Noncommutative Formal Power Series and Noncommutative Rational Functions

NONCOMMUTATIVE FORMAL POWER SERIES AND NONCOMMUTATIVE RATIONAL FUNCTIONS

Victor Vinnikov (Ben Gurion University)

1 2 NC polynomials and nc rational functions

I am interested in the free skew field (the skew field of noncommutative rational functions) which is the universal skew field of fractions of the free algebra.

My motivation comes from operator algebras, operator theory, and systems and control, where we are interested in some kind of free noncommutative functions that can take as arguments certain tuples of arbitrary (not necessarily commuting) operators on (say) a Hilbert space. NC polynomials and nc rational functions 3

Notation:

Gd — the free monoid with d generators g1,..., gd; elements of Gd are arbitrary words w = gi` ··· gi1 and the monoid operation is concatenation; the neutral element is the empty word ∅; we use the noncommutative multipower notation xw = xi` ··· xi1. Khx1, . . . , xdi — the free associative algebra over a ﬁeld K with d generators x1, . . . , xd (the semigroup algebra of the free monoid); elements of Khx1, . . . , xdi are arbitrary noncommutative polynomials p = P p xw (ﬁnite sum) over in x , . . . , x . w∈Gd w K 1 d 4 NC polynomials and nc rational functions

A reminder on noncommutative localization: a skew field of fractions of a given ring is a skew field containing the ring and generated by it in the sense that no proper skew subfield contains the ring. If a noncommutative integral domain R satisfies the so called right Ore condition, (1) ∀ a, b ∈ R, b =6 0 ∃ c, d ∈ R, d =6 0: ad = bc, then one can construct a skew field of fractions analogously to the commutative case as the ring of right quotients, i.e., of formal fractions cd−1, d =6 0. In general, a skew field of fractions of a noncommutative integral domain might or might not exist.

If a skew ﬁeld of fractions exists, it might not be unique. NC polynomials and nc rational functions 5

The correct definition (P.M. Cohn): A skew field of fractions K of a ring R is called a universal skew field of fractions if for every homomorphism φ: R → L to a skew field L there exists a local subring K0 ⊆ K containing R and a homomorphism θ : K0 → L extending φ such that

ker θ = {x ∈ K0 : x is not invertible in K0}.

Remarks: • The universal skew field of fractions is, when it exists, unique up to isomorphism. • An alternative characterization of the universal skew field of fractions is as follows: every matrix over R, which becomes invertible under some homomorphism from R to a skew field, is invertible over K. 6 NC polynomials and nc rational functions

The free algebra Khx1, . . . , xdi admits a universal skew ﬁeld of fractions — namely, the skew ﬁeld K

There are several other constructions of the free skew ﬁeld, most important (P.M. Cohn) as the localization with respect to full matrices over Khx1, . . . , xdi, since Khx1, . . . , xdi is a ﬁr (free ideal ring) hence a Sylvester domain.

Alternatively, since the free group Ud on d generators can be ordered, we can consider the skew field K((Ud)) of power series with exponents U , f = P f xw (well or- d w∈Ud w dered support, Malcev–Neumann). It turns out that the sub-skew-field of K((Ud)) generated by Khx1, . . . , xdi is the universal skew field of fractions (Reutenauer). Main disadvantages: no easy characterization of those general- ized series that correspond to nc rational functions; no easy relation to evaluations. Rational nc power series 7

We can consider the local ring O0 of nc rational functions with 0 ∈ dom r. A nc rational expression which is regular at 0 determines a nc formal power series. This correspondence is deﬁned recursively using addition and multiplication of nc formal power series and inversion of a nc formal power series with an invertible constant term (the coeﬃcient of z∅). Fur- thermore, r1 and r2 are equivalent if and only if the corresponding nc formal power series coincide.

Therefore O0 can be identiﬁed with the ring of rational nc formal power series — the smallest subring of Khhx1, . . . , xdii containing Khx1, . . . , xdi and closed under the inversion (of invertible elements). 8 Rational nc power series

TFAE (1) P r xw is a rational nc power series. w∈Gd w (2) (Kleene, Sch¨utzenberger) P r xw is a recognizable nc power series, i.e., w∈Gd w L×L there exist A1,...,Ad ∈ K for some integer L, L×1 1×L B1,...,Bd ∈ K , C ∈ K so that rwgj = w CA Bj for all w ∈ Gd, j = 1, . . . , d ⇐⇒ the nc rational function r ∼ P r xw admits a (Fornasini– w∈Gd w Marchesini) realization (2) −1 r(x) = D+C(I −A1x1−· · ·−Adxd) (B1x1+···+Bdxd)

(D = f∅ = f(0)). (3) (Fliess) The corresponding inﬁnite Gd × Gd Hankel matrix = [r ] has ﬁnite rank. H uv u,v∈Gd

Furthermore (Ball–Groenewald–Malakorn), the realization (2) is essentially unique if it is minimal (L is as small as possible ⇐⇒ it is controllable: span im AwB = i=1,...,d, w∈Gd i L w K , and observable: ∩w∈Gd ker CA = {0}), and (Kaliuzhnyi- Verbovetzkyi–V) in this case: ∞ a n×nd dom r = X = (X1,...,Xd) ∈ K : n=1

det(ILn − X1 ⊗ A1 − · · · − Xd ⊗ Ad) =6 0 . Rational nc power series 9

Comments: • The proof of the last result uses the universal deriva- tions (nc difference-differential operators) on the free skew field,

∆j : K) → K) ⊗ K,)

j = 1, . . . , d, deﬁned by ∆j(xi) = δij(1⊗1), ∆j(r1r2) = ∆j(r1)(1 ⊗ r2) + (r1 ⊗ 1)∆j(r2). • If r ∈ O , r ∼ P r xw, then r = ∆wr(0,..., 0) 0 w∈Gd w w and the nc power series expansion of r at 0 is an in- stance of (Brooke) Taylor – (Joseph L.) Taylor series expansion of a nc function. • Applications: automata and formal languages; systems and control (multidimensional system theory, robust control, semideﬁnite programming for dimen- sion independent problems); multivariable operator theory; free probability. • Limitations: by translation we can apply the theory of rational nc power series to the local ring Oλ of nc rational functions with λ = (λ1, . . . , λd) ∈ (dom r). But we cannot cover nc rational functions −1 such as (x1x2 − x2x1) . 10 Rational nc power series: general matrix centre

However: consider the local ring OY of nc rational s×s d functions with Y = (Y1,...,Yd) ∈ (K ) ∈ dom r. Then we have power series expansion of r around Y which d is given by, for X ∈ (Ksm×sm) , X sw (3) r(X) ∼ (X − Im ⊗ Y ) rw.

w∈Gd s×s l Here, the coefficient rw is a l-linear mapping (K ) −→ Ks×s, where l is the length of the word w, or alternatively ⊗l a linear mapping (Ks×s) −→ Ks×s. ( sw denotes a multipower of a d-tuple of sm × sm matrices viewed as m × m matrices over the tensor algebra ⊗l of Ks×s, hence it is a m × m matrix over (Ks×s) ; we can apply rw to every entry of this matrix yielding a matrix in s×s m×m ∼ sm×sm (K ) = K — which is where the value r(X) lies.) Important difference with the case of a scalar centre: the coefficients rw are not arbitrary multilinear mappings; they have to satisfy certain compatibility conditions with respect to Y . NC formal power series with a matrix centre Y , of the form (3), with the coefficients satisfying these conditions, form a ring Khhx1, . . . , xdiiY with an obvious convolution product.

Furthermore, OY can be identiﬁed with the ring of rational nc formal power series with centre Y — the smallest subring of Khhx1, . . . , xdiiY containing Khx1, . . . , xdi and closed under the inversion (of invertible elements). Rational nc power series: general matrix centre 11

tfae (Kaliuzhnyi-Verbovetzkyi–V) (1) P (X − I ⊗ Y ) sw r is a rational nc power w∈Gd m w series with centre Y . (2) P (X − I ⊗ Y ) sw r is a recognizable nc power w∈Gd m w s×s Ls×Ls series, i.e., there exist A1,..., Ad : K → K s×s Ls×s for some integer L and B1,..., Bd : K → K s×Ls linear mappings, C ∈ K so that rwgj = (Im ⊗ w C)A Bj for all w ∈ Gd, j = 1, . . . , d ⇐⇒ the nc rational function r ∼ P (X − I ⊗ Y ) sw r w∈Gd m w d admits a realization: for X ∈ (Ksm×sm) ,

r(X) = Im ⊗ D + (Im ⊗ C) −1 ILsm − (X1 − Im ⊗ Y1)A1 − · · · − (Xd − Im ⊗ Yd)Ad (X1 − Im ⊗ Y1)B1 + ··· + (Xd − Im ⊗ Yd)Bd .

(D = f∅ = f(0)). 12 Rational nc power series: general matrix centre

Further work in progress: • Minimal realization: uniqueness and

sm×smd (dom r)sm = X = (X1,...,Xd) ∈ K : det ILsm−(X1−Im⊗Y1)A1−· · ·−(Xd−Im⊗Yd)Ad =6 0 . • An analogue of the ﬁnite Hankel rank condition for rationality.

• Construction of the K) as the direct limit of the rings of rational nc power series with centre Y over all Y . • Applications. A diﬀerent approach to the realization problem for nc rational functions that are not regular at 0: Volˇciˇc, arXiv preprint.