Left-covariant Differential Calculi on SL_{q}(N)

K. Schmüdgen and A. Schüler, Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Federal Republic of Germany ^{∗}

Abstract

We study N^{2} − 1 dimensional left-covariant differential calculi on the quantum group SL_{q}(N). In this way we obtain four classes of differential calculi which are algebraically much simpler as the bicovariant calculi. The algebra generated by the left-invariant vector fields has only quadratic-linear relations and posesses a Poincaré-Birkhoff-Witt basis. We use the concept of universal (higher order) differential calculus associated with a given left-covariant first order differential calculus. It turns out that the space of left-invariant k-forms has the dimension \binom{N^{2}−1}{k} as in the case of the corresponding classical Lie group SL(N).

Keywords: noncommutative differential calculus, quantum Lie algebra MSC (1991): 17 B 37, 46 L 87, 81 R 50.

∗e-mail: schmuedgen@mathematik.uni-leipzig.de, schueler@rz.uni-leipzig.de

1 Introduction

After the seminal work [6] of S. L. Woronowicz, bicovariant differential calculi on quantum groups (Hopf algebras) have been extensively studied in the literature. There is a well developed general theory of such calculi. Bicovariant differential calculi on the quantum group SL_{q}(N), N ≥ 3, have been recently classified in [3]. All calculi occuring in this classification have dimension N^{2}, i. e. their dimension does not coincide with the dimension N^{2} − 1 of the corresponding classical Lie group. On the other hand, the first example of a non-commutative differential calculus on a quantum group was Woronowicz’ 3D-calculus on SU_{q}(2) [5]. This is a three dimensional left-covariant calculus which is not bicovariant. The 3D-calculus is algebraically much simpler and in many respects nearer to the classical differential calculus on SU_{q}(2) than the four dimensional bicovariant calculi on SU_{q}(2). This motivates to look for N^{2} −1 dimensional left-covariant differential calculi on SL_{q}(N). The main aim of our approach is to follow the classical situation as close as possible. Let us briefly explain the basic idea of the approach given in this paper. As in [3], we assume that the differentials du_{j}^{i} of the matrix entries u_{j}^{i} generate the left module of 1-forms. Hence the differential d can be expressed as da = ∑(X_{ij} ∗ a)ω(u_{j}^{i}) for a ∈ SL_{q}(N), where ω(u_{j}^{i}) := ∑_{n} κ(u_{n}^{i})du_{j}^{n} are the left-invariant Maurer-Cartan forms and X_{ij} are linear functionals on SL_{q}(N) such that X_{ij}(1) = 0. In our approach the functionals X_{ij} will be chosen from the algebra of regular functionals U_{R} as defined in [2], 2. 2. We assume that the vector space of leftinvariant 1-forms has dimension N^{2} − 1 and that X_{ij}(u_{s}^{r}) = δ_{ir}δ_{js} for i ⊧ j. The main property of these calculi is that there are only quadratic-linear relations between the left-invariant vector fields. The corresponding formulas show that these calculi are very close to the classical differential calculi on SL(N) in many respects. Their enveloping algebras of the Lie algebra have Poincaré-Birkhoff-Witt property. The dimensions of the vector spaces of left-invariant k-forms Γ _{inv}^{∧k} is equal to the corresponding dimension \binom{N^{2}−1}{k} in the classical case. Moreover, the associated higher order calculi (Γ , d) give the ordinary calculus on SL(N) in the limit q → 1. This paper is organized as follows. Section 2 contains some general results about leftcovariant differential calculi on quantum groups which will be needed later. In Section 3 we develop four classes of left-covariant first order differential calculi (Γ_{r}, d), r = 1, …, 4. We start with the choice of left-invariant functionals X_{𝔦} and compute all relations between them. In Sections 4 we briefly describe the construction of the universal higher order differential calculus associated with a given left-covariant first order calculus on a quantum group. We apply this construction to the calculi (Γ_{r}, d), r = 1, …, 4. Throughout this paper q is a nonzero complex number such that q^{2} ⊧ 1 and we abbreviate λ := q − q^{−1}.

2 Left-covariant Differential Calculi on Quantum Groups

Our basic reference concerning differential calculi on quantum groups is [6]. Let A be a fixed Hopf algebra with comultiplication Δ, counit ε, antipode κ and unit element 1. Sometimes we use Sweedler’s notation Δ^{(n)}(a) = a_{(1)} ⊗ a_{(2)} ⊗ ⋯ ⊗ a_{(n+1)}. A first order differential calculus (briefly, a FODC) over A is a pair (Γ,d) of an A-bimodule Γ and a linear mapping d: A → Γ such that d(ab) = da·b + a·db for a, b ∈ A and Γ = lin{adb : a, b ∈ A}. A FODC (Γ , d) is called left covariant if there is a linear mapping Δ_{L}: Γ → A⊗Γ for which Δ_{L}(adb) = Δ(a)(id ⊗ d)Δ(b), a, b ∈ A. Suppose that (Γ , d) is a left-covariant FODC over A. Recall that the canonical projection of Γ into Γ_{inv} = {ω ∈ Γ : Δ_{L}(ω) = 1 ⊗ ω} is defined by P_{inv}(da) = κ(a_{(1)})da_{(2)}, cf. [5]. We abbreviate ω(a) = P_{inv}(da). Then R := {x ∈ ker ε : ω(x) = 0} is the right ideal of ker ε associated with (Γ , d). Recall that A^{∘} denotes the Hopf algebra of representative linear functionals on A. If A is Hopf ^{∗}-algebra then A^{∘} becomes a Hopf ^{∗}-algebra via f^{∗}(a) = \overline{f(κ(a)^{∗})}, f ∈ A^{∘}, a ∈ A. The vector space X := {X ∈ A^{∘} : X(1) = 0 and X(a) = 0 for a ∈ R} is called the quantum Lie algebra of the FODC (Γ , d).

Lemma 1. A vector space X of linear functionals on A is the quantum Lie algebra of a left-covariant FODC (Γ , d) if and only if X(1) = 0 and ΔX − 1⊗X ∈ X ⊗A^{∘} for all X ∈ X.

Proof. The necessity of the condition ΔX −1⊗X ∈ X ⊗A^{∘} follows at once from formula (5.20) in [6]: Δ(X_{i}) = ∑_{j} X_{j}⊗f_{i}^{j} + 1⊗X_{i}. To prove the sufficiency part, let us note that the above conditions imply that R := {a ∈ ker ε : X(a) = 0 for X ∈ X } is a right ideal of ker ε. From the general theory (cf. Theorem 1.5 in [6]) we conclude easily that R is the right ideal associated with some left-covariant FODC (Γ , d) over A. Since X is finite dimensional, X^{⊥⊥} = X. Hence, X is the quantum Lie algebra of (Γ , d). □ Obviously, if X is a quantum Lie algebra, so is X^{∗}.

3 Left-covariant first order Differential Calculi on SL_{q}(N)

Let ℓ^{+} = (ℓ^{+}_{j}^{i}) and ℓ^{−} = (ℓ^{−}_{j}^{i}) be the N×N matrices of linear functionals ℓ^{+}_{j}^{i} and ℓ^{−}_{j}^{i} on A = SL_{q}(N) as defined in [2]. Recall that ℓ^{±} is uniquely determined by the properties that ℓ^{±} : A → M_{N}(C) is a unital algebra homomorphism and ℓ^{±}_{j}^{i}(u_{m}^{n}) = p^{∓1}(R̂^{±1})_{mj}^{in} for i, j, n, m=1, …, N, where p is an N-th root of q, R̂ is the R-matrix of the quantum group SL_{q}(N), and u = (u_{m}^{n}) is the fundamental matrix corepresentation of SL_{q}(N). In this section we define N^{2} − 1 dimensional left-covariant FODC (Γ_{r}, d), r = 1, …, 4, over A = SL_{q}(N). We set

X_{ij} = −λ^{−1}ℓ^{+}_{j}^{j}ℓ^{−}_{i}^{j}, X_{ji} = q^{2δ_{jN}} λ^{−1}g_{i}ℓ^{−}_{i}^{i}ℓ^{+}_{j}^{i}, i < j,

X_{i} = q^{−1}λ^{−1}(g_{i} − 1), i = 1, …, N − 1, g_{i} = (ℓ^{+}_{i}^{i}ℓ^{−}_{N}^{N})^{2} for r = 1,
X_{ij} = −q^{−2δ_{i1}}λ^{−1}h_{j}ℓ^{+}_{j}^{j}ℓ^{−}_{i}^{j}, X_{ji} = λ^{−1}ℓ^{−}_{i}^{i}ℓ^{+}_{j}^{i}, i < j,
X_{j} = qλ^{−1}(1 − h_{j}), j = 2, …, N, h_{j} = (ℓ^{+}_{1}^{1}ℓ^{−}_{j}^{j})^{2} for r = 2,
X_{ij} = q^{2δ_{jN}} λ^{−1}κ(ℓ^{−}_{i}^{j})ℓ^{−}_{i}^{i}g_{i} X_{ji} = −λ^{−1}κ(ℓ^{+}_{j}^{i})ℓ^{+}_{j}^{j}, i < j,
X_{i} = q^{−1}λ^{−1}(g_{i} − 1), i = 1, …, N − 1, for r = 3,
X_{ij} = λ^{−1}κ(ℓ^{−}_{i}^{j})ℓ^{−}_{i}^{i}, X_{ji} = −q^{−2δ_{i1}}λ^{−1}κ(ℓ^{+}_{j}^{i})ℓ^{+}_{j}^{j}h_{j}, i < j,
X_{j} = qλ^{−1}(1 − h_{j}), j = 2, …, N, for r = 4.

Let X_{r}, r = 1, …, 4, denote the linear span of the functionals X_{ij}, i ⊧ j, i, j = 1, …, N, and X_{n}, n = 1, …, N − 1 (2, …, N), for r = 1, 3 (r = 2, 4). Computing the coproducts of these functionals we obtain for r = 1 and i < j:

ΔX_{ij} = 1 ⊗ X_{ij} + ∑_{m=i}^{j−1} X_{mj} ⊗ ℓ^{+}_{j}^{j}ℓ^{−}_{i}^{m},

ΔX_{ji} = g_{i} ⊗ X_{ji} + ∑_{m=i+1}^{j} X_{mi} ⊗ g_{i}ℓ^{−}_{i}^{i}ℓ^{+}_{j}^{m},
ΔX_{i} = 1 ⊗ X_{i} + X_{i} ⊗ g_{i}, i = 1, …, N − 1,

for r = 2 and i < j:

ΔX_{ij} = h_{j} ⊗ X_{ij} + ∑_{m=i}^{j−1} X_{mj} ⊗ h_{j}ℓ^{+}_{j}^{j}ℓ^{−}_{i}^{m},

ΔX_{ji} = 1 ⊗ X_{ji} + ∑_{m=i+1}^{j} X_{mi} ⊗ ℓ^{−}_{i}^{i}ℓ^{+}_{j}^{m},
ΔX_{j} = 1 ⊗ X_{j} + X_{j} ⊗ h_{j}, j = 2, …, N.

These formulas show that ΔX − 1⊗X ∈ X_{r}⊗A^{∘} for all X ∈ X_{r}, r = 1, 2. The computations for r = 3, 4 are quite similar. Therefore, by Lemma 1 the vector space X_{r}, r = 1, …, 4, defines a left-covariant FODC over A. Using the pairing between SL_{q}(N) and U_{R} we verify that X_{ij}(u_{l}^{k}) = δ_{ik}δ_{jl} for i ⊧ j and X_{i}(u_{l}^{k}) = δ_{kl}(δ_{ik} − q^{−2}δ_{kN}), i = 1, …, N − 1, r = 1, 3, and X_{j}(u_{l}^{k}) = δ_{kl}(δ_{jk} − q^{2}δ_{1k}), j = 2, …, N, r = 2, 4.

Now we will consider the ∗-structures. In case SL_{q}(N, R), |q| = 1, we get X _{r}^{∗} = X_{r}, r = 1, …, 4. Hence, all four calculi are ∗-calculi. In case SU_{q}(N), q ∈ R, we get X _{1}^{∗} = X_{3} and X _{2}^{∗} = X_{4}. Hence, none of them are ∗-calculi with respect to SU_{q}(N).

We shall study the calculus X_{3} in more detail. Let {ω_{ij}, ω_{n} : i ⊧ j, n = 1, …, N − 1} (resp. {u_{j}^{i}, u_{n}^{n} − 1 : i ⊧ j, n = 1, …, N − 1}) be the dual basis of Γ_{inv} (resp. of ker ε/R) to the basis {X_{ij}, X_{n}} of X. Let ϑ denote the Heaviside symbol, i. e. ϑ(m) = 1 if m > 0 and ϑ(m) = 0 if m ≤ 0.

The commutation rules between matrix entries and 1-forms are as follows:

ω_{ij}u_{k}^{l} = q^{δ_{ik}+δ_{jk}−2δ_{Nk}}u_{k}^{l}ω_{ij} + λϑ(k − j)u_{j}^{l}ω_{ik},
ω_{ji}u_{k}^{l} = q^{−δ_{ik}+δ_{jk}}u_{k}^{l}ω_{ji} − λϑ(i − k)u_{i}^{l}ω_{jk},
ω_{i}u_{k}^{l} = q^{2δ_{ik}−2δ_{kN}} u_{k}^{l}ω_{i} + qλϑ(k − i)u_{i}^{l}ω_{ik}.

Note that the commutation rules are closed in the upper (i < j) and lower (i > j) triangle of left invariant 1-forms, respectively, while the commutation rules on the diagonal ω_{i} are not closed. The associated right ideal R_{3} of ker ε is generated by the following set of quadraticlinear terms:

u_{j}^{i}u_{l}^{k} for i ⊧ j, i ⊧ l, k ⊧ j, k ⊧ l; u_{j}^{i}u_{i}^{j} for i ⊧ j; u_{a}^{i}u_{j}^{a} for i > a, j > a and for i < a, j < a; u_{a}^{i}u_{j}^{a} − λu_{j}^{i} for i < a < j and for i > a > j;

u_{a}^{a}u_{j}^{i} − q^{2δ_{ia}−2δ_{aN}} u_{j}^{i} for i < j, u_{a}^{a}u_{j}^{i} − u_{j}^{i} for i > j;

(u_{i}^{i} − q^{−2δ_{iN}})(u_{i}^{i} − q^{2−2δ_{iN}}), i = 1, …, N; (u_{i}^{i} − 1)(u_{j}^{j} − q^{−2δ_{jN}}) for i < j, V − ε(V).

Note that V := ∑_{i=1}^{N} q^{2δ_{iN}} u_{i}^{i} is the only linear element of A that is annihilated by X_{3}. We abbreviate q_{j} := q^{2δ_{jN}} and q_{nij} := q^{2(δ_{ni}−δ_{nj}+δ_{jN})}. Throughout let i < j < k < l. Then the commutation rules between the {X_{𝔦}}_{𝔦∈I}, I = {(i, j), n: i ⊧ j, n = 1, …, N − 1} are as follows:

X_{ij}X_{il} − qq_{l}X_{il}X_{ij} = 0, X_{il}X_{jl} − qX_{jl}X_{il} = 0, X_{ij}X_{jl} − qq_{l}X_{jl}X_{ij} − qq_{l}λX_{il}X_{j} = q_{l}X_{il}, X_{ij}X_{kl} − q_{l}X_{kl}X_{ij} = 0, X_{jk}X_{il} − q_{l}X_{il}X_{jk} = 0, X_{ik}X_{jl} − q_{l}X_{jl}X_{ik} − q_{l}λX_{il}X_{jk} = 0,

X_{ji}X_{li} − qX_{li}X_{ji} = 0, X_{li}X_{lj} − qX_{lj}X_{li} = 0, X_{ji}X_{lj} − q^{−1}X_{lj}X_{ji} = −q^{−1}X_{li}, X_{ji}X_{lk} − X_{lk}X_{ji} = 0, X_{kj}X_{li} − X_{li}X_{kj} = 0, X_{ki}X_{lj} − X_{lj}X_{ki} − λX_{li}X_{kj} = 0,

X_{ij}X_{li} − q^{−1}q_{l}^{−1}X_{li}X_{ij} + q_{l}^{−1}λX_{lj}X_{i} = −q^{−1}q_{l}^{−1}X_{lj}, X_{il}X_{lj} − q^{−1}q_{l}^{−1}X_{lj}X_{il} = X_{ij}, X_{ij}X_{lj} − q^{−1}q_{l}^{−1}X_{lj}X_{ij} = 0, X_{ij}X_{lk} − q_{l}^{−1}X_{lk}X_{ij} = 0, X_{il}X_{kj} − X_{kj}X_{il} = 0, X_{ik}X_{lj} − q_{l}^{−1}X_{lj}X_{ik} + q_{l}^{−1}λX_{lk}X_{ij} = 0,

X_{ji}X_{il} − qX_{il}X_{ji} = X_{jl}, X_{li}X_{jl} − q^{−1}q_{l}X_{jl}X_{li} + q_{l}λX_{ji}X_{l} = −q^{−1}q_{l}X_{ji}, X_{ji}X_{jl} − q^{−1}X_{jl}X_{ji} = 0, X_{ji}X_{kl} − X_{kl}X_{ji} = 0, X_{li}X_{jk} − q_{l}X_{jk}X_{li} = 0, X_{ki}X_{jl} − X_{jl}X_{ki} + λX_{kl}X_{ji} = 0,

X_{il}X_{li} − q_{l}^{−1}X_{li}X_{il} = X_{i} − X_{l},

X_{n}X_{m} − X_{m}X_{n} = 0, X_{n}X_{ij} − q_{nij}X_{ij}X_{n} = q^{−1}λ^{−1}(q_{nij} − 1)X_{ij}, X_{n}X_{ji} − q_{nij}^{−1}X_{ji}X_{n} = q^{−1}λ^{−1}(q_{nij}^{−1} − 1)X_{ji}.

Summarizing the preceding we obtain

1. X_{r}, r = 1, …, 4, is an N^{2} − 1 dimensional left-covariant FODC on SL_{q}(N).

2. The relations of the algebra U_{r} generated by X_{r} are quadratic-linear. To be more precise, there are complex matrices σ = (σ_{𝔨𝔩}^{𝔦𝔧}) and C = (C_{𝔦𝔧}^{𝔨}), 𝔦, 𝔧, 𝔨, 𝔩 ∈ I, such that

X_{𝔦}X_{𝔧} − ∑_{𝔨,𝔩} σ_{𝔨𝔩}^{𝔦𝔧}X_{𝔨}X_{𝔩} = ∑_{𝔪} C_{𝔦𝔧}^{𝔪}X_{𝔪}, 𝔦, 𝔧 ∈ I.

The matrix σ is diagonalizable and has eigenvalues −1 and 1 with multiplicities \binom{N^{2}−1}{2} and \binom{N^{2}}{2}, respectively. The structure constants C are antisymmetric, i. e. Cσ^{t} = −C.

3. The algebra U_{r} has a linear basis

{Xm_{1}_{𝔦_{1}} … Xm_{n}_{𝔦_{n}} : 𝔦_{1} < 𝔦_{2} < … < 𝔦_{n}, 𝔦_{k} ∈ I, m_{k} ∈ N_{0}, k = 1, …, n}, n = N^{2} − 1,

with respect to an arbitrary order of the index set I.

4 Higher Order Calculi

Given a left-covariant FODC (Γ , d) on A. Let R be the associated right ideal of (Γ , d) and S: R → Γ_{inv}⊗Γ_{inv} the linear mapping defined by S(r) = ω(r_{(1)}) ⊗ ω(r_{(2)}). Let S_{2} be the subbimodule of Γ ⊗_{A} Γ generated by S(R). Furthermore, let S be the ideal of the tensor algebra Γ^{⊗} = ∑_{n} Γ^{⊗n} generated by S_{2}. Then Γ^{∧} = Γ^{⊗}/S is called universal exterior algebra associated with (Γ , d). By construction, Γ^{∧} is a left-covariant bimodule and a Z-graded algebra.

Proposition 2. Let (Γ , d) be a left-covariant FODC over A and let Γ^{∧} be the universal exterior algebra built over (Γ , d). Then there exists one and only one linear mapping d: Γ^{∧} → Γ^{∧} such that

1. d increases the grade by one.

2. d extends d: A → Γ.

3. d fulfills the graded Leibniz rule.

4. d^{2} = 0.

5. d is left-covariant, i. e. Δ_{L}(dθ) = (id ⊗ d)Δ_{L}θ, θ ∈ Γ^{∧}.

(Γ^{∧}, d) is called universal differential calculus associated with the left-covariant FODC (Γ , d). Woronowicz’ construction of higher order calculi for bicovariant FODC can be obtained by taking a quotient of the universal differential calculus (Γ^{∧}, d). The mapping d|_{Γ_{inv}} is given by dω(a) = −ω(a_{(1)}) ∧ ω(a_{(2)}). We apply the above construction to the left-covariant FODC X_{r}, r = 1, …, 4. Fix an arbitrary linear order of I.

Theorem 3. Let (Γ^{∧}, d) be the universal left-covariant differential calculus associated with one of the FODC (Γ_{r}, d), r = 1, …, 4. Then

dim(Γ _{inv}^{∧k}) = \binom{N^{2} − 1}{k}, k ≥ 1.

More precisely, the set {ω_{𝔦_{1}} ∧ ω_{𝔦_{2}} ∧ … ∧ ω_{𝔦_{k}} : 𝔦_{1} < … < 𝔦_{k}, 𝔦_{j} ∈ I, j = 1, …, k} is a basis of the vector space Γ _{inv}^{∧k}.

In particular, for k = 2 Theorem 3 states that the space of left-invariant 2-forms is equal to ker (σ + id ) (where σ is the linear mapping Γ_{inv} ⊗ Γ_{inv} → Γ_{inv} ⊗ Γ_{inv} defined by ω_{𝔦}⊗ω_{𝔧} ↦ ∑_{𝔨,𝔩} σ_{𝔦𝔧}^{𝔨𝔩}ω_{𝔨}⊗ω_{𝔩}.) In contrast, there are examples of three dimensional quadratically closed FODC on SU_{q}(2) whose space of left-invariant 2-forms is only two dimensional.

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