The Diffiety Institute Preprint Series

Preprint DIPS–2 March 1 — May 8, 2002

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The Diffiety Institute

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On n-ary generalizations of the Lie algebra 𝔰𝔩_{2}(𝕜)

by

Arthemy V. Kiselev

On n-ary generalizations of the Lie algebra 𝔰𝔩_{2}(𝕜)

ARTHEMY V. KISELEV

Abstract. Polynomials a_{j} ∈ 𝕜_{N}[x], 1 ≤ j ≤ N, of degree deg a_{j} = N are shown to be closed w.r.t. action of the N-ary bracket [a_{1}, …, a_{N}] = W (a_{1}, …, a_{N}), where W denotes the Wronskian determinant. This bracket is proved to induce the homotopical N-Lie algebra structure on the polynomials 𝕜_{N}[x] of degree N, so that an extended Jacobi identity (with 2^{N−1} summands) holds; the case N = 2 is the Lie bracket in 𝔰𝔩_{2}(𝕜) satisfying the Jacobi identity.

The property of the Wronskian determinants to compose the homotopical N-Lie algebras is proved to hold for arbitrary analytic functions a_{j} ∈ 𝕜[[x]], thus extending the former result. The identity [[W i⃗, W j⃗]]^{RN} = 0 is discussed, where i⃗ = (0, 1, …, k) ∈ ℤ_{+}^{k+1}, j⃗ = (0, 1, …, l) ∈ ℤ_{+}^{l+1}, ∂ is a derivation, W i⃗ = ∂^{i_{1}} ∧ … ∧ ∂^{i_{k+1}} and W j⃗ = ∂^{j_{1}} ∧ … ∧ ∂^{j_{l+1}} are the Wronskians and [[·, ·]]^{RN} is the Richardson–Nijenhuis bracket. The notion of the Wronskian determinant is generalized to the case of n independent variables: (x^{1}, …, x^{n}) ∈ 𝕜^{n}, so that the resulting concept preserves the homotopical N-Lie Jacobi identity. Mathematics Subject Classification (2000): 13M10, 15A15, 15A24, 26C05.

Introduction. In this paper^{1}, following [1, 2, 3, 4] we consider a special case of N-ary generalizations of the Lie algebra 𝔰𝔩_{2}(𝕜), namely, the homotopical N-Lie algebra of the Nth degree polynomials 𝕜_{N}[x], where 2 ≤ N ∈ ℕ and the field 𝕜 has characteristic 0.

Remark 1. We make a conventional remark concerning the underlying field 𝕜. In all the illustrative cases involving the differential calculus we assume 𝕜 = ℝ or 𝕜 = ℂ thus investigating smooth functions a_{j} ∈ C^{∞}(𝕜). Generally, we deal with analytic functions a_{j} ∈ 𝕜[[x]] treating them as formal series over the field 𝕜 of characteristic 0. In the ndimensional case x = (x^{1}, …, x^{n}) ∈ 𝕜^{n}, the analytic functions are a_{j} ∈ 𝕜[[x^{1}, …, x^{n}]]. Still, we preserve the notation ‘C^{∞}(𝕜^{n})’ from the case 𝕜 = ℝ bearing in mind its applications in mathematical physics.

First let us introduce some notation. Let A be an associative commutative algebra over the field 𝕜 such that char 𝕜 = 0, and let ∂ be a derivation of A. Further on, we consider the non-graded (even) case only. Let σ ∈ S_{m}^{k} ⊂ S_{m} be the permutations such that σ(1) < σ(2) < ⋯ < σ(k) and σ(k + 1) < σ(k + 2) < ⋯ < σ(m) and let [·, …, ·] ∈ Hom(⋀^{N} A, A) be an N-linear skew-symmetric bracket: [a_{Σ(1)}, …, a_{Σ(N)}] = (−1)^{Σ}[a_{1}, …, a_{N}] for any rearrangement Σ ∈ S_{N}.

Definition 1. The algebra A is said to be the homotopical N-Lie algebra, if the Jacobi identity

(1)  ∑_{σ∈S_{2N−1}^{N}}(−1)^{σ}[[a_{σ(1)}, …, a_{σ(N)}], a_{σ(N+1)}, …, a_{σ(2N−1)}] = 0

1Submitted March 1, 2002; Revised May 8, 2002.

holds for any a_{j} ∈ A, 1 ≤ j ≤ 2N − 1.

Let Δ ∈ Hom(⋀^{k} A, A) and ∇ ∈ Hom(⋀^{l} A, A). By Δ[∇] ∈ Hom(⋀^{k+l−1} A, A) we denote the action Δ[·]: Hom(⋀^{N} A, A) → Hom(⋀^{N+k−1} A, A) of Δ on ∇:

Δ[∇](a_{1}, …, a_{k+l−1}) = ∑_{σ∈S_{k+l−1}^{l}}(−1)^{σ} Δ(∇(a_{σ(1)}, …, a_{σ(l)}), a_{σ(l+1)}, …, a_{σ(k+l−1)}),

where a_{j} ∈ A. Then, the Jacobi identity given in Eq. (1) is

(2)  Δ[Δ] = 0.

By [[Δ, ∇]]^{RN} ∈ Hom(⋀^{k+l−1} A, A) we denote the Richardson–Nijenhuis bracket of Δ and ∇:

[[Δ, ∇]]^{RN} =^{def} Δ[∇] − (−1)^{(k−1)(l−1)}∇[Δ].

Let Δ ∈ Hom(⋀^{k} A, A) and a_{j} ∈ A, 1 ≤ j ≤ k; suppose 1 ≤ l ≤ k. The inner product Δ_{a_{1},...,a_{l}} ∈ Hom(⋀^{k−l} A, A) is defined by

Δ_{a_{1},...,a_{l}}(a_{l+1}, …, a_{k}) =^{def} Δ(a_{1}, …, a_{k}).

Now we describe the scheme that generates the (N, k, r)-Lie algebra structures [1] and treates the homotopical 2N-Lie algebras as the (2N, 1, 0)-Lie algebras.

Definition 2 ([1]). Choose integers N, k, and r such that 0 ≤ r ≤ k < N, and let a_{1}, …, a_{r}, b_{1}, …, b_{k} ∈ A. The skew-symmetric map Δ ∈ Hom(⋀^{N} A, A) is said to determine the Lie algebra structure of the type (N, k, r) on the 𝕜-vector space A, if the (N, k, r)-Jacobi identity

(3)  [[Δ_{a_{1},...,a_{r}}, Δ_{b_{1},...,b_{k}}]]^{RN} = 0

holds for any a⃗ and b⃗. By Lie^{(N,k,r)}(A) we denote the set of all type (N, k, r) structures Δ ∈ Hom(⋀^{N} A, A) on A.

Remark 2. We claim that

(4)  Lie^{(N,0,0)}(A) = Lie^{(N,1,0)}(A)

for any even N; this is a typical instance of the heredity structures [1]. Really, the following two conditions are equivalent:

[[Δ, Δ]]^{RN} = 0 ⇔ [[Δ, Δ]]^{RN}_{a} = −2 [[Δ, Δ_{a}]]^{RN} = 0 ∀a ∈ A,

due to Corollary 1.1 in [1]:

[[Δ, Δ]]^{RN}_{a} = (−1)^{N−1}[[Δ, Δ_{a}]]^{RN} + [[Δ_{a}, Δ]]^{RN}.

Finally, [[Δ, Δ_{a}]]^{RN} = 0 for any a ∈ A.

Thus, the Jacobi identity of the type (N, 0, 0)

[[Δ, Δ]]^{RN} = 2Δ[Δ] = 0

implies Eq. (1) for any even N. The preceding papers [5, 6] should be regarded w.r.t. the isomorphism (4).

If N is odd, then the expression [[Δ, Δ]]^{RN} ≡ 0 is trivial, and we consider Eq. (2) separately from the condition (3).

In the sequel, we study the Jacobi identity (1) of the form (2), where Δ ∈ Hom(⋀^{N} A, A). We conclude that for even Ns the results will be equivalent to the properties of the type (N, 1, 0) structures, while for odd Ns our results are independent from the scheme proposed in Definition 2.

1. Consider the space 𝕜_{N}[x] ∋ a_{j} of polynomials a_{j} of degree not greater than N; on this space, there is the N-linear skew-symmetric bracket

(5)  [a_{1}, …, a_{N}] = W (a_{1}, …, a_{N}) ,

where W denotes the Wronskian determinant. Since N-ary bracket (5) is N-linear, we consider monomials const ·x^{k} only. We choose {a_{j}^{0}} = {x^{k}} or {a_{j}^{0}} = {x^{k}/k!}, where 0 ≤ k ≤ N and 1 ≤ j ≤ 2N − 1, for standard basis in 𝕜_{N}[x]; exact choice depends on the situation: the monomials x^{k} are used to demonstrate the presence or absence of certain degrees in N-linear bracket (5), and the monomials x^{k}/k! are convenient in calculations since they are closed w.r.t. the derivations. The set of monomials a_{j} = x^{k} is bi-graded: ord a_{j} = j and deg x^{k} = k.

Let p⃗ ∈ ℤ_{+}^{s}, ℕ ∋ s ≥ 1. By |p⃗| ∈ ℤ_{+} we denote the ‘city-block’ norm: |p⃗| = ∑_{j=1}^{s} p_{j}. By definition, put

(6)  Δ_{ℤ,t}^{s−1} ≡ {p⃗ = (p_{1}, …, p_{s}) ∈ ℤ_{+}^{s} | |p⃗| = t}.

We see that Δ_{ℤ,t}^{s−1} is the intersection ℤ_{+}^{s} ∩ Δ_{t}^{s−1}, where Δ_{t}^{s} is the standard s-dimensional simplex: Δ_{t}^{s} = {p⃗ ∈ ℝ_{+}^{s+1} | ∑_{j=1}^{s+1} p_{j} = t}.

Lemma 1. Let 0 ≤ k < N; then

(1) the inequality 0 ≤ deg W (1, …, _{\frac{x^{k}}{k!}} , …, \frac{x^{N}}{N!} ) ≤ N holds, i.e., the monomials are closed w.r.t. the Wronskian determinant;

(2) ([6]) the relation W (1, …, _{\frac{x^{k}}{k!}} , …, \frac{x^{N}}{N!} ) = const(N, k) · x^{N−k} holds, i.e., the Wronskian determinant is a monomial;

(3) the coefficient const(N, k) is

(7)  const(N, k) = \frac{1}{(N − k)!} · \{\frac{(−1)^{N−k+1}}{2} + \frac{1}{2} ∑_{l=1}^{N−k} l! · ∑_{p⃗∈Δ_{ℤ,N−k−l}^{l−1}} ∏_{j=1}^{l}(j)^{p_{j}}\},

while for k = N the Wronskian coefficient in (7) equals 1.

Proof. 1)^{2} The Wronskian determinant is

∂i⃗(aj⃗) = ∂^{i_{1}} ∧ … ∧ ∂^{i_{N}} (a_{1}, …, a_{N}) = det ∥∂^{i_{α}}a_{β}∥.

In the case of monomials we have

∂^{0} ∧ … ∧ ∂^{N−1}(x^{j_{1}}, …, x^{j_{N}}) = ∑_{σ∈S_{N}}(−1)^{σ} ∂^{0}(x^{j_{σ(1)}}) · . . . · ∂^{N−1}(x^{j_{σ(N)}}).

By definition, put xj⃗ = {x^{j_{1}}, …, x^{j_{♯j⃗}}} and |xj⃗| = |j⃗|. Suppose j⃗ ∈ [0, N]^{N} ∩ ℤ^{N}. We claim that for every non-trivial summand S_{σ,j⃗}:

S_{σ,j⃗} =^{def} (−1)^{σ} ∂^{0}(x^{j_{σ(1)}}) · . . . · ∂^{N−1}(x^{j_{σ(N)}}),

2The proof of 1) was reported by A. M. Verbovetsky (private communication).

the degree deg S_{σ,j⃗} has the upper bound N:

(8)  0 ≤ deg S_{σ,j⃗} = |xj⃗| − |W^{0,1,...,N−1}| = j_{1} + … + j_{N} − \frac{N(N − 1)}{2} ≤ N.

Really, we have

0 + 1 + … + N − 1 = \frac{N(N − 1)}{2} ≤ j_{1} + … + j_{N} ≤ 1 + 2 + … + N = \frac{N(N + 1)}{2}

and, since the derivative ∂: 𝕜_{N′}[x] → 𝕜_{N′−1}[x], 1 ≤ N′ ≤ N, ker ∂ = 𝕜, decreases the degree deg by 1, we obtain (8).

This proves 1). It is not hard to prove 2) just now, but we need some additional notation and recurrence relation (11) in order to prove 3), so we apply another technique.

2) Since the degrees deg a_{i} ≠ deg a_{j}, i ≠ j, lest the Wronskian would be zero identically, rearrange the elements a_{j} such that ord a_{i} < ord a_{j} ⇔ deg a_{i} < deg a_{j}. We have

(9) W (1, …, ˆ \frac{x^{k}}{k!}, …, \frac{x^{N}}{N!} ) = W (1, …, \frac{x^{k−1}}{(k − 1)!}) · W (x, …, \frac{x^{N−k}}{(N − k)!}) ,

where the first factor in the r.h.s. of (9) equals 1 and has the degree 0. Denote the second factor, the determinant of the (N − k) × (N − k) matrix, by W_{m}, m ≡ N − k. We claim that W_{m} is a monomial: deg W_{m} = m, and prove this fact by induction on m ≡ N − k. For m = 1, deg det(x) = 1 = m. Let m > 1; the decomposition of W_{m} w.r.t. the last row gives (10)

W_{m} = W (x, …, \frac{x^{m}}{m!} ) = x · W (x, …, \frac{x^{m−1}}{(m − 1)!}) − 1 · W (x, …, \frac{x^{m−2}}{(m − 2)!}, \frac{x^{m}}{m!} ) ,

where the degree of the first Wronskian in r.h.s. of (10) is m − 1 by the inductive assumption. Again, decompose the second Wronskian in r.h.s. of (10) w.r.t. the last row and proceed so iteratively using the induction hypothesis. We obtain the recurrence relation

(11)  W_{m} = ∑_{l=1}^{m−1} W_{m−l} · \frac{x^{l}}{l!} − (−1)^{m} \frac{x^{m}}{m!} ,  m ≥ 1,

whence deg W_{m} = m. We see that the initial Wronskian (9) is a monomial itself of degree m = N − k with yet unknown coefficient.

3) Now we calculate the coefficient W_{m}(x)/x^{m} ∈ 𝕜 in the Wronskian determinant (9). Consider the generating function

(12)  f(x) ≡ ∑_{m=1}^{∞} W_{m}(x)

such that

W_{m}(x) = \frac{x^{m}}{m!} \frac{d^{m}f}{dx^{m}} (0), 1 ≤ m ∈ ℕ.

Note that exp(x) ≡ ∑_{m=0}^{∞} x^{m}/m!; treating (12) as the formal sum of equations (11), we have

f(x) = f(x) · (exp(x) − 1) − exp(−x) + 1,

whence

(13)  f(x) = \frac{exp(−x) − 1}{exp(x) − 2} = −\frac{1}{2} · (exp(−x) + \frac{1}{exp(x) − 2}) .

Note that f(0) = 0 and f(x) is smooth in the vicinity of zero. Then, differentiating (13) m times w.r.t. x, we obtain

(14) \frac{d^{m}f}{dx^{m}} (x) = \frac{(−1)^{m+1}}{2} exp(−x) − \frac{1}{2} · ∑_{l=1}^{m} \underbrace{(−1)^{l}l!}_{A} ·\frac{exp(lx)}{(exp(x) − 2)^{l+1}} · ∑_{p⃗∈Δ_{ℤ,m−l}^{l−1}} \overbrace{∏_{j=1}^{l}(j)^{p_{j}}}^{B},

where the underbraced factor A comes from differentiating (exp(x)−2)^{−1} l times w.r.t. x and \binom{2(m−l)−1}{m−l−1} overbraced summands B of the form 1^{p_{1}} · 2^{p_{2}} · . . . · l^{p_{l}} come from differentiating the exponents exp(x), …, exp(lx) m − l times w.r.t. x, see (6) for definition of the set Δ_{ℤ,m−l}^{l−1}. Considering (14) at the point x = 0, we finally have

W_{m}(x) = \frac{x^{m}}{m!} · \{\frac{(−1)^{m+1}}{2} + \frac{1}{2} ∑_{l=1}^{m} l! · ∑_{p⃗∈Δ_{ℤ,m−l}^{l−1}} ∏_{j=1}^{l}(j)^{p_{j}}\}

for any integer m ≥ 1, whence follows (7). Also, we see that W_{m}(x) ⇝ (ln 2)^{−m} · x^{m} as m → ∞, since series (12) converges within |x| < ln 2.

The proof is complete. □

From Lemma 1 follows

Theorem 1 ([6]). Polynomials a_{1}, …, a_{N} ∈ 𝕜_{N}[x] of degree deg a_{j} ≤ N, 1 ≤ j ≤ N, are closed with respect to action of N-ary bracket (5): [a_{1}, …, a_{N}] = W (a_{1}, …, a_{N}) ∈ 𝕜_{N}[x], while the structural constants are (7).

2. Now we start to construct the homotopical N-Lie generalizations of the Lie algebra 𝔰𝔩_{2}(𝕜). In this section, we introduce some notation and describe the number ♯ of summands in the relations we deal with. Our starting point is the following

Example 1 ([5]). The polynomials 𝕜_{2}[x] = {αx^{2} + βx + γ | α, β, γ ∈ 𝕜} of degree 2 with bracket (5) form a Lie algebra isomorphic to 𝔰𝔩_{2}(𝕜). The commutation relations can be easily checked in the basis ⟨1, −2x, −x^{2}⟩:

[−2x, 1] = 2,  [−2x, −x^{2}] = 2x^{2},  [1, −x^{2}] = −2x,
[h, e] = 2e,  [h, f] = −2f,  [e, f] = h,

whence the representation ϱ: 𝔰𝔩_{2}(𝕜) → 𝕜_{2}[x] is ϱ(e) = 1, ϱ(h) = −2x, and ϱ(f) = −x^{2}.

Definition 3. The system (a_{1}, …, a_{2N−1}) of monomials is called the standard system of monomials a⃗(k_{1}, k_{2}) if 0 ≤ deg a_{j} ≤ N for all 1 ≤ j ≤ 2N − 1 and the set {a_{j}} of elements a_{j}, ungraded by ord, is

(15) {a_{j}} = {1, …, x^{N}} ∪ {1, …, x^{N}} \ {x^{k_{1}}, x^{k_{2}}, x^{N−k_{1}}} where 0 ≤ k_{1}, k_{2} ≤ N.

By J N,∞ ; a⃗(x) we denote the l.h.s. in (1) with N-ary bracket (5) acting on analytic functions a_{j} of x ∈ 𝕜, and by J_{a⃗(k_{1},k_{2})}^{N}(x) we denote the same l.h.s. in (1) with N-ary bracket (5) acting on the standard system of monomials a⃗(k_{1}, k_{2}).

Lemma 2 ([6]). The number of summands ♯J_{a⃗(k_{1},k_{2})}^{N}(x) = 2^{N−1}. Also, the number of summands ♯J N,∞ ; a⃗(x) = \binom{2N−1}{N−1} = \binom{2N−1}{N}, see [5].

Proof. Consider three cases: k_{1} ≠ k_{2} ≠ N − k_{1}, k_{2} = k_{1}, and k_{2} = N − k_{1}.

Case 1. Let k_{1} ≠ k_{2} ≠ N − k_{1}; then there are two types of the internal Wronskians in (1). We say that the summands with the internal Wronskians W (…, x̂^{k_{1}}, x^{k_{2}}, x^{N−k_{1}}, …)

belong to type 1, and the summands containing W (…, x^{k_{1}}, x^{k_{2}}, { \atop x^{N−k_{1}}}, …) belong to type 2, where k_{2} ≠ N − k_{1} and k_{2} ≠ k_{1} by assumption. Other elements x^{m} are met twice in the set {a_{j}} and their choice for position into the internal (resp., external) Wronskian is arbitrary. So, we have 2 types × 2^{N−2} arrangements = 2^{N−1} summands.

Case 2. Let k_{1} = k_{2} ≠ N −k_{1}; then there are 2^{N−1} type 1 internal Wronskians W (…,

x̂^{k_{1}}, x^{N−k_{1}}, …), necessarily containing the element x^{N−k_{1}}; the type 2 summands are not realized for this a⃗(k_{1}, k_{1}) since these Wronskians are zero identically.

Case 3. This case k_{2} = N − k_{1} repeats literally Case 2 after the transposition type 1 ↔ type 2.

Thus, in all cases the number of summands ♯J_{a⃗(k_{1},k_{2})}^{N}(x) equals 2^{N−1}.

Obviously, the number of summands ♯J^{N,∞}(x) = ♯S_{2N−1}^{N} = \binom{2N−1}{N}, if no three elements a_{m_{1}}, a_{m_{2}}, and a_{m_{3}} ∈ a⃗ have equal degrees. □

Corollary 1. By the Stirling formula,

\frac{♯J N,∞ ; a⃗(x)}{♯J_{a⃗(k_{1},k_{2})}^{N}(x)} = \frac{\binom{2N−1}{N−1}}{2^{N−1}} ⇝ \frac{2^{N}}{\sqrt{πN}} as N → ∞.

We see that the numbers of summands grow exponentially but the case 0 ≤ deg a_{j} ≤ N provides exponentially less summands than the general case of analytic functions.

3. Our goal is to prove that J_{a⃗(k_{1},k_{2})}^{N}(x) = J^{N}(x; k_{1}, k_{2}) ≡ 0, not depending on rearrangements of a⃗, if the elements x^{k_{1}}, x^{k_{2}}, and x^{N−k_{1}} ∉ a⃗.

Lemma 3 ([6]). Let the orders ord and deg coincide: ord a_{i} ≤ ord a_{j} ⇔ deg a_{i} ≤ deg a_{j}, i.e.,

(a_{1}, …, a_{2N−1}) = (1, 1, x, x, …, x̂^{k_{1}}, x̂^{k_{2}}, { \atop x^{N−k_{1}}}, …, x^{N}, x^{N}) ← 2N − 1 elements.

Then J_{a⃗(k_{1},k_{2})}^{N}(x) ≡ 0.

Proof. Consider a pair of neighbouring elements a_{m}, a_{m+1} such that deg a_{m} = deg a_{m+1} = k; for N ≥ 2, such a pair exists. Now consider two kinds of summands in (1): the ones with direct order, i.e., (m = σ(i) and m + 1 = σ(j)) ⇒ i < j, and the ones with the reverse order, i > j. Obviously, the algebraic expressions W (W (…, x^{k}, …) , …, x^{k}, …) are identical to each other, but we claim that these two kinds of summands in J_{a⃗(k_{1},k_{2})}^{N}(x) differ by a sign −1 that comes from (−1)^{σ}. Really, all elements a_{j} with j < m of j > m + 1 do not acquire additional inversions spoiling the sign (−1)^{σ} after the transposition a_{m} ↔ a_{m+1}, yet the unique additional inversion between a_{m+1} and a_{m} appeares in all arrangements σ ∈ S_{2N−1}^{N}, giving the sign −1. So, to every summand in (1) there corresponds the unique opposite one, and the pairs of opposite summands do not intersect. Thus, J_{a⃗(k_{1},k_{2})}^{N}(x) = 0. □

Now we maximize the number of summands in J_{a⃗(k_{1},k_{2})}^{N}(x) in order to note its skewsymmetry w.r.t. the transpositions a_{j} ↦ a_{Σ(j)}, Σ ∈ S_{2N−1}.

Remark 3. The l.h.s. of Jacobi identity (1) equals

(16) J N ; a⃗ = \frac{1}{N!(N − 1)!} ∑_{σ∈S_{2N−1}}(−1)^{σ}[[a_{σ(1)}, …, a_{σ(N)}], a_{σ(N+1)}, …, a_{σ(2N−1)}],

where all elements σ ∈ S_{2N−1} are taken into consideration; see [2, 4].

Remark 4. The r.h.s. in (16) is skew-symmetric w.r.t. any rearrangement Σ of the elements a_{j} ∈ a⃗:

\frac{1}{N!(N − 1)!} ∑_{σ∈S_{2N−1}}(−1)^{σ}[[a_{(σ∘Σ)(1)}, …, a_{(σ∘Σ)(N)}], a_{(σ∘Σ)(N+1)}, …, a_{(σ∘Σ)(2N−1)}] =

= \frac{(−1)^{Σ}}{N!(N − 1)!} ∑_{σ∈S_{2N−1}}(−1)^{σ}[[a_{σ(1)}, …, a_{σ(N)}], a_{σ(N+1)}, …, a_{σ(2N−1)}].

Consequently, the l.h.s in (16) is skew-symmetric also with no restrictions on degrees deg a_{j} ∈ ℤ.

From Remarks 3 and 4 follows

Lemma 4 ([6]). Let 0 ≤ k_{1}, k_{2} ≤ N. Suppose a⃗(k_{1}, k_{2}) is a standard system of monomials (15) and Σa⃗(k_{1}, k_{2}) = {a_{Σ(1)}, …, a_{Σ(2N−1)}} is its arbitrary rearrangement, Σ ∈ S_{2N−1}. Suppose J_{a⃗(k_{1},k_{2})}^{N}(x) ≡ 0; then J_{Σa⃗(k_{1},k_{2})}^{N}(x) ≡ 0.

From Lemmas 3 and 4 we obtain

Theorem 2 ([6]). Let an integer N ≥ 2, 1 ≤ j ≤ 2N − 1, and the elements a_{j} be such that 0 ≤ deg a_{j} ≤ N; then J N ; a⃗(x) ≡ 0.

From Theorems 1 and 2 we have the main theorem:

Theorem 3. Polynomials 𝕜_{N}[x] of degree not greater than N form the homotopical N-Lie algebra with N-linear skew-symmetric bracket (5) for any integer N ≥ 2.

Thus, we have generalized the Lie algebra 𝔰𝔩_{2}(𝕜) of quadratic polynomials 𝕜_{2}[x] to the homotopical N-Lie algebra of the Nth degree polynomials 𝕜_{N}[x] for arbitrary integer N ≥ 2. Still, the dimension n of the base 𝕜 ≡ 𝕜^{1} ∋ x equals 1. In Section 6, we generalize the concept of the homotopical N-Lie algebra of polynomials 𝕜_{N}[x] to the case x ∈ 𝕜^{n}, where integer n ≥ 1 is arbitrary.

4. The case when the monomials x^{k} have an arbitrary degree k ∈ ℕ (the Taylor series) or k ∈ ℤ (the Laurent series) is a natural succession of the preceding research.

Remark 5 ([5]). Let a⃗ be a system of arbitrary analytic functions; then J 3,∞ ; a⃗(x) ≡ 0. The proof is by direct calculation of 20 Wronskians.

Conjecture 1. Choose an integer N ≥ 2 and let a⃗ be a system of 2N − 1 analytic functions; then the relation J N,∞ ; a⃗(x) ≡ 0 is expected to hold.

Remark 6. In fact, the homotopical N-Lie algebra of the polynomials of degree N must not be the unique subalgebra in the homotopical N-Lie algebra of analytic functions. E.g., let k ∈ ℤ and choose an integer p ≥ 0; then the monomials x^{pk+1} are closed w.r.t. the (2, 1, 0)-Lie bracket, the Wronskian (5) with N = 2; see also [8] for mod p-Lie algebras.

We also note that the monomials x^{k}, k ≥ k_{0} > 0, form the homotopical N-Lie subalgebra L_{k_{0}} = {span⟨x^{k}⟩, [·, …, ·] | k ≥ k_{0} > 0} in the homotopical N-Lie algebra L_{ℕ} of the Taylor polynomials. The homotopical N-Lie factoralgebra L_{ℕ}/L_{k_{0}} is k_{0}- dimensional for any k_{0} ∈ ℕ.

Conjecture 2. The homotopical N-Lie algebra of monomials x^{k}, 0 ≤ k ≤ N, is expected to be the unique finite-dimensional subalgebra in the homotopical N-Lie algebra of analytic functions.

The preceding results were submitted before January 25, 2002, for publication [6] in the Proceedings of the Young Scientists Conference at the Faculty of Mechanics and Mathematics, Lomonosov MSU, April 8–13, 2002. Recently, Conjecture 1 was proved in [7]. In this section, we sketch the proof for consistency.

Consider the generalized Wronskians W i⃗ = ∂^{i_{1}} ∧ … ∧ ∂^{i_{N}} ∈ Hom(⋀^{N} A, A). Let a multiindex i⃗ ∈ ℤ_{+}^{N} be such that 0 ≤ i_{1} < … < i_{N}. By Hom_{t}(⋀^{N} A, A) we denote the linear span of the generalized Wronskians W i⃗ such that |i⃗| = t. By definition, put |W i⃗| = |i⃗|.

Lemma 5 ([7]). Let t < N(N − 1)/2, then Hom_{t}(⋀^{N} A, A) = 0.

Proof [7]. If 0 ≠ ∂^{i_{1}} ∧ … ∧ ∂^{i_{N}} ∈ Hom_{t}(⋀^{N} A, A), then t = ∑_{j=1}^{N} i_{N} ≥ 0 + 1 + 2 + … + (N − 1) = N(N − 1)/2. □

Lemma 6 ([8]). Let i⃗ and j⃗ be multiindexes: i⃗ ∈ ℤ_{+}^{k} and j⃗ ∈ ℤ_{+}^{l}; then the relation

W i⃗[W j⃗] ∈ Hom_{|i⃗|+|j⃗|}(⋀^{k+l−1} A, A)

holds.

Corollary 2 ([7]). Let k and l be positive integers, then the identity

W^{0,1,...,k}[W^{0,1,...,l}] = 0

holds.

Proof [7]. First note that |W^{0,1,...,k}| = k(k + 1)/2 for any k. Thence,

W^{0,1,...,k}[W^{0,1,...,l}] ∈ Hom_{\frac{k(k+1)+l(l+1)}{2}} (⋀^{k+l+1} A, A).

Nevertheless, k^{2} + k + l^{2} + l < (k + l + 1)(k + l) for any k and l. Consequently, by Lemma 5,

Hom_{\frac{k^{2}+k+l^{2}+l}{2}} (⋀^{k+l+1} A, A) = 0 and W^{0,1,...,k}[W^{0,1,...,l}] = 0.

This completes the proof. □

Taking into account the definition of J N,∞ ; a⃗, we observe that the identity J N,∞ ; a⃗(x) ≡ 0 in Conjecture 1 follows from Corollary 2.

Remark 7. We also see that Theorem 3 follows from Theorem 1 in virtue of Corollary 2.

From Corollary 2 we obtain

Theorem 4. Let k and l be positive integers, then the relation

[[W^{0,1,...,k}, W^{0,1,...,l}]]^{RN} = 0

holds.

5. We see that the proof of Theorem 3 is non-inductive, i.e., a property of the polynomials of degree not greater than N′ < N to form the homotopical N′-Lie algebra is not used if ℕ ∋ N ≥ 3. In this section, we study the scheme that generates the Wronskian W^{0,1,...,N+1} starting from the Wronskians W^{0,1,...,N−1} and W^{N,N+1}.

In [1], the notion of the exterior multiplication ∧ in Hom(⋀^{∗} A, A) is introduced.

Definition 4. Let Δ ∈ Hom(⋀^{k} A, A) and ∇ ∈ Hom(⋀^{l} A, A) be two operators; by definition, put

(Δ ∧ ∇)(a_{1}, …, a_{k+l}) = ∑_{σ∈S_{k+l}^{k}}(−1)^{σ} Δ(a_{σ(1)}, …, a_{σ(k)}) · ∇(a_{σ(k+1)}, …, a_{σ(k+l)})

for any a_{1}, …, a_{k+l} ∈ A.

We see that the Wronskians W^{0,1,...,N+1} = ∂^{0} ∧ … ∧ ∂^{N+1} are obtained in the fasion of Definition 4, e.g., W^{0,1,...,N+1} = W^{0,1,...,N−1} ∧ W^{N,N+1}.

Definition 5. The structure Δ ∈ Hom(⋀^{N} A, A) is called a multi-derivation, if the Leibnitz rule

Δ(ab, a_{2}, …, a_{N}) = a Δ(b, a_{2}, …, a_{N}) + Δ(a, a_{2}, …, a_{N}) b

is valid for any a, b, a_{j} ∈ A.

Taking into account Remark 2, we restrict ourselves to the case of even N = 2N′.

Proposition 1 ([1]). Let Δ ∈ Lie^{(2k,0,0)}(A) and ∇ ∈ Lie^{(2l,0,0)}(A) be multi-derivations. If [[Δ, ∇]]^{RN} = 0, then Δ ∧ ∇ ∈ Lie^{(2k+2l,0,0)}(A).

From Corollary 2 we know that the Wronskians W^{0,1,...,N+1} do satisfy the identity (2) for any even N, in particular. Thus, the conclusion of Proposition 1 is valid; still, we stress that the Wronskian determinants are not multi-derivations; besides, by a simple argument we observe the discouraging fact that, generally, [[W^{0,1,...,N−1}, W^{N,N+1}]]^{RN} ≠ 0. So, the example of the Wronskians demonstrates that there is another mechnism that generated higher order structures by using the exteriour multiplication rather than the reasoning in Proposition 1.

The task to describe all pairs of the generalized Wronskians W i⃗, W j⃗ ∈ Hom(⋀^{∗} A, A), such that the identity [[W i⃗, W j⃗]]^{RN} = 0 holds, remains an independent problem.

6. In this section, we generalize the concept of the Wronskian determinants to the multidimensional case of the base 𝕜^{n}: further on, we consider the kth order jets J^{k}(n, 1) over the bundle π: 𝕜^{n} ⊕ 𝕜 → 𝕜^{n}, where the base dimension is n ≥ 1 and the algebra A is the associative commutative algebra C^{∞}(𝕜^{n}) of smooth functions.

In order to construct a natural n-dimensional base generalization of the Wronskians, we pass to the geometrical standpoint and make an experimental observation first.

Remark 8 ([5]). Consider the infinite jets J^{∞}(π) over the bundle π: 𝕜 ⊕ 𝕜 → 𝕜. Let x ∈ 𝕜 be the independent base variable, u be the dependent fiber variable, D_{x} be the total derivative w.r.t. x, and u^{(k)} ≡ D_{x}^{k} u be the coordinates in J^{∞}(π) for any k ≥ 0. By d_{C} we denote the Cartan differential, d_{C}: C^{∞}(J^{∞}(π)) → CΛ(J^{∞}(π)), that maps u^{(k)} ↦ du^{(k)} − D_{x}u^{(k)} dx. The Wronskian determinants (5) can be interpreted as action of the N-forms d_{C}u ∧ … d_{C}u^{(N−1)} ∈ CΛ^{∗}(J^{N−1}(π)) ⊂ CΛ^{∗}(J^{∞}(π)) upon the evolutionary vector fields ϶_{a_{j}}≡ ∑_{k=0}^{∞} D_{x}^{k}(a_{j}) ∂/∂u^{(k)}:

[a_{1}, a_{2}] = du ∧ d(u′) (϶_{a_{1}}, ϶_{a_{2}}),
[a_{1}, a_{2}, a_{3}] = du ∧ d(u′) ∧ d(u′′) (϶_{a_{1}}, ϶_{a_{2}}, ϶_{a_{3}}),  etc.

for any a_{j} ∈ C^{∞}(𝕜).

Remark 9. Consider the ternary bracket □_{3} ∈ Hom(⋀^{3} C^{∞}(ℝ^{2}), C^{∞}(ℝ^{2})):

□_{3}(a_{1}, a_{2}, a_{3}) = d_{C}u ∧ d_{C}u_{x} ∧ d_{C}u_{y}(϶_{a_{1}}, ϶_{a_{2}}, ϶_{a_{3}}) = |\begin{matrix}a_{1} & a_{2} & a_{3} \\ D_{x}(a_{1}) & D_{x}(a_{2}) & D_{x}(a_{3}) \\ D_{y}(a_{1}) & D_{y}(a_{2}) & D_{y}(a_{3}) \end{matrix}| .

For the bracket □_{3}, the homotopical ternary Jacobi identity □_{3}[□_{3}] = 0 of the form (2) holds. We prove this fact by direct calculations using the Jet software [9].

Proposition 2 ([10]). The dimension of the jets space J^{k}(n, 1) vertical part J^{k}(n, 1)/𝕜^{n} is

dim \frac{J^{k}(n, 1)}{𝕜^{n}} = dim J^{k}(n, 1) − n = ∑_{i=0}^{k} \binom{n + i − 1}{n − 1} = \binom{n + k}{n}.

We also note that the dimension N ≡ \binom{n+k}{n} is such that the inequality

dim J^{2k}(n, 1) − n − 1 > 2(dim J^{k}(n, 1) − n − 1)

is valid; in what follows, we need to substract the dimension dim J^{0}(n, 1) = n + 1 in order to deal with non-trivial multiindexes σ ≠ ∅ such that |σ| > 0 and u_{σ} ∉ T^{∗}J^{0}(n, 1).

Choose arbitrary positive integers n and k; then N ≡ \binom{n+k}{n} is the dimension dim(J^{k}(n, 1)/𝕜^{n}). Let A = C^{∞}(𝕜^{n}) be the algebra of smooth functions a_{j} ∈ A, 1 ≤ j ≤ N. Now we define the N-linear skew-symmetric bracket □ ∈ Hom(⋀^{N} A, A): by definition, put

(17)  □(a_{1}, …, a_{N}) = ⋀_{l=0}^{k}( ⋀_{|σ|=l} d_{C} · D_{σ}u) (϶_{a_{1}}, …, ϶_{a_{N}}).

Theorem 5. The N-linear skew-symmetric bracket □ ∈ Hom(⋀^{N} A, A) defined in (17) satisfies the homotopical N-Lie Jacobi identity

(2)  □[□] = 0.

Proof. In contrast with the reasoning in Section 4, we deal with Dσ⃗ = D_{σ_{1}} ∧ … ∧ D_{σ_{N}}, where σ_{j} is a multiindex (♯x^{1}, …, ♯x^{n}) ∈ ℤ_{+}^{n} for any j, 1 ≤ j ≤ N = \binom{n+k}{n}. By definition, put |Dσ⃗| = |σ⃗| = ∑_{j−1}^{N} |σ_{j}|. We see that |□[□]| = 2|□|, c.f. Lemma 6.

Now we note that the non-trivial skew-symmetric (2N − 1)-linear bracket □_{min} ∈ Hom(⋀^{2N−1} A, A) with the minimal norm is

(18)  □_{min} = □ ∧ (  ∑_{j̄∈Λ^{N−1}(J^{2k}(n,1)/J^{k}(n,1))} const(j̄) · D_{σj̄}),

where const(j̄) ∈ 𝕜 are some constant coefficients.

We claim that |□_{min} | > 2|□|, and thence □[□] = 0. Really, consider the r.h.s. in (18) and note that |Δ ∧ ∇| = |Δ| + |∇|. The set of N different derivatives in the first wedge factor □ admits the canonical splitting:

□ = Dτ⃗ = 1 ∧ \underbrace{D_{τ_{2}} ∧ … ∧ D_{τ_{N}}}_{N − 1 factors},

where τ⃗ contains all multiindexes in J^{k}(n, 1), and those underbraced derivatives are in bijective correspondence with N −1 different derivatives in any summand in the second wedge factor (there is the correspondence due to the equal numbers of elements). Still,

1 ≤ |D_{τ_{i}}| = |τ_{i}| ≤ k < k + 1 ≤ |σ_{j}| = |D_{σ_{j}}| ≤ 2k ∀i ≠ 1, ∀j.

Indeed, if a multiindex σ_{j} is such that u_{σ_{j}} ∈ T^{∗}(J^{2k}(n, 1)/J^{k}(n, 1)), then σ_{j} is longer than any multiindex τ_{i} such that u_{τ_{i}} ∈ T^{∗}J^{k}(n, 1). Consequently, the norm |·| of the second wedge factor in the r.h.s. of (18) is strictly greater than |□|, and thus □[□] is trivial. This completes the proof. □

In addition, we give an example of the binary homotopical 3-Lie algebra of polynomials:

Example 2. The polynomials span_{𝕜}⟨1, x, y, xy⟩ ⊂ 𝕜_{2}[x, y] endowed with the ternary bracket 1 ∧ D_{x} ∧ D_{y} form a homotopical 3-Lie algebra.

The commutation relations of this algebra are

[1, x, y] = 1,  [1, x, xy] = x,  [1, y, xy] = −y, and [x, y, xy] = −xy,

and we see that the case under consideration is somehow different from the scheme of Example 1.

7. The nontrivial one-dimensional central extension of the holomorphic vector fields a(x) d/dx Lie algebra Vect^{1} on the circumpherence x^{2} = 1 of unit raduis is well known [11] to be the Virasoro algebra. In the basis ⟨d_{n} = −x^{n+1} d/dx, c⟩, the generating relations are

[d_{m}, d_{n}] = (m − n)d_{m+n} + δ_{m+n,0} \frac{m^{3} − m}{12} · c,
[d_{m}, c] = 0,  ∀m ∈ ℤ.

Now we discuss a possibility to construct a nontrivial one-dimensional central extension of the homotopical 3-Lie algebra. Choose d_{n} ≡ x^{n+3/2} d/dx for the basis and write down the decomposition of the ternary bracket [ · , · , ·] w.r.t. the extended basis:

[d_{k}, d_{l}, d_{m}] = π(k, l, m)d_{k+l+m} + f(k, l, m) · c,

where π(k, l, m) ≡ (k − l)(l − m)(m − k) and

f(a_{σ(1)}, a_{σ(2)}, a_{σ(3)}) = (−1)^{σ}f(a_{1}, a_{2}, a_{3}),

due to the skew-symmetry of the bracket for any permutation σ ∈ S_{3} and any set of arguments a⃗, ♯a = 5. The condition c ∈ Z for c to belong to the center of the extended algebra is naturally understood as

[d_{i}, d_{j}, c] ≡ 0, ∀i, j ∈ ℤ.

For any a⃗, the Jacobi identity defined in (1) implies the functional equation [5]

(19)  ∑_{σ∈S_{5}^{2}}(−1)^{σ}π(a_{σ(1)}, a_{σ(2)}, a_{σ(3)}) · f(a_{σ(1)} + a_{σ(2)} + a_{σ(3)}, a_{σ(4)}, a_{σ(5)}) = 0

for the function f. Nontrivial solutions to Eq. (19) are unknown. Final remarks.

1. To each Bäcklund transformation between equations of the f-Gordon type u_{xy} = f(u), where d^{2}f/du^{2} = const·f, Shadwick assigned [12] the one-dimensional representation of 𝔰𝔩_{2}(ℝ) in smooth vector fields a(x) d/dx. Given a representation of the homotopical N-Lie generalization of 𝔰𝔩_{2}(ℝ) in monomials x^{k}, 0 ≤ k ≤ N, can we generalize Bäcklund transformations as relations between N PDE? We assume that these relations are non-trivial, i.e., they do not split into pairs of initial Bäcklund transformations (in general, those binary initial Bäcklund transformations may even not exist).

If such N-ary Bäcklund transformations existed, the question of their deformations by using the Frölicher-Nijenhuis bracket [13, 14] would naturally arise.

2. Suppose a⃗ is a system of ♯a⃗ = 2N − 1 analytic functions a_{j} and a⃗ contains a standard subsystem of monomials (15). Consider the N-ary bracket (5) satisfying the homotopical N-Lie Jacobi identity (1), as N → ∞. In this case, it is interesting to study whether bracket (5) has an integral limit or any relation to super-power series. Quantization of bracket (5) is a matter of further study, also.

3. In fact, generalizations of the PDE conformal symmetry algebras are provided by Theorems 3, 4, and 5. So, one can try to describe standard objects in CFT in the general homotopical N-Lie case now; e.g., non-trivial one-dimensional central extensions of the homotopical 3-Lie algebra of smooth functions depend on solutions of Eq. (19).

4. Treating a_{j} ∈ a⃗ as state functions of N + 1 (or 2N − 1) particles, we see that bracket (5) (resp., the Jacobi identity (1)) describes a collective effect, their interaction. Intuitively, in descriptions of the condensed matter models in statistical physics, the role of large N ≫ 2 is supposed to grow as the temperature θ → 0. Also, the homotopical N-Lie dynamics with different Ns may describe different phase states and phase state transformations, e.g., a process starting from binary interactions and passing through near ordered matter to far ordered states. Study of the 𝔰𝔩_{2}(𝕜)-based models (or a(x) d/dx-based models, a ∈ C^{∞}(𝕜)) with respect to their homotopical N-Lie generalizations would be of great interest.

Acknowledgement. The author thanks A. V. Ovchinnikov, A. M. Verbovetsky, and Prof. E. B. Vinberg for discussion and remarks, and also Prof. I. S. Krasil’shchik for pointing out the paper [7]. The work was partially supported by the INTAS Grant YS 2001/2-33.

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Lomonosov Moscow State University, Faculty of Physics, Department of Mathematics. Vorobjevy Gory, 119992 Moscow, Russia., Mail to: Independent University of Moscow, Math College, Bol’shoj Vlas’evki per. 11, Moscow 121002 Russia.

E-mail address: arthemy@mccme.ru