A Note on Cyclic Gradients

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1 A Note on Cyclic Gradients Dan Voiculescu arxiv:math/ v1 [math.ra] 4 May 2000 To the memory of Gian-Carlo Rota The cyclic derivative was introduced by G.-C. Rota, B. Sagan and P. R. Stein in [3] as an extension of the derivative to noncommutative polynomials. Here we show that there are simple necessary and sufficient conditions for an n-tuple of polynomials in n noncommuting indeterminates to be a cyclic gradient (see Theorem 1 and similarly for a polynomial to have vanishing cyclic gradient (see Theorem 2. Our interest in cyclic gradients stems from free probability theory and random matrices (see the Remark at the end [1],[2],[4],[5],[6]. This note should also reduce the paucity of results on cyclic derivatives in several variables pointed out in [3, page 73]. Let K n = K X 1,..., X n be the ring of polynomials in noncommuting indeterminates X 1,...,X n with coefficients in the field K of characteristic zero. The partial generalized difference quotients are the derivations j : K n K n K n such that j X k = 0 if j k and j X j = 1 1. The here is over K and K n K n is given the bimodule structure such that a(b c = ab c, (b cd = b cd. The partial cyclic derivatives are then δ j = µ j : K n K n where µ(a b = ba. We shall denote by N : K n K n the number operator, i.e. the linear map so that N1 = 0, NX i1... X ik = kx i1...x ik. Also, CK n will denote the cyclic subspace, i.e. the vector subspace spanned by all cyclic symmetrizations of monomials CX i1... X ip = X ij+1...x ip X i1...x ij, p 1 and C1=0 1 j p (the constants are not in the cyclic subspace. 1

2 Theorem 1 Let P 1,...,P n K n. The following conditions are equivalent: (i there is P K n such that δ j P = P j (1 j n. (ii [X j, P j ] = 0. (iii (iv δ k ( X j P j CK n. X j P j = (N + IP k. (Here I denotes the identity map of K n to itself. Proof. It is easily seen that it suffices to prove the theorem for homogeneous P 1,..., P n of the same degree, i.e. we may assume NP j = sp j (1 j n for some s 0. Also the case of constants being obvious we will concentrate on s 1. (i (ii To check that [X j, δ j P] = 0 it suffices to do so when P is a monomial X i0...x is. Then δ j P = i p=j X ip+1...x is X i0...x ip 1 so that ( Xip...X is X i0...x ip 1 X ip+1...x is X i0...x ip [X j, δ j P] = i p=j and hence (ii (iii Let [X j, δ j P] = 1 p s The coefficient of X i0... X is in ( Xip...X is X i0...x ip 1 X ip+1...x is X i0...x ip = 0. P j = i 1,...,i s c j i 1...i s X i1... X is [X j, P j ] is c i 0 c is i 0...i s 1. Hence (ii gives c i 0 = c is i 0...i s 1. On the other hand, if c i0...i s denotes the coefficient of X i0...x is in X j P j, clearly c i0...i s = c i 0, so that (ii implies c i0...i s = c isi 0...i s 1, i.e. cyclicity. i j n 2

3 (iii (iv As before, let c j i 1...i s and c i0...i s denote the coefficients of P j and j X j P j respectively. Then c i 0 = c i0...i s and the cyclicity condition gives c i0...i s = c isi 0...i s 1. We have ( δ k X j P j k = c i0...i s X ir+1...x is X i0...x ir 1 i 0...i s {r:i r=k} = i 0...i s {r:i r=k} = 0 r s c k i r+1...i si 0...i r 1 X ir+1...x is X i0... X ir 1 i 0...i r 1 i r+1...i s c k i r+1...i si 0...i r 1 X ir+1...x is X i0...x ir 1 = (s + 1P k. (iv (i Since the P j are homogeneous of the same degree and the field characteristic is zero, this is obvious. There is also a simple description of the noncommutative polynomials with vanishing cyclic gradient. Theorem 2 We have Ker δ = Proof. (i Ker δ Ker C. We have (ii Clearly, [X k, K n ] + C1 = C1 + [K n, K n ] = Ker C Cp = 1 j m X j δ j p = 0 [X k, K n ] + C1 [K n, K n ] + C1 Also, since 1 Ker C and [X i1...x ir, X ir+1...x ir+s ] is the difference of two cyclic permutations of X i1... X ir+s, we have [K n, K n ] + C1 Ker C. To see that Ker C [X k, K n ] + C1, remark that Ker C is spanned by homogeneous elements and that Cp = 0, where p is homogeneous of degree m iff p is a linear combination of differences X i1...x im X i2...x im X i0. 3

4 (iii To see that [X k, K n ] + C1 Ker δ, it suffices to show that [X k, X i1...x is ] Ker δ. This is clearly so, since [X k, X i1...x is ] = X k X is...x is X i1...x is X k and the cyclically equivalent elements X k X i1...x is, X i1...x is X ik gradient. have the same cyclic Putting together the two theorems, we have an exact sequence δ 0 [K n, K n ] K n (K n n θ K n where θ((p j = j [X j, P j ]. Remark. The motivation for this note is from free entropy and large deviations for random matrices. Let (M, τ be a von Neumann algebra with normal faithful trace-state τ and X k = Xk M (1 k n which are algebraically free. Let J k = J (X k : C X 1,...,X k 1, X k+1,..., X n be the noncommutative Hilbert transforms defined in [4], in connection with free entropy. On the other hand, the upper bound for large deviations for n-tuples of random matrices found in [2] fits well with free entropy except for a term involving cyclic gradients and about which it is not known whether it is not actually zero. The precise question is, whether the n-tuple (J k (when it exists is a limit in 2-norm of cyclic gradients of polynomials in the noncommuting variables X 1,...,X n? The theorem we proved here provides a partial affirmative answer: If (J k are noncommutative polynomials in X 1,...,X n, then there is a noncommutative polynomial P in X 1,..., X n such that J k = δ k P (1 k n. Indeed, by Corollary 5.12 in [5] we have k [J k, X k ] = 0. Hence the commutator condition (ii in the Theorem is satisfied. Acknowledgments. This research was conducted by the author for the Clay Mathematics Institute. Partial support was also provided by National Science Foundation Grant DMS

5 References [1] P.Biane, R.Speicher. Free diffusions, free entropy and free Fisher information, preprint. [2] T.Cabanal-Duvillard, A.Guionnet. Large deviations upper bounds and noncommutative entropies for some matrices ensembles, preprint. [3] G.-C.Rota, B.Sagan, P.R.Stein. A cyclic derivative in noncommutative algebra. Journal of Algebra 64, (1980. [4] D.Voiculescu. The analogues of entropy and of Fisher s information measure in free probabilitiy theory, V: noncommutative Hilbert transforms. Invent. Math. 32, no. 1, (1998. [5] D.Voiculescu. The analogues of entropy and of Fisher s information measure in free probabilitiy theory, VI: liberation and mutual free information. Advances in Mathematics 146, (1999. [6] D.Voiculescu. Lectures on free probability theory. Notes for a course at the Saint-Flour Summer School on Probability Theory, preprint (1998. Department of Mathematics University of California Berkeley, California dvv@math.berkeley.edu 5