bilinear form. Therefore, the Casimir element ∑_{w∈𝔖_{n}} q^{−l(w)}T_{w} ⊗ T_{w^{−1}} is central:

∑_{w∈𝔖_{n}} q^{−l(w)}T_{i}T_{w} ⊗ T_{w^{−1}} = ∑_{w∈𝔖_{n}} q^{−l(w)}T_{w} ⊗ T_{w^{−1}}T_{i}.

Since A_{λ} is simple, the bilinear form restricted on A_{λ} should satisfy

(E_{λ}^{ij}, E_{λ}^{kl}) = δ_{k}^{j}δ_{l}^{i}k_{λ}

for certain coefficient k_{λ}. k_{λ} is given explicitly by k_{λ} = ∏_{x∈[λ]}[h_{λ}(x)]_{q}^{−1}, where [n]_{q} := (q^{n} − 1)/(q − 1). Thus {E_{λ}^{ij}, λ ⊢ n, 1 ≤ i, j ≤ d_{λ}} and {k_{λ}^{−1}E_{λ}^{ji}, λ ⊢ n, 1 ≤ i, j ≤ d_{λ}} are dual basis. Therefore

∑_{w∈𝔖_{n}} q^{−l(w)}T_{w} ⊗ T_{w^{−1}} = ∑_{{λ⊢n \atop 1≤i,j≤d_{λ}}} k_{λ}^{−1}E_{λ}^{ij} ⊗ E_{λ}^{ji}.  (1)

Hecke Operators Let V be a finite dimensional vector space over a field 𝕂 and R be a invertible operator on V ⊗ V. R is called Hecke operator if the following conditions are satisfied:

R_{1}R_{2}R_{1} = R_{2}R_{1}R_{2} where R_{1} := R ⊗ id_{V} , R_{2} := id_{V} ⊗ R,
(R + 1)(R − q) = 0.

It is easy to see that a Hecke operator induces a represetation ρ_{n} of H_{n,q} oo V^{⊗n}, for n > 1: ρ_{n}(T)i) = R_{i} := id_{V}^{⊗i−1} ⊗ R ⊗ id_{V}^{⊗n−i−1}. For any w ∈ 𝔖_{n}, we denote R_{w} := ρ_{n}(T_{w}).

1. Matrix Quantum Semigroups of Type A

1.1. Quadratic Algebras Let V be a vector space over 𝕂 of finite dimension. Let R be a subspace of V ⊗V. Let A = A(V, R) be the factor algebra of the tensor algebra on V by the two-sided ideal generated by elements of R:

A(V, R) := T (V)/(R).

Such an algebra is called quadratic algebra [4]. R is called the space of relations of A. The two-sided ideal, generated by R, is usually denoted by R(A). Since the relation on A are homogeneous, A inherits the grading from T (V)

A = ⨁_{n=0}^{∞} A_{n}, A_{n} = V^{⊗n}/R^{n}(A)

R^{n}(A) = ∑_{i=1}^{n−1} R(A)_{i}^{n}, R(A)_{i}^{n} := V^{i−1} ⊗ R ⊗ V^{n−i−1}, 1 ≤ i ≤ n − 1.

The Poincaré (or Hilbert) series of A is by definition the formal power series P_{A}(t) := ∑_{k=0}^{∞} dim_{𝕂}A_{k}. The dual quadratic algebra to A, A^{!} is defined to be A^{!} := T (V^{∗})/(R(A)^{⊥}). It is easy to see that (A_{n}^{!})^{∗} = ⋂_{i=1}^{n−1} R(A)_{i}^{n}. 1.2. Quantum Spaces and Quantum Endomorphism Rings Let R be a Hecke operator on V. We define the following quadratic algebras

S = S_{R} := T (V)/(Im (R − q)),  (2)
Λ = Λ_{R} := T (V)/(Im (R + 1)).  (3)

S_{R} and Λ_{R} are considered as the function algebra and the exterior algebra on a quantum space. The function algebra on the quantum semi-group of endomorphisms of this quantum space is defined to be a quadratic algebra on V^{∗} ⊗ V:

E = E_{R} := T (V^{∗} ⊗ V)/(Im (R̄ − 1)).  (4)

where R̄ := S_{(23)}(R^{∗−1} ⊗ R)S_{(23)}, acting on (V^{∗} ⊗ V)^{⊗2}. One can check that

R(E) = S_{(23)} (R(S)^{⊥} ⊗ R(S) ⊕ R(Λ)^{⊥} ⊗ R(Λ)) .

Using S_{(23)} we shall identify R̄ with R^{∗−1} ⊗ R acting on V^{∗⊗2} ⊗ V^{⊗2}.

E is a bialgebra with the coproduct Δ(E_{n}) ⊂ E_{n}⊗E_{n}, i.e., E_{n} is a coalgebra. Λ_{n}, S_{n} are E_{n}-comodules, they are factor comodules of V^{⊗n}. Thus E_{n}^{∗} is an algebra and acts on these spaces. On the other hand, the operator R induces an action ρ_{n} on V^{⊗n}. We have a quantum analogue of Schur’s double centralizers theorem [2]:

E_{n}^{∗} ≅ End_{H_{n}}(V^{⊗n})  (5)
ρ_{n}(H_{n}) ≅ End_{E_{n}^{∗}}(V^{⊗n}) = End^{E_{n}}(V^{⊗n}).  (6)

As a consequence, simple E_{n} comodules are parameterized by primitive idempontents of H_{n}, which, in their order, are parameterized by partitions of n. For each primitive idempotents E_{λ} of H_{n}, ρ_{n}(E_{λ}) is either zero of a simple E_{n}-comodule, conjugated idempontents define isomorphism comodules. In particular, as E_{n}- comodules

Λ_{n} ≅ Im ρ_{n}(Y_{n}), S_{n} ≅ Im ρ_{n}(X_{n}),

where X_{n} = ([n]_{q}!)^{−1} ∑_{w∈𝔖_{n}} T_{w} , Y_{n} = ([n]_{q^{−1}}!)^{−1} ∑_{w∈𝔖_{n}} q^{−l(w)}T_{w}, the symmetrizer and anti-symmetrizer operators. Moreover we have an isomorphism of algebras

S ≅ ⨁_{n=0}^{∞} ρ_{n}(X_{n}) Λ ≅ ⨁_{n=0}^{∞} ρ_{n}(X_{n})  (7)

where the product on the spaces on the right-hand sides of these isomorphisms are given by

x ∈ ρ(X_{n}), y ∈ ρ(X_{m}) ⟶ x · y := ρ_{m+n}(X_{m+n})(x ⊗ y)  (8)
x ∈ ρ(Y_{n}), y ∈ ρ(Y_{m}) ⟶ x · y := ρ_{m+n}(Y_{m+n})(x ⊗ y).  (9)

This is the second realization of S and Λ, first considered by Gurevich [1]. Further, we have

Im ρ_{n}(X_{n}) = ⋂_{i=1}^{n−1} R(Λ)_{i}^{n}, Im ρ_{n}(Y_{n}) = ⋂_{i=1}^{n−1} R(S)_{i}^{n}.  (10)

This means S is isomorphic to Λ^{!∗}, Λ is isomorphic to S^{!∗} as graded space, here ∗ means graded dual.

A problem arises naturally, to get an analogous realizations for the algebra E. In studying the Poincaré series of E, Sudbery [5] introduced an operator Φ,

acting on E_{1}^{⊗3}. It is an analogue of the symmetrizer X_{n}, defined for −R̄. The study was further developed in [2], where a complete description of the Poincaré series of E was given, in terms of the Poincaré series of Λ. The following operator plays a role:

Φ^{n} := ∑_{w∈𝔖_{n}}(−1)^{l(w)}R̄_{w}.  (11)

As it has been mentioned, we identity R̄_{w} with R_{w}^{∗−1} ⊗ R_{w}, acting on V^{∗⊗n} ⊗ V^{⊗n}. For n = 3, Φ^{3} is the one introduced by Sudbery. It was shown that

Im Φ^{n} = ⋂_{i=1}^{n−1} R(E)_{i}^{n},  (12)

for transcendent q, from which follows

P_{E}(t) = P_{S} ∘ P_{Λ},  (13)

“∘” is the product in the λ-ring 𝕂_{0}[[t]] of power series with constant coefficient equal to 1. Although, later, Equation (13) was proved for all q not root of unity [3], (12) wasn’t proved for non-transcendent q. One of the aims of the next section is to prove (12).

2. Another Realization for The Algebra E

2.1. The Operator Φ^{n} All needed information on the operator Φ^{n} can be obtained form the following observation. Let us denote S := qR^{−1}. Then S is a Hecke operator, too. We have R̄ = −q^{−1}S^{∗} ⊗ R. Let σ_{n} be the representation of H_{n}^{op} on V^{∗⊗n}, σ(T_{i}) = S_{i}^{∗}, thus, σ_{n}(T_{w}) = (S_{w})^{∗} = S^{∗}_{w^{−1}}. Applying σ_{n} ⊗ ρ_{n} on the identity (1), we get

∑_{s∈𝔖_{n}} q^{−l(w)}S^{∗}_{w^{−1}} ⊗ R_{w} = ∑_{s∈𝔖_{n}} σ_{n}(E_{λ}^{ij}) ⊗ ρ_{n}(E_{λ}^{ji}).  (14)

Therefore

∑_{s∈𝔖_{n}}(−1)^{l(w)}R̄_{w} = ∑_{s∈𝔖_{n}} q^{l(w)}S^{∗}_{w^{−1}} ⊗ R_{w}
= ∑_{s∈𝔖_{n}} k_{λ}^{−1}σ_{n}(E_{λ}^{ij})^{∗} ⊗ ρ_{n}(E_{λ}^{ji}).  (15)

Denote Φ_{λ} := ∑_{1≤i,j≤d_{λ}} k_{λ}^{−1}σ_{n}(E_{λ}^{ij}) ⊗ ρ_{n}(E_{λ}^{ji}). Then we have

Φ_{λ}^{2} = d_{λ}k_{λ}^{−1}Φ_{λ}.  (16)

Thus, the spectrum of Φ^{n} is contained in {d_{λ}k_{λ}^{−1}|λ ⊢ n} ∪ 0. Note that if q = 1, d_{λ}k_{λ}^{−1} = n!.

V^{⊗n} decomposes into H_{n}-modules as follows:

V^{⊗n} ≅ ⨁_{λ⊢n} V_{λ}, V_{λ} ≅ ⨁_{{λ⊢n \atop 1≤i≤l_{λ}}} S_{i,λ}.  (17)

The module V_{λ} is determined uniquely, while S_{i,λ} ≅ S_{λ}, ∀i, l_{λ} = dim Im ρ_{n}(E_{λ}). Therefore, as H_{n}^{op} ⊗ H_{n}-module, V^{∗⊗n} ⊗ V^{⊗n} decomposes as

V^{∗⊗n} ⊗ V^{⊗n} ≅ ⨁_{λ,μ⊢n} V_{λ}^{∗} ⊗ V_{μ} ≅ ⨁_{λ,μ⊢n ; {1≤i≤l_{λ} \atop 1≤j≤lμ}} S_{j,μ}^{∗} ⊗ S_{i,λ}.  (18)

Notice, that as an H_{n}-module, S_{λ} is isomorphic to the subspace of H_{n} spanned by {E_{λ}^{mj}|1 ≤ m ≤ d_{λ}}, for each fixed j. Analogously, {E_{λ}^{jm}|1 ≤ m ≤ d_{λ}} is isomorphic to S_{λ}^{∗} as H_{n}^{op}-module, for each fixed j. Therefore, we have Lemma 2.1 If l_{λ} := dim Im ρ_{n}(E_{λ}) ≠ 0, i.e., the E_{n}-comodule corresponding to λ is not zero, then the operator Φ_{λ} has rank 1 on S_{λ}^{∗} ⊗ S_{λ} and vanishes on S_{μ}^{∗} ⊗ S_{ν}, for μ ≠ λ or ν ≠ λ.

Proof. Fix i, j and let {E_{λ}^{im} ⊗ E_{λ}^{ni}|1 ≤ m, n ≤ l_{λ}} be a basis of S_{λ}^{∗} ⊗ S_{λ}. Then

Φ_{λ}(E_{λ}^{im} ⊗ E_{λ}^{ni}) = δ_{n}^{m}k_{λ}^{−1} ∑_{1≤k≤l_{λ}} E_{λ}^{ik} ⊗ E_{λ}^{ki}

Consequently, as E_{n}^{op} ⊗ E_{n}-comodules

Im (Φ_{λ}) ≅ Im σ_{n}(E_{λ}) ⊗ Im ρ_{n}(E_{λ}),
Im (Φ^{n}) ≅ ⨁_{λ⊢n} Im σ_{n}(E_{λ}) ⊗ Im ρ_{n}(E_{λ}).  (19)

2.2. The Operator Ψ^{n} In (4), we see that E_{2} is isomorphic to Im (R̃ + q), where R̃ := S_{(23)} (R^{∗} ⊗ R) S_{(23)}. Let us define

Ψ^{n} := ∑_{s∈𝔖_{n}} q^{−l(w)}R̃_{w}.

We shall also identify R̃ with R^{∗} ⊗R. Let τ be the representation of H_{n}^{op} on V^{∗⊗n}, induced by R^{∗}. In analogy to (15), we have

∑_{s∈𝔖_{n}} q^{l(w)}R̃_{w} = ∑_{s∈𝔖_{n}} k_{λ}^{−1}τ_{n}(E_{λ}^{ij}) ⊗ ρ_{n}(E_{λ}^{ji}).

Like for Φ_{λ}, we have:

Lemma 2.2 If l_{λ} ≠ 0, then the operator Ψ^{n} has rank 1 on S_{λ}^{∗} ⊗ S_{λ} and vanishes on S_{ν}^{∗} ⊗ S_{ν}, for μ ≠ λ or ν ≠ λ. Consequently, we have and isomorphism of E_{n}^{op} ⊗ E_{n} comodules

Im Ψ^{n} = ⨁_{λ⊢n} Im τ_{n}(E_{λ}) ⊗ Im ρ_{n}(E_{λ}).  (20)

On the other hand, according to (6),

E_{n}^{∗} ≅ ⨁_{λ⊢n} End_{A_{λ}}(V^{⊗n}) ≅ ⨁_{λ⊢n} Im ρ_{n}(E_{λ}) ⊗ Im ρ_{n}(E_{λ})^{∗}.

Consequently, we have

E_{n} ≅ ⨁_{λ⊢n} Im ρ_{n}(E_{λ})^{∗} ⊗ Im ρ_{n}(E_{λ}).  (21)

Therefore

E_{n} ≅ Im Ψ^{n},  (22)

as E_{n}^{cop} ⊗ E_{n}. The operator Ψ^{n} is not a projector, so that, in order to get a second realization for E_{n}, we need the following operator.

Ψ̄^{n} := ∑_{λ} k_{λ}d_{λ}^{−1}τ_{n}(E_{λ}^{ij})^{∗} ⊗ ρ_{n}(E_{λ}^{ji}).

From (16) we see that Ψ̄^{n} is a projector, having the same image as Ψ^{n}. We define a multiplication on ⨁_{n=0}^{∞} Im Ψ^{n}, setting, for a ∈ V^{∗⊗n} ⊗ V^{⊗n} and b ∈ V^{∗⊗m} ⊗ V^{⊗m}, a · b := Ψ̄^{n+m}(a ⊗ b).

Theorem 2.3 The algebra E is isomorphic as a graded algebra to ⨁_{n=0}^{∞} Im Ψ^{n}.

Proof. We have (R̃ + q)(R̄ − 1) = (R̄ − 1)(R̃ + q) = 0. Therefore, if x ∈ Im (R̄_{i} − 1) then (R̃ + q)x = 0, consequently, Ψ^{n} vanishes on ∑_{i=1}^{n−1} R(E)_{i}^{n}. Then vanishes Ψ̄^{n} on ∑_{i=1}^{n−1} R(E)_{i}^{n}, too. Taking (22) into account, we conclude that ∑_{i=1}^{n−1} R(E)_{i}^{n} is precisely the Kernel of Ψ̄^{n}.

Thus, the linear map ψ : T (V^{∗} ⊗ V) ⟶ ⨁_{n=0}^{∞} Im Ψ̄^{n}, a ⟼ Ψ̄^{n}(a) has the following properties

\begin{matrix}a ∈ R(E) ⟹ ψ(a) = 0 \\ a = ψ̄(a) mod R(E). \end{matrix}  (23)

Therefore, ψ factorizes through R(E) to a map ψ : E ⟶ ⨁_{n=0}^{∞} Im Ψ̄^{n}. Let a ∈ (V^{∗} ⊗ V)^{⊗m}, b ∈ (V^{∗} ⊗ V)^{⊗n}. According to (23), we have

Ψ̄^{m}(a) = a mod R(E)^{m} Ψ̄^{n}(b) = a mod R(E)^{n},

hence

Ψ̄^{m}(a) ⊗ Ψ̄^{n}(b) = a ⊗ b mod R(E)^{m+n}.

Consequently,

Ψ̄^{m+n}(Ψ̄^{m}(a) ⊗ Ψ̄^{n}(b)) = Ψ̄^{m+n}(a ⊗ b),  (24)

meaning that ψ is an algebra homomorphism and therefore isomorphism E ⟶ ⨁_{n=0}^{∞} Im Ψ̄^{n}.

We now proceed to show that

Im Φ^{n} = ⋂_{i=1}^{n=1} R(E)_{1}^{n}.  (25)

The inclusion “⊂” is obviously, for Φ^{n} = (R̄_{i} − 1)Φ_{i}^{n} for certain operator Φ_{i}^{n}.

Let us denote B_{n} := ⋂_{i=1}^{n=1} R(E)_{1}^{n}. To show the equation, we compare the dimension of B_{n} and Im Φ^{n}. Accoriding to (19), the dimension of Im Φ^{n} is equal to ∑_{λ⊢n} dim Im σ_{n}(E_{λ})dim Im ρ_{n}(E_{λ}). From the defition of the operator S, we see that σ_{n}(E_{λ}) ≅ ρ_{n}(E_{λ′}) as vector spaces. Therefore dim Im Φ^{n} = ∑_{λ⊢n} l_{λ′}l_{λ}.

On the other hand, since E is a Koszul algebra [2]

P_{E}(t)P_{E^{!}}(−t) = 1.  (26)

Since dim (E_{n}) = ∑_{λ⊢n} l_{λ}^{2}, we also have

dim ⋂_{i=1}^{n−1} R(E)_{i}^{n} = dim R(E_{n}^{!})^{∗} = dim R(E_{n}^{!}) = ∑_{λ⊢n} l_{λ′}l_{λ}.  (27)

Thus, dim Im Φ^{n} = dim B_{n}. Whence (25) follows.

Remark 2.4 The proof of (25) above is not self-contained, it uses the fact that E is a Koszul algebra. The other proof might be as follows. One shows directly the exactness of the complex

0 → Im Φ^{n} → Im Φ^{n−1} ⊗ E_{1} → Im Φ^{n−2} ⊗ E_{2} → ⋯ → E_{n} → 0  (28)

for n > 1, from which both (25) and the Koszul property of E would follow. The exactness of this complex seems to have close connection with the Littlewood-Richardson coefficients.

Finally, let us denote by F the quadratic algebra on V^{∗} ⊗ V with relation R(F) := Im (R̃ + q). Thus, F can be considered as the exterior algebra over the matrix quantum semi-group. We have

R(F) = S_{(23)} (R(S)^{⊥} ⊗ R(Λ) ⨁ R(Λ)^{⊥} ⊗ R(S)) .  (29)

The vector space V^{∗} ⊗ V is self dual with respect to the pairing (ϕ ⊗ x, ψ ⊗ y) := (ϕ, y)(ψ, x). With respect to this pairing, F is canonically isomorphic to E^{!}. Therefore F is Koszul algebra and

P_{F}(t) = P_{E}(−t)^{−1} = P_{Λ}(t) ∘ P_{S}(t),  (30)

or equivalently

dim F_{n} = ∑_{λ⊢n} l_{λ}l_{λ′}.  (31)

Decompose (V^{∗} ⊗ V)^{⊗n} (which is identified with V^{∗⊗n} ⊗ V^{⊗n}) into simple H_{n}^{op} ⊗ H_{n}-modules (with the action given by τ_{n} ⊗ ρ_{n}). Notice that, as in the case of E, F_{n} and R(F)^{n} are H_{n}^{op} ⊗ H_{n}-modules. Recall that the operator Φ^{n} also has rank ∑_{λ⊢n} l_{λ}l_{λ′} on V^{∗⊗n} ⊗V^{⊗n} and vanishes on R(F)^{n}. Comparing the dimension we see that R(F)^{n} is precisely the kernel of Φ^{n}. Analogously, the image of Ψ^{n} is ⋂_{i=1}^{n−1} R(F)_{i}^{n}. Thus we proved an analogous of Theorem 2.3:

Theorem 2.5 The algebra F is a Koszul algebra and can be realized as ⨁_{n=0}^{∞} Im Φ̄^{n}.

3. The Quantum Spaces of Homomorphisms

Let R and S be Hecke operators on V and W, respectively. We define the quadratic algebra M := M_{RS} on W^{∗} ⊗ V with

R(M) := S_{(23)} ((R_{S}(S) ⊥ ⊗R_{R}(S) ⊕ R_{S}(Λ)^{⊥} ⊗ R_{R}(Λ)) ,  (32)

where, as usual, R_{S}(S) := Im (S − q), R_{R}(S) := Im (R − q), and so on. We have

R(M) = Im (S_{(23)}(S^{∗−1} ⊗ R)S_{(23)}).  (33)

As in the previous section, we shall identify the two vector space (W^{∗} ⊗ V)^{⊗n} and W^{∗⊗n} ⊗ V^{⊗n}.

Notice that W^{∗} ⊗ V ≅ Hom (V, W)^{∗}. The algebra M can be interpreted as the function algebra on the quantum space of homomorphisms from the quantum space associated to R to the one associated to S. Let x_{1}, ⋯, x_{m} be a basis of V and ξ^{1}, ⋯, ξ^{n} be a basis of W^{∗}. Then {z_{j}^{i} := ξ^{i} ⊗ x_{j}} form a basis of W^{∗} ⊗ V. The algebra M = M_{RS} is then isomorphic to

𝕂⟨z_{j}^{i}|1 ≤ i ≤ m, 1 ≤ j ≤ n⟩/(SZ_{1}Z_{2} − Z_{1}Z_{2}R)  (34)

where Z_{1} := Z ⊗ id(n), Z_{2} := id(m) ⊗ Z. M has the following properties: let A := (a_{1}, a_{2}, …, a_{m}) be a point of S_{S}, i.e., (R − q)(A ⊗ A) = 0, and T = (t_{j}^{i}) be a point of M_{RS}, such that a_{i} and t_{l}^{k} commute. Then A · T is a point of S_{R}. Analogously, if B is a point of Λ_{S} then B · T is a point of Λ_{R}.

There is also an interpretation of M_{RS} from the categorical view-point. Remember that the algebra E can be constructed as the Coend of the functor 𝔉 from the braided monoidal category V, generated by one object v and one morphism τ : v^{⊗2} ⟶ v^{⊗2}, into the category of vector space, such that 𝔉(v) = V and 𝔉(τ) = R. We then have, for any vector space X,

Nat (𝔉, 𝔉 ⊗ X) ≅ Hom (E, X).  (35)

Let us now consider another function 𝔊, with 𝔊(v) = W and 𝔊(τ) = S. Then we have

Nat (𝔉, 𝔊 ⊗ X) ≅ Hom (M_{RS}, X).  (36)

The exterior algebra on the quantum space of homomorphism is defined to be

N = N_{RS} := T (W^{∗} ⊗ V)/Im (S_{(23)}(S^{∗} ⊗ R)S_{(23)} + q).  (37)

There exist coactions of E^{cop,op} ⊗ E on M and N.

Here, we are interested in the Poincaré series, the Koszul property and the second realization of M and N. First notice that N_{RS} is canonically isomorphic to M_{SR}^{!}. Therefore it is sufficient to study M_{RS}.

Since, on each simple module S_{λ}^{∗} ⊗ S_{μ} the action of Φ_{RS}^{n}, if it not trivial, does not depend on the operators R and S. Hence we have

R(M)^{n} = ⨁ R(M)^{n} ∩ (S_{λ}^{∗} ⊗ S_{μ}), and
M_{n} ∩ (S_{λ}^{∗} ⊗ S_{μ}) = (S_{λ}^{∗} ⊗ S_{μ}) /(S_{λ}^{∗} ⊗ S_{μ}) ∩ R(M)^{n}  (38)

The operator Φ_{RS}^{n} is defined as Φ^{n} with R^{∗−1} ⊗ R replaced by S^{∗−1} ⊗ R. Ψ_{RS}^{n} is defined analogously. Lemmas 2.1 and 2.2 hold for Φ_{RS}^{n} and Ψ_{RS}^{n}, respectively. We shall denote by Φ̄_{RS}^{n} and Ψ̄_{RS}^{n} the corresponding projectors. According the Theorem 2.3, we have, for λ, μ, such that S_{λ}^{∗} ⊗ S_{μ} contributes in W^{∗⊗n} ⊗ V^{⊗n},

(S_{λ}^{∗} ⊗ S_{μ}) ∩ R(M)_{i}^{n} = Ker Φ_{RS}^{n} ∩ (S_{λ}^{∗} ⊗ S_{μ}),  (39)
dim (S_{λ}^{∗} ⊗ S_{λ})/((S_{λ}^{∗} ⊗ S_{λ}) ∩ R(M)) = 1.  (40)

These equations implies that dim M_{n} = ∑_{λ⊢n} l_{λ}k_{λ′}, where k_{λ} = dim Im ρ_{S}(E_{λ}) and l_{λ} = dim Im ρ_{R}(E_{λ}). Since Ψ̄_{RS}^{n} is a projector, the map ϕ̄_{RS} : T (W^{∗} ⊗ V) ⟶ ⨁_{n=0}^{∞} Im Φ̄_{RS}^{n} induces an isomorphism of algebra M ⟶ ⨁_{n=0}^{∞} Im Φ̄_{RS}^{n}.

On the other hand,the fact, that the action of S_{i}^{∗−1} ⊗ R_{i} on (S_{λ}^{∗} ⊗ S_{μ}) does not depend on R and S, implies that the lattice induced by R(M)_{i}^{n} ∩ (S_{λ}^{∗} ⊗ S_{μ}), 1 ≤ i ≤ n − 1 is distributive in (S_{λ}^{∗} ⊗ S_{μ}) (cf. [2]). Consequently, the lattice generated by R(M)_{i}^{n}, 1 ≤ i ≤ n − 1 is distributive in W^{∗⊗n} ⊗ V^{⊗n}, that is M_{RS} is a Koszul algebra.

We collect the results obtained in a theorem.

Theorem 3.1 The algebra M_{RS} and N_{RS} are Koszul algebras, their Poincaré series are given by

PM(t) = P_{S_{R}} ∘ P_{S_{S}}(t), P_{N} (t) = P_{Λ_{S}} ∘ P_{Λ_{R}}(t).  (41)

There are realizations of M and N as subspaces of T (W^{∗} ⊗ V):

M ≅ ⨁_{n=0}^{∞} Im Φ̄^{n}, N ≅ ⨁_{n=0}^{∞} Im Ψ̄^{n}  (42)

further

Im (Φ̄_{RS}^{n}) = ⋂_{i=1}^{n−1} R(N)_{i}^{n}, Im (Ψ̄_{RS}^{n}) = ⋂_{i=1}^{n−1} R(M)_{i}^{n}.  (43)

References

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[2] Phung Ho Hai. Koszul Property and Poincaré Series of Matrix Bialgebra of Type A_{n}. Journal of Algebra, 192(2):734–748, 1997.

[3] Phung Ho Hai. Poincaré Series of Quantum Spaces Associated to Hecke Operators. Acta Math. Vietnam., to appear, 1998. Available at

   xxx.lanl.gov/dvi/q-alg/9711020.

[4] Yu.I. Manin. Quantum Groups and Non-commutative Geometry. GRM, Univ. de Montreal, 1988.

[5] A. Sudbery. Matrix-element bialgebras determined by quadratic coordinate algebras. J. Algebra, 158(2):375–399, 1993.