Efficient Decomposition of Associative Algebras

Transcription

1 Effcent Decomposton of Assocatve Algebras W Eberly Department of Computer Scence Unversty of Calgary Calgary, Alberta Canada, T2N 1N4 emal: eberly@cpscucalgaryca M Gesbrecht Department of Computer Scence Unversty of Mantoba Wnnpeg, Mantoba Canada, R3T 2N2 emal: mwg@csumantobaca Abstract We present new, effcent algorthms for some fundamental computatons wth fnte-dmensonal (but not necessarly commutatve) assocatve algebras For a semsmple assocatve algebra A over a fnte feld or number feld F, we show how to compute a bass for the centre of A and the complete Wedderburn decomposton of A as a drect sum of smple algebras If A s gven by a generatng set of matrces n F m m then our algorthm requres about O(m 3 ) operatons n F, plus the cost of factorng a polynomal n F[x] of degree O(m), and the cost of generatng a small number of random elements from A We also show how to compute a complete set of orthogonal prmtve dempotents n any assocatve algebra over a fnte feld 1 Introducton Determnng the structure of an assocatve algebra A and ts modules s a fundamental problem n abstract and appled algebra Here, a fnte dmensonal assocatve algebra, or algebra for short, s a fnte dmensonal vector space over a feld F equpped wth a multplcaton under whch the space forms an assocatve (though not necessarly commutatve) rng wth dentty In ths paper we gve very effcent algorthms for some fundamental computatons wth assocatve algebras over fnte felds and number felds Recall that the (Jacobson) radcal Rad(A) of an algebra A s the ntersecton of all maxmal left deals n A A s sad to be semsmple f Rad(A) = 0 and smple f A has no nontrval two-sded deals The Wedderburn Structure Theorem (Wedderburn, 1907) shows that for any semsmple A, A = A 1 A 2 A k (11) Research was supported n part by Natural Scences and Engneerng Research Councl of Canada research grant OGP Research was supported n part by Natural Scences and Engneerng Research Councl of Canada research grant OGP and Unversty of Mantoba research grant A complete verson of ths paper s avalable va anonymous ftp on ftpcsumantobaca n /pub/mwg/wedderburnpsz Appears n the Proceedngs of the ACM Internatonal Symposum on Symbolc and Algebrac Computaton, 1996 (ISSAC 96), pp for smple algebras A 1,, A k A, and each A = D t t where D s a dvson algebra (non-commutatve feld) wth an extenson feld E of F as ts centre We gve a fast probablstc algorthm to compute a bass for the centre of A and a representaton of the complete Wedderburn decomposton of A as a drect sum of smple algebras We suppose throughout that A s gven by some generatng set L A whch allows us to produce random elements of A effcently We assume that we can select a random element α unformly from A usng O(R(A)) operatons n F A more precse defnton s gven below After ntroducng some specal types of elements needed n the computaton n Secton 21, we show how to dentfy the centre of A n Secton 22 Over a number feld or large fnte feld wth at least 12n 6 elements we present an algorthm for ths whch requres an expected number of O((m 3 + F(m) + R(A)) log(1/ǫ)) operatons n F, where F(m) operatons n F are suffcent to factor a polynomal n F[x] of degree m The algorthm s probablstc of the Monte Carlo type; ǫ > 0 s a user-specfed tolerance and the answer s guaranteed correct wth probablty at least 1 ǫ In Secton 23 we show how to decompose the centre to obtan a decomposton of the entre algebra Assumng that the centre was correctly dentfed, we fnd central dempotents and a transton matrx whch reveals the smple components wth an expected number of O(m 3 + F(m) + R(A)) operatons n F Ths algorthm s probablstc of the Las Vegas type: t always returns the correct answer Snce one expects to requre matrx multplcaton n such a computaton, the cost of ths algorthm s surprsngly low Asymptotcally more exact results, n terms of the cost of matrx and polynomal multplcaton, are also presented n Theorems 28 and 210 Over a small fnte feld F, wth fewer than 12n 6 elements, thngs are somewhat trcker because certan types of elements used n the above algorthms are no longer guaranteed to exst We can make use of the algorthms for large felds by carefully extendng the ground feld to one whch s suffcently large To recover the structure correctly we must work n O(log m) of these extensons We nstead take a dfferent approach descrbed below When A s a (not necessarly semsmple) assocatve algebra over a fnte feld, we show how to compute a complete set of orthogonal prmtve dempotents Recall that an dempotent s an element ω A such that ω 2 = ω Two dempotents are orthogonal f ther product s zero, and a nonzero dempotent s prmtve f t cannot be represented as a sum of two or more nonzero orthogonal dempotents In Secton 31 we ntroduce decomposable elements whch al- 170

2 low the generaton of non-trval dempotents We show how to use these effcently and that there are many of them n any algebra These elements are smlar to the Fttng elements employed for a smlar purpose by Schneder (1990), but are much easer to fnd In Secton 32 we show how to use decomposable elements to compute a complete set of orthogonal prmtve dempotents Ths algorthm requres an expected number of O((m 3 +m 2 log q + R(A)) log 2 (m) log(1/ǫ)) operatons n a fnte feld F of sze q, for an algebra presented as above Ths s agan a Monte Carlo algorthm: for a user specfed ǫ the algorthm returns the correct answer wth probablty at least 1 ǫ Fnally, n Sectons 33 and 34 we return to the semsmple case for small felds In Secton 33 we show how to construct bases for the smple components and central smple dempotents from any set of prmtve orthogonal dempotents Ths gves a Monte Carlo algorthm for fndng these dempotents whch requres an expected number of O((m 3 + m 2 log q + R(A)) log 2 (m) log(1/ǫ)) operatons n F and returns the correct answer wth probablty at least 1 ǫ We then show n Secton 34 that there s an effcent test for the correctness of the output of the Monte Carlo algorthms n the fnte feld case for semsmple algebras Ths algorthm actually computes an explct somorphsm between each smple component and a full matrx algebra over an extenson feld of F Ths yelds a Las Vegas type probabltc algorthm (e, the output s always correct) for the decomposton of semsmple algebras over fnte felds whch requres an expected number of O((m 3 + m 2 log q + R(A)) log 2 (m)) operatons n F 11 Hstorcal Perspectve The study of assocatve algebras goes back to the semnal work of Perce (1881), the beautful structure theorem of Wedderburn (1907), and the exploraton of the radcal (see Jacobson (1956)) As well as standng as a feld of actve research n ts own rght, the mportance of ths theory n the study of groups and ther representatons has been developed snce Noether (1929) The computatonal study of assocatve algebras s obvously consderably younger The frst general algorthms for computng ther structure are due to Fredl & Rónya (1985), who gve polynomal-tme algorthms to fnd the Jacobson radcal and to decompose a semsmple algebra as a drect sum of smple algebras Subsequent work by Rónya (1987, 1990, 1992) examned addtonal questons over number felds, and n partcular showed that fndng a complete set of prmtve, orthogonal dempotents n a smple algebra over a number feld had the same complexty as factorng ntegers, e, t s (currently) ntractable Whle theoretcally of great nterest, these algorthms are probably not practcal For commutatve algebras, Gann et al (1988) gve an effcent algorthm to decompose an assocatve algebra over Q as a drect sum of local algebras Much more practcal work on a closely related problem was nstgated by Parker (1984), who gves a probablstc algorthm (the Meat-Axe ) to test for rreducblty of an A-module and to splt reducble A-modules, where A s a matrx algebra over a fnte feld Whle the algorthm s apparently not analysed n general, t appears to works very well for algebras over very small fnte felds (typcally F 2) Ths was extended to work over any ground feld n Holt & Rees (1994) for all but one famly of modules Ths dffculty has apparently now been overcome as well The Krull-Schmdt theorem guarantees that every A- module M can be unquely decomposed as a drect sum of ndecomposable A-modules (up to somorphsm) In hs survey paper, Mchler (1990) proposes the open problem of fndng an effcent algorthm to fnd ths decomposton n the case when A s an algebra over a fnte feld F It clearly suffces to fnd a set of orthogonal prmtve dempotents ω 1,, ω s EndA(M) such that ω = 1, whch 1 s Mchler (1990) also proposes as an open problem We propose a very effcent algorthm for the computaton of a set of orthogonal prmtve dempotents of an algebra n Secton 32 Ths problem was frst addressed n Schneder (1990) for small fnte felds by the selecton of Fttng elements n EndA(M) whch allow ts decomposton In Secton 31 we present the smlar noton of decomposable elements whch also allow the effcent decomposton of the algebra, but are much easer to fnd n general 12 Notaton We wll generally te the complexty of our results to that of matrx multplcaton We assume that O(MM(m)) operatons n a feld F are suffcent to multply two matrces n F m m Usng the standard algorthm requres MM(n) = n 3 whle the currently best known algorthm of Coppersmth & Wnograd (1990) allows MM(n) = n 2376 Also assume that O(M(m)) operatons n F are suffcent to multply two polynomals n F[x] of degree m Usng the standard algorthm allows M(m) = m 2, whle the algorthm of Schönhage & Strassen (1971) and Schönhage (1977) allows M(m) = m log mlog log m For notatonal convenence n the statement of complextytheoretc results, f a sub-cubc algorthm for matrx multplcaton s used we assume that M(m) = mlog mlog log m and that MM(m) = m θ for some θ > 2 Fnally, we assume that a polynomal of degree m n F[x] can be factored usng O(F(m)) operatons n F If F s a fnte feld wth q elements, then Berlekamp s (1970) algorthm allows F(m) = MM(m) + m 2 log q operatons n F If F s a number feld then the algorthm of Landau (1985) wll factor a polynomal of degree m wth a polynomal number of operatons 13 Selectng random elements of A To prove correctness of our probablstc algorthms, we requre some techncal condtons on the presumed ablty to select a random element α from A Recall that ths s assumed to be possble wth O(R(A)) operatons n F If A s fnte and small we assume that α s selected unformly from A When A s large or nfnte we assume that there exsts a bass γ 1,, γ n A for A and subset S of F of suffcently large sze κ Elements of A are unformly generated from aγ for 1 k unformly and ndependently selected elements a S In practce, almost any reasonable scheme for generatng random elements of A wll work However, the only scheme we know of for generatng such elements wth provable unformty requres a bass of n matrces n F m m for A In ths case R(A) = nm 2 The requrement for perfect unformty can be relaxed somewhat whle stll mantanng provably correct algorthms, though not suffcently to yeld an asymptotc mprovement n performance 2 The Wedderburn Decomposton over Large Felds In ths secton we present an algorthm to compute the Wedderburn decomposton of a semsmple algebra A over a fnte feld or number feld F We assume throughout that A decomposes as a sum of full matrx rngs over dvson algebras as n (11), so that for 1 k the centre of A s E and [D : E ] = e If F s fnte then D = E for all 171

3 Specal elements of A (separators, splttng pars, and complemented splttng pars) whch can be used to decompose A effcently are ntroduced Secton 21 We show that these are easy to fnd f one can choose elements randomly from A In Secton 22 we employ these elements to gve a Monte Carlo algorthm for the centre of A Ths algorthm wll produce an element α A and polynomals f 1,, f l F[x] such that f 1(α),, f l(α) A form a bass for Centre(A) If F 2n 2, then the algorthm requres an expected number of O(MM(m) log m log(1/ǫ)) operatons n F, or an expected number of O(m 3 log(1/ǫ)) operatons n F usng standard matrx and polynomal arthmetc, and computes the correct answer wth probablty greater than 1 ǫ for a user specfed ǫ In Secton 23 we use the bass for the centre to obtan a set of central prmtve dempotents and a semsmple transton matrx whch nduces a change of bass under whch the smple components are revealed The central dempotents are gven by an element γ A and polynomals h 1,, h k such that h (γ) A s the central dempotent correspondng to A (e, h (γ) s the dentty n A and zero n all other components) Ths s a Las Vegas algorthm, assumng that a bass for the centre s correctly provded and that F > 2n 2, and requres an expected number of O(MM(m)log m + F(m) + R(A)) operatons n F, or O(m 3 + F(m) + R(A)) operatons n F usng standard matrx and polynomal arthmetc 21 Separators and Splttng Pars Ths secton ntroduces some specal types of elements of an algebra whch are useful n determnng the structure of that algebra These are employed n later sectons to obtan algorthms for computaton of the centre of a semsmple algebra, and the central dempotents of an assocatve algebra, over a suffcently large ground feld As shown, for example, by Perce (1982), the dmenson of D over ts centre E s a perfect square The dmenson of A over E s therefore a perfect square as well; suppose t s n 2 for 1 k Then, n = e 1n e kn 2 k The frst proposton wll serve as the bass for a proof of correctness for an algorthm for the decomposton of semsmple algebras over felds, and motvates the defntons that follow t Lemma 21 Suppose A s a semsmple algebra over a fnte feld or number feld F Let k, A, e, and n be as above () The mnmal polynomal of any element of A has degree at most e 1n e kn k over F () If an element α of A has a squarefree mnmal polynomal of maxmal degree e 1n e kn k over F, then for all γ A, γ commutes wth α f and only f γ F[α] Elements of A whose mnmal polynomals are squarefree and of maxmal degree are very useful and capture much of the structure of a semsmple algebra Defnton 22 An element α of a semsmple algebra A s a separator for A f α s nvertble and has a squarefree mnmal polynomal of maxmal degree e 1n e kn k over F By the above proposton the centre of A s contaned n F[α] for any separator α of A If the ground feld F s suffcently large then there exsts a separator α for A such that the centre of the algebra s equal to F[α] However, t s not clear how such a separator could be found wthout decomposng the algebra A or computng a bass for ts centre frst Fortunately, we wll see that a par of separators whch determnes the centre s easer to fnd, f the ground feld s suffcently large Defnton 23 A par (α,β) of elements of a semsmple algebra A s a splttng par for A f α and β are both separators for A, and f the centre of A s exactly the set of elements commutng wth both α and β If (α, β) s a splttng par for A and the mnmal polynomal of α over the ground feld F has degree l, then the centre of A could be obtaned by solvng the system of lnear equatons (y 0+y 1α+ +y l 1α l 1 )β β(y 0+y 1α+ +y l 1α l 1 ) = 0 for the unknowns y 0,, y l 1 n F Every soluton determnes an element y 0 +y 1α+ +y l 1α l 1 of F[α] that commutes wth β Snce β s a separator for A ths mples that y 0 +y 1α+ +y l 1α l 1 F[β], so that t belongs to F[α] F[β], the centre of A Conversely, every element of the centre belongs to {y 0 + y 1α + + y l 1α l 1 : y 0,, y l 1 F} and specfes a soluton for the above system Whle t s plausble that ths method s faster than prevous general methods for computaton of the centre, t requres that we form and solve a system of m 2 lnear equatons n l unknowns We can do consderably better than ths by projectng from the space of matrces to the space of vectors We show that wth suffcently hgh probablty the desred relatonshps stll hold Defnton 24 A par (α,β) of elements of a semsmple matrx algebra A F m m over a fnte feld or number feld F s a complemented splttng par f (α, β) s a splttng par for A and, furthermore, there exsts a par of vectors (u, v) each n F m 1 such that, for all µ F[α] and ν F[β], (µu = νu and µv = νv) µ = ν F[α] F[β] (21) Any par of vectors (u, v) F m 1 satsfyng condton (21) s sad to complement the splttng par (α,β) Theorem 25 Suppose A F m m s a semsmple algebra of dmenson n over a fnte feld or number feld F If F > 6n 6 then a complemented splttng par for A exsts Theorem 26 Let A F m m be a semsmple matrx algebra over a fnte feld or number feld F Let S be a fnte subset of F wth sze κ 4n 6 /ǫ, for ǫ > 0 If the elements α, β are randomly and ndependently chosen from A as n Secton 13, and u, v are chosen unformly and ndependently from S m 1, then (α, β) s a complemented splttng par for A that s complemented by (u, v) wth probablty at least 1 ǫ 22 Computaton of the Centre over Large Felds The centre of A can be computed effcently f a complemented splttng par (and a par of vectors that complements t) s avalable Lemma 27 If A F m m s a semsmple matrx algebra over a fnte feld or number feld F then, gven a complemented splttng par (α, β) for A and a par of vectors (u, v) that complements t, a set of polynomals f 1,, f l F[x] that each has degree less than m can be computed such that f 1(α),, f l(α) s a bass for the centre of A, usng a determnstc algorthm n tme O(mn(m 3, MM(m)log m)) 172

4 Indeed, by Defntons 23 and 24, t suffces to fnd a bass for the set of solutons of a homogeneous system of 2m lnear equatons n 2m unknowns n order to compute these polynomals f a complemented splttng par s avalable Theorems 25 and 26 mply that the a bass for the centre of A can be computed effcently, provded that F s suffcently large and A s gven by a bass over F Theorem 28 Let A F m m be a semsmple matrx algebra over a fnte feld or number feld F wth at least 12n 6 elements An element α of A, and polynomals f 1, f l F[x] that have degree less than m such that f 1(α),, f l(α) s a bass for the centre of A, can be computed usng a Monte Carlo algorthm that s guaranteed to produce the correct answer wth probablty at least 1 ǫ for a user specfced parameter ǫ > 0, wth O((MM(m)log m + R(A)) log(1/ǫ)) operatons n F, or O((m 3 + R(A)) log(1/ǫ)) operatons n F usng standard matrx and polynomal arthmetc Proof Snce F has sze greater than 6n 6, Theorem 25 mples that a complemented splttng par for A exsts The algorthm generates random elements α, β A and u, v F m 1 as n Theorem 26 Wth probablty at least 1/2, (α, β) s a complemented splttng par for A that s complemented by (u, v) In ths case, Lemma 27 mples that these values can be used to compute the centre for A determnstcally at the cost stated n the theorem To ncrease the probablty of success to 1 ǫ, we frst attempt to determne the degree of the mnmal polynomal of a separator n A Choose k 1 elements n A and compute the mnmal polynomal of each, and let l be the degree of the mnmal polynomal whch s squarefree of maxmal degree Ths s the degree of the mnmal polynomal of any separator, and hence the dmenson of Centre(A) over F, wth probablty at least 1 1/2 k 1 Now attempt to choose k 2 complemented splttng pars (α, β) and vectors u, v as above, dscardng any where the mnmal polynomal of α or β has degree less than l or s not squarefree Assumng l s correct (e, t really s the degree of the mnmal polynomal of a separator) these are all separators Hence F[α] F[β] s a superset of Centre(A), as s the canddate subspace computed for the centre n Lemma 27 The correct centre s a canddate subspace of mnmal dmenson over E, and we return ths correctly wth probablty 1 (1/2) k 1+k 2 Choosng k 1, k 2 = 1 + log 2 (1/ǫ) yelds the desred result 23 Identfcaton of Smple Components Recall that correspondng to the decomposton (11) there are orthogonal central dempotents ω 1,, ω k A such that ω ω k = 1 A, ω rω s = 0 f r s, and ω j s the dentty element of the smple algebra Aω j = Aj, for 1 j k In ths secton an asymptotcally fast algorthm wll be gven for the dentfcaton of the smple components of A Polynomal-tme algorthms for ths computaton have prevously been gven by Fredl & Rónya (1985) The algorthm to be gven here s based on a later algorthm of Eberly (1991) The new algorthm s asymptotcally faster than prevous algorthms Defnton 29 A nonsngular matrx X F m m s a semsmple transton matrx for a semsmple matrx algebra A F m m over a fnte feld or number feld F, wth k smple components A 1,, A k, f the followng condtons are satsfed () For all elements η of A, XηX 1 s a block dagonal matrx In partcular, there exst postve ntegers m 1,, m k such that m m k = m and, for all η A, η 1 0 η = X 1 X, 0 η k where η j F m j m j for 1 j k () The central prmtve dempotents of A are ω 1,, ω k, where D j,1 0 ω j = X 1 X, 0 D j,k D j,j s the dentty matrx of order m j and D j,l s the zero matrx of order m l f 1 l k and l j Theorem 210 Suppose A F m m s a semsmple algebra of dmenson n 2 wth k smple components over a fnte feld or number feld wth at least 2n 2 elements Gven a bass for A and α A and g 1,, g l F[x] such that g 1(α),, g l(α) A form a bass for Centre(A), we can compute () a semsmple transton matrx X for A, () postve ntegers m 1,, m k such that m 1+ +m k = m and such that for all η A, XηX 1 has dagonal blocks of orders m 1,, m k, () an element γ Centre(A) that s a separator for the commutatve algebra Centre(A) over F, and (v) polynomals h 1,, h k F[x] wth degree less than the dmenson of the centre of A over F such that h 1(γ),, h k(γ) are the central prmtve dempotents of A, usng a Las Vegas algorthm that requres an expected number of O(MM(m)log m+f(m)) operatons n F, or an expected number of O(m 3 + F(m)) operatons n F usng standard matrx and polynomal arthmetc Proof Lemmas 21 and 31 of Eberly (1991) mply that an element of the cenre whose mnmal polynomal s squarefree and has degree equal to the dmenson of the centre over F can be selected as a random lnear combnaton of g 1(α),, g l(α) In partcular, f values b 1,, b l are chosen unformly and ndependently from a fnte subset S of F wth sze at least 2n 2, and g = b 1g b lg l F[x], then γ = g(α) = b 1g 1(α)+ + b lg l(α) has these propertes, and s a separator for Centre(A), wth probablty at least 1/2 Snce we know [Centre(A) : F], we can correctly fnd such an element wth an expected number of 2 choces Suppose the mnmal polynomal of γ s f F[x] If γ s a separator for the centre then f s squarefree wth degree l and, snce the centre s a drect sum of k felds, f has k dstnct monc rreducble factors Suppose f = k f s a factorzaton of f n F[x] A result of Gesbrecht =1 (1995) (Theorem 74) mples that a ratonal Jordan form can be computed for γ effcently That s, one can fnd a nonsngular matrx X such that γ 1 0 X 1 γx = 0 γ k 173

5 s block dagonal, and the th block γ has mnmal polynomal f, for 1 k Set m to be the order of γ (so, γ F m m ) for 1 k Fnally, let h 1,, h k F[x] be polynomals wth degree less than the degree of the mnmal polynomal of γ (and, therefore, less than the dmenson of the centre of A over F) such that { 1 (mod fj) f 1 j k and j =, h 0 (mod f j) f 1 j k and j Then t s easly checked that D j,1 0 h (γ) = ω j = X 1 0 D j,k X, where D j,l F m l m l s as descrbed n Defnton 29 Thus, ω 1,, ω k are the central smple dempotents of A, X s a semsmple transton matrx for A, and the output of ths process s correct, provded that the attempt to fnd a separator γ for the centre was successful Snce l m, the coeffcents of a polynomal g = b 1g b lg l can be computed usng O(m 2 ) operatons The matrx γ = g(α) can be computed usng a Las Vegas algorthm of Gesbrecht (1995) n tme O(MM(m) log m) or expected tme O(m 3 ) usng standard matrx and polynomal arthmetc The mnmal polynomal of ths matrx can be computed n at most ths cost, usng the Las Vegas algorthm of Gesbrecht (1995), and ths can be factored wth cost F(m) The transton matrx X can be computed from γ and these polynomals usng Gesbrecht s (1995) algorthm for the ratonal Jordan form of a matrx, agan at the same cost The ntegers m 1,, m k can be obtaned by computng and nspectng the matrx X 1 γx Fnally, the polynomals h 1,, h k can be computed usng dvde and conquer n tme O(mM(m)log k), or tme O(km 2 ) usng standard arthmetc 3 Fndng Prmtve Orthogonal Idempotents In ths secton we gve an algorthm whch fnds a complete set of prmtve orthogonal dempotents for any algebra A over a fnte feld F The dea s to make use of decomposable elements n A These are elements whose mnmal polynomals n F[x] have at least two relatvely prme factors n F[x] A decomposable α A allows us to compute orthogonal dempotents ω 1,, ω l A such that ω ω l = 1 A and A = Aω 1 Aω 2 Aω l, (31) a drect sum of left deals Ths dea s smlar n sprt to the use of fttng elements by Schneder (1990) Fttng elements are zero dvsors whch are not nlpotent and also allow the decomposton of A as a sum of left deals However, whereas fttng elements are relatvely rare n A when F s large Schneder shows they have densty about 1/ F decomposable elements have a hgh densty In Secton 31 we present a new algorthm whch fnds decomposable elements effcently, and constructs a correspondng set of orthogonal dempotents Ths algorthm, wth hgh probablty, produces balanced decomposable elements, such that the decomposton (31) s nto deals of about the same sze In Secton 32 t s shown how to terate ths algorthm to fnd a complete set of prmtve orthogonal dempotents effcently In Secton 33, we use these technques to decompose semsmple algebras over small fnte felds as a drect sum of smple algebras, much as we dd for large fnte felds and number felds n Secton 2 Fnally, n Secton 34 we show how to compute an explct somorphsm between each smple component of a semsmple A and a full matrx algebra over an extenson feld of F If these somorphsms are successfully computed, we obtan a certfcate that A s ndeed semsmple Wth ths check, our decomposton algorthm for semsmple algebras over fnte felds s of the Las Vegas type, e, t never produces an ncorrect answer 31 Fndng balanced decomposable elements effcently In ths secton we ntroduce the noton of balanced decomposable elements, show how to fnd them effcently and how to construct dempotents from them We assume throughout ths secton that A s a subalgebra of F m m of dmenson n (not necessarly semsmple) For any algebra A, a decomposable element α A s defned as an element whose mnmal polynomal f F[x] has a factorzaton f = f 1f 2 f l nto two or more monc, parwse relatvely prme f F[x] \ F In ths case we can construct dempotents ω 1,, ω l A as follows For 1 l, use the Chnese remander theorem to construct h 1 mod f, h 0 mod f j for j, and assgn ω = h (α) A It follows easly that each ω s an dempotent, that ω ω j = 0 for j (e, they are parwse orthogonal) and that ω ω l = 1 A We call ω the dempotent that corresponds to f Lemma 31 Gven a decomposable α A, we can compute () the mnmal polynomal f F[x] of α and the factorzaton f = f 1f 2 f l nto powers of dstnct rreducble polynomals n F[x], () polynomals h 1,, h l F[x] such that ω = h (α), 1 l, are parwse orthogonal dempotents wth ω = 1 A, 1 l () m 1,, m l N such that m = m 1 + +m l and such that ω has rank m as a matrx n F m m, (v) a matrx U F m m such that D 1 ˆω = U 1 ω U = D D l Fm m, (32) where D j F m j m j s the dentty matrx when j = and the zero matrx when j, wth a Las Vegas algorthm usng an expected number of O(MM(m)log m+m(m)log q) operatons n F, or O(m 3 + m 2 log q) operatons n F usng standard matrx and polynomal arthmetc Proof Frst, compute the ratonal Jordan form of α F m m, and a transton matrx U F m m to ths form Recall that ths s a block dagonal matrx J F m m, such that J 1 U 1 αu = J = F m m J l 174

6 Each ratonal Jordan block J F m m s assocated to a dstnct rreducble factor g F[x] of the mnmal polynomal f F[x] of α, and J = C g c 1 C g c k F m m, where C c g j s the companon matrx of g c j F[x] We assume that c 1 c 2 c k, so f = g c 1 and the mnmal polynomal of α s f = g c 11 1 g c l1 l Gesbrecht (1995) gves a Las Vegas algorthm to compute ths ratonal Jordan form along wth a transton matrx U F m m and the mnmal polynomal f F[x] usng an expected number of O(MM(m)log m + M(m)log q) operatons n F, or O(m 3 +m 2 log q) operatons n F usng standard arthmetc For 1 l, let h F[x] wth h 1 mod f, h 0 mod f j for j These can be computed usng a dvde and conquer applcaton of the Chnese remander algorthm wth O(mM(m)log l) operatons n F, or O(m 2 l) operatons usng standard arthmetc For 1 l, ω = h (α) Under the change of bass nduced by U, each ω has the propertes descrbed n (v) It wll be useful to fnd dempotents whch partton A nto components of approxmately the same sze The followng theorem addresses ths concern for smple algebras Recall that any smple algebra over F s somorphc to E r r over an extenson feld E of F Theorem 32 Let A F m m be a smple algebra of dmenson n wth A = E r r, where E = F q µ and F = F q The number of α A wth f = mn F(α) such that there exsts a factorzaton f = f 1f 2 for relatvely prme f 1, f 2 F[x] wth correspondng dempotents ω and 1 ω, and n/2 dm F(Aω) 3n/4 s at least q n /22 Ths theorem s a great understatement of the number of such balanced reducble elements n A At the very least, the estmate of the densty of balanced reducble elements should easly be mproved to somethng approachng 1/5 by a computer aded enumeraton of cases creatng dffcultes, namely when F = F q for very small q 32 Fndng prmtve orthogonal dempotents In ths secton we descrbe an algorthm to compute a complete set of prmtve, orthogonal dempotents ω 1,, ω s A such that ω = 1 A The dea s to terate 1 s the algorthm descrbed n Lemma 31 on randomly chosen elements a A Suppose we have computed parwse orthogonal dempotents ω 1,, ω l A and a transton matrx U F m m as n (32), so that U 1 ω U s zero except for a d d dentty block on the dagonal The two-sded Perce decomposton of A wth respect to these dempotents s A = ω Aω j 1 l 1 j l The man dea behnd the algorthm s that we only have to work n the subalgebra ω Aω 1 l To see why ths s true, note that f ω s prmtve then ω Aω s a local algebra and can be decomposed no further Conversely, f ω s not prmtve, that s, ω = ω 1 + ω 2 for orthogonal dempotents ω 1, ω 2 A, then ω 1 and ω 2 are n ω Aω snce ω ω 1ω = ω 1 and ω ω 2ω = ω 2 Thus, we need only decompose ω Aω to refne the dempotent ω Suppose we have already computed a transton matrx U F m m and orthogonal dempotents ω 1,, ω l A as n Lemma 31 Let  = U 1 AU and ˆω = U 1 ω U  for 1 l Clearly A = Â, and the element ˆω s flled wth zeros except for a d d dentty matrx n the th dagonal block It s computatonally easy to compute a lnear map ψ : A ˆω ˆω 1 l Smply map β A to b 11 b 12 b 1l β U 1 b 21 b 22 b 2l βu = b l1 b ll b b b ll 1 l ˆω ˆω (33) where b j F d d j A randomly chosen β A wll yeld randomly and ndependently chosen components b ˆω ˆω A refnement of each of the ω s can be computed by decomposng the algebra ˆω ˆω as n Lemma 31 (assumng for now that we can fnd a decomposable element), for 1 l Assume we obtan ˆω = ˆω ˆω l for parwse orthogonal dempotents ˆω 1,, ˆω l ˆω ˆω Also assume that V F d d s the obtaned transton matrx, that ˆω j has rank d j, and that ω j = V 1 ˆω jv s a d d matrx whch s all zero except for a d j d j dentty matrx n the jth block on the dagonal If V 1 V = F m m, then W = UV s a transton matrx for A to ths refned set of dempotents That s, f ω j = W 1 ω jw A, we have ω j = 1 A and ω = ω j 1 l 1 j l 1 j l for 1 l Theorem 33 Let A F m m be an algebra of dmenson n over F = F q Then we can compute () a transton matrx U F m m, and () d 1,, d s Z >0 such that m = d d s, such that the followng holds For 1 s let D 1 ˆω = F m m, ω = U ˆω U 1 A, D s V l 175

7 where D j F d d s the dentty matrx f = j and the zero matrx f j Then ω 1,, ω s are prmtve, parwse orthogonal dempotents n A and ω ω s = 1 A The computaton can be performed wth a Monte Carlo algorthm, that returns a correct answer wth probablty at least 1 ǫ for a use specfed parameter ǫ > 0, usng an expected number of O((MM(m)log m+m(m)log q + R(A)) log 2 (m) log(1/ǫ)) operatons n F, or O((m 3 + m 2 log q + R(A)) log 2 (m) log(1/ǫ)) operatons usng standard matrx and polynomal arthmetc Proof Frst, we note that the above algorthm s correct At each teraton of the algorthm, suppose we have a transton matrx U F m m and d 1,, d l the ranks of orthogonal dempotents ω 1,, ω l A Choose a random α A, and fnd ψ(α) ˆωˆω Suppose α s the mage of 1 l α n ˆω ˆω The α s are also random and ndependent As descrbed above, use Lemma 31 to refne each ˆω, and compute a new transton matrx W If ω s not prmtve, then ω Aω s non-local, and such reducble elements α exst (f ω = ω 1 + ω 2 for orthogonal dempotents ω 1, ω 2, then these dempotents are reducble) We prove fast convergence of the algorthm on a complete set of dempotents by examnng how A/ Rad(A) decomposes By the Wedderburn-Malcev prncpal theorem (see Perce (1982), Secton 116), there exsts a semsmple subalgebra S of A such that S = A/Rad(A) and A = S Rad(A) (a drect sum as addtve groups) Moreover, f ω A s a prmtve dempotent and ω = ω + ρ for ω S and ρ Rad(A), then ω s a prmtve dempotent n S Suppose that S = S 1 S 2 S k for smple algebras S 1,, S k, and that S r = E r, for an extenson feld E of F wth [E : F] = µ Wthn any smple component, by Lemma 32 wth probablty 1/22 we choose a reducble element n S and obtan an dempotent ω S such that µ r 2 /2 dm S ω 3µ r 2 /4 Snce (3/4) 35 log r 1, we wll have constructed a complete set of prmtve dempotents for S wth 35 log r such reducble elements Hence, wth the choce of 77 log r elements from S we wll obtan a complete set of prmtve dempotents for S wth probablty at least 1/2 Snce there are at most m smple components, wth 77 log 2 m teratons, we obtan a set of prmtve dempotents for all smple components wth probablty at least 1/2, and wth 77 log 2 (m) log(1/ǫ) teratons we expect to fnd a complete set of prmtve dempotents for A wth probablty at least 1 ǫ Each teraton of the algorthm uses O(MM(m)log m+ M(m)log q + R(A)) operatons n F or O(m 3 + m 2 log q + R(A)) operatons n F usng standard arthmetc, by Lemma Computng smple components of semsmple algebras over fnte felds In ths secton we employ the algorthms of Sectons 31 and 32 to fnd the central dempotents and smple components of a semsmple algebra over a fnte feld Unlke the algorthms presented n Secton 2, these work over any fnte feld (even very small ones) We assume throughout ths secton that A F m m s a semsmple algebra of dmenson n A = S 1 S 2 S k (34) where S = E t t for an extenson feld E of F Usng the algorthm from Theorem 33 we can compute a transton matrx U F m m, d 1,, d s Z >0 such that m = d d s and D 1 ˆω =, ω = U ˆω U 1 A, (35) D s where D j F d d s the dentty matrx f = j and the zero matrx f j, so that ˆω F m m for all, ω 1,, ω s are prmtve, parwse orthogonal dempotents n A, and ω ω s = 1 A Snce A s semsmple, s = t 1 + t l and each of the ω s correspond to a matrx n exactly one of S 1,, S k consstng of a sngle 1 n some dagonal element and zeros everywhere else It wll be suffcent to group the ω s together accordng to whch smple component they belong to, and produce a new transton matrx whch reflects ths re-orderng Lemma 34 Suppose A decomposes as n (34) and suppose ω 1,, ω s A are prmtve, orthogonal dempotents wth sum 1 A If ω and ω j belong to dfferent smple components, then for all α A, ω αω j = 0 If ω and ω j belong to the same smple component S t, then ω αω j 0 wth probablty 1 1/ E t Now suppose U s a transton matrx as n (35) and α s a randomly chosen element of A Then A 11 A 12 A 1l U 1 A αu = 21 A 22 (36) A l1 A ll where A j F d d j and U 1 ω αω ju F m m s the matrx whch s 0 except for A j Theorem 35 Let A F m m be a semsmple algebra as n (34) Gven ntegers d 1,, d s and transton matrx U to a complete set of prmtve orthogonal dempotents as n (35), we can fnd a semsmple transton matrx W F m m (see Defnton 29) and a permutaton/re-labelng d 11,, d 1t1,, d k1,, d ktk of d 1,, d s such that () a complete set of prmtve, orthogonal dempotents ω j (1 k, 1 j t ) s formed by D 11 D 1t1 ω j = F m m, D k1 D ktk ω j = W ω jw 1 A, (37) where D st F dst dst s the dentty matrx f = s and j = t and the zero matrx otherwse () For 1 k, ω = 1 j t ω j s a central dempotent for A so that A = ω 1A ω 1 ω 2A ω 2 ω ka ω k (a drect sum as algebras) and ω A ω = S 176

8 Ths computaton can be performed by a Monte Carlo algorthm that returns a correct answer wth probablty at least 1 ǫ, for a user specfed parameter ǫ > 0, usng an expected number of O(MM(m) log(1/ǫ)) operatons n F Proof We smply choose random elements of α A and compute U 1 αu, whch has block form as n (36) After each random choce of an α, note whch pars of dempotents are lnked by notng non-zero blocks n U 1 αu: A j 0 mples ω and ω j are n the same smple component The probablty that two dempotents n the same component are not recognzed as such s at most 1/ F 2 1/4 Thus, after log 4 (m) attempts we should have dentfed all such lnkages wth probablty at least 1/2 Iteratng ths log(1/ǫ) tmes ensures that the probablty of success s at least 1 ǫ Once we have determned whch dempotents belong to whch smple components, we can construct W from U by a smple permutaton Combnng ths theorem wth Theorem 33 we obtan the followng corollary Corollary 36 Let A F m m be a semsmple algebra over a fnte feld F = F q as n (34) We can fnd a semsmple transton matrx W F m m and d 11,, d 1t1,, d k1,, d ktk Zwth sum m such that () a complete set of prmtve, orthogonal dempotents ω j (1 k, 1 j t ) s formed as n (37), and () for 1 k, ω = 1 j t ω j s a central dempotent for A so that A = ω 1A ω 1 ω 2A ω 2 ω ka ω k (a drect sum as algebras) and ω A ω = S, wth a Monte Carlo algorthm that returns a correct answer wth probablty at least 1 ǫ, for a user specfed parameter ǫ > 0, usng an expected number of O((MM(m)log m + M(m)log q + R(A)) log 2 (m) log(1/ǫ)) operatons n F, or O((m 3 + m 2 log q + R(A)) log 2 (m) log(1/ǫ)) operatons n F usng standard matrx and polynomal arthmetc 34 Las Vegas Decomposton of Sem-Smple Algebras over Fnte Felds Now let A F m m be an algebra over a fnte feld F = F q whch we beleve to be smple Suppose we are gven parwse orthogonal dempotents ω 1,, ω r A, wth ω ω r = 1 A We wll suppose as well that the dmenson of A over F s known and that we have a means of choosng random elements from A In ths secton, we descrbe an algorthm that ether provdes a proof that A s smple wth prmtve dempotents ω 1,, ω r or reports falure When appled to each of the smple components obtaned by the Monte Carlo algorthm of Secton 33, ths completes an asymptotcally effcent Las Vegas algorthm for the decomposton of semsmple matrx algebras over fnte felds Assume ω = dag( 1,, r) F m m, where j F s s s the dentty matrx f = j and the zero matrx otherwse In partcular, rank ω = s for 1 r unless ether A s not smple of ω 1,, ω r are not all prmtve (so that we should report falure f the rank s smaller than s) Suppose then that rank ω = s for all Every a A can be wrtten as a 11 a 1r a = A, a r1 a rr where a j F s s, and ω aω j s zero except for the (, j) block, whch equals a j We frst fnd a change of bass for A so that ω 1Aω 1 s n a normal form If A s smple wth dempotents ω 1,, ω r then ω 1Aω 1 s a fnte feld of (unknown) dmenson µ over F, such that µ s Choose a random element c A and compute the Frobenus form of the leadng s s submatrx c 11 F s s (the nonzero part of ω 1c ω 1): f A s smple then there exsts an nvertble u F s s such that C f λ = u 1 c 11u = F s s where C f F µ µ s the companon matrx of the mnmal polynomal f F[x] of c 11, µ = deg µ and µ µ Wth probablty at least 1/2 we have µ = µ If λ has two or more dstnct companon matrces n ts Frobenus form, or f s not rreducble, then report falure In these cases ω 1Aω 1 s not a feld and hence ω 1 s not prmtve or A s not smple It s convenent to fnd an element a A such that ω 1aω 0 and ω aω 1 0 for 2 r If A s smple wth prmtve dempotents ω 1,, ω r then, for fxed, j (1, j r) and randomly chosen b A, ω bω j 0 wth probablty at least 1 1/ F 1/2 Thus wth an expected number of O(log r) random choces of elements of such b we can construct b 1, b 1 A such that ω 1b 1ω 0 and ω b 1ω 1 0 for 2 r If A s smple wth orthogonal prmtve dempotents ω 1,, ω r then each b 1, b 1 has rank s for 2 s r If ths s not the case then the algorthm should report falure Otherwse, snce ω bω j s zero except possbly for the (, j)th block, we can add together approprate non-zero blocks of these b 1k s and b k1 s to construct a Let U = u a 1 12 u C f Fm m a 1 1r u and A = U 1 AU Note that snce ω 1,, ω r commute wth U, these are also dempotents n A and are prmtve and orthogonal f and only f they are prmtve and orthogonal n A Consder the elements a = U 1 au and ω 1k = ω 1a ω k of A for 2 k r By constructon ω 1k s zero except for the (1, k) block, whch equals u 1 a 1ka 1 1k u = 1s Also, ω 1U 1 cuω 1 generates a fnte feld of degree µ over F Let Λ = ω 1U 1 cuω 1 A and recall that, wth probablty at least one half, µ = µ If ths s the case (and, agan, A s smple wth prmtve orthogonal demptotents ω 1, ω r) then ω 1A ω 1 = ω 1F[Λ]ω 1 and, snce ω ka ω 1 s nonzero, ω ka ω 1 = (ω ka ω 1)(ω 1F[Λ]ω 1) If A s a smple wth prmtve dempotents ω 1,, ω r then for 2 k r there exsts an x A such that ω 1aω k ω kxω 1 = ω 1 Equvalently, there exsts a y A such that ω 1k ω ky ω 1 = ω 1, e, such that the (k,1) block of y k1 of y equals 1 s We must check that such a y A exsts for each k (2 k r) Suppose a k1 F s s s the (k,1) block of a ; f the algorthm has not already faled then ths matrx s nvertble and we can effcently check whether (a k1) 1 F[λ] If t s, then we can safely conclude that the desred element y belongs to A, and we can conclude that the element ω k1 whose (k,1) block s 1 s (and whch s zero elsewhere) belongs 177

9 to A ; f t s not, then the algorthm should report falure If µ = µ then the probablty of falure at ths step s less than one half Fnally, assumng that the algorthm has not faled, we can construct a bass for a smple subalgebra of A as follows For 2, j r let ω j = ω 1 ω 1j A, the matrx whch s zero except for the (, j) block whch s equal to 1 s It s easly shown that the set {ω 1Λ k ω 1j : 1, j r,0 k µ} s a bass for a smple subalgebra S of E r r, where E = F[λ] s an extenson feld of degree µ over F If dm S = dm A then clearly S = A and A s a smple algebra; otherwse, once agan, falure should be reported If reportng falure s equated wth reportng that ether A s not smple fo the ω s are not prmtve n A then ths establshes the followng Theorem 37 Let A be an algebra and ω 1,, ω r A be parwse orthogonal dempotents wth ω = 1 The 1 r algorthm descrbed above ether reports that A s smple wth prmtve dempotents ω 1,, ω r or reports that ether A s not smple or the ω s are not prmtve n A In ether case the algorthm requres an expected number of O(mn(m 3, MM(m)log m) + R(A)log r) operatons n F It returns the correct answer wth probablty bounded away from zero on all nputs, and t never reports that A s smple wth prmtve dempotents ω 1,, ω r f ths s not the case By applyng the above algorthm to each smple component n a decomposton of a presumed semsmple algebra, we obtan an effcent proof that the decomposton s ndeed correct Combnng ths wth the algorthm summarzed n Theorem 38 we obtan the followng theorem Theorem 38 Let A F m m be a semsmple algebra over a fnte feld F = F q as n (34) We can fnd a semsmple transton matrx W F m m and d 11,, d 1t1,, d k1,, d ktk Zwth sum m such that () a complete set of prmtve, orthogonal dempotents ω j (1 k, 1 j t ) s formed as n (37), and, () for 1 k, ω = 1 j t ω j s a central dempotent for A so that A = ω 1A ω 1 ω 2A ω 2 ω ka ω k (a drect sum as algebras) and ω A ω = S usng a Las Vegas algorthm that requres an expected number of O((MM(m) log m+m(m)log q+r(a)) log 2 (m)) operatons n F, or O((m 3 +m 2 log q+r(a)) log 2 (m)) operatons n F usng standard matrx and polynomal arthmetc References E R Berlekamp Factorng polynomals over large fnte felds Math Comp 24, pp , 1970 D Coppersmth and S Wnograd Matrx multplcaton va arthmetc progressons J Symb Comp 9, pp , 1990 W Eberly Decompostons of algebras over fnte felds and number felds Computatonal Complexty 1, pp , 1991 K Fredl and L Rónya Polynomal tme solutons of some problems n computatonal algebra In 7th Ann Symp Theory of Comp, pp , Provdence, RI, USA, 1985 P Gann, V Mller, and B Trager Decomposton of algebras In Proc ISSAC 88, vol 358 of Lecture Notes n Computer Scence, Rome, Italy, 1988 Sprnger-Verlag M Gesbrecht Nearly optmal algorthms for canoncal matrx forms SIAM J Comp 24, pp , 1995 D F Holt and S Rees Testng modules for rreducblty J Australan Mathematcal Socety 57, pp 1 16, 1994 N Jacobson Structure of Rngs, vol 37 Amercan Math Soc Colloquum Publ (Provdence, USA), 1956 S Landau Factorng polynomals over algebrac number felds SIAM J Comput 14, pp , 1985 G Mchler Some problems n computatonal representaton theory J Symbolc Computaton 9, pp , 1990 E Noether Hyperkomplexe Grössen und Darstellungstheore Math Zet 30, pp , 1929 R A Parker The computer calculaton of modular characters (the meat-axe) In Computatonal Group Theory: Proceedngs of the London Mathematcal Socety Symposum on Computatonal Group Theory, pp , London, 1984 Academc Press B O Perce Lnear assocatve algebra Amercan Journal of Mathematcs 4, pp , 1881 R Perce Assocatve Algebras Sprnger-Verlag (Hedelberg), 1982 L Rónya Smple algebras are dffcult In Proc 19th ACM Symp on Theory of Comp, pp , New York, 1987 L Rónya Computng the structure of fnte algebras J Symb Comp 9, pp , 1990 L Rónya Algorthmc propertes of maxmal orders n smple algebras over Q Computatonal Complexty 2, pp , 1992 G J A Schneder Computng wth endomorphsm rngs of modular representatons J Symbolc Computaton 9, pp , 1990 A Schönhage Schnelle Multplkaton von Polynomen über Körpern der Charakterstk 2 Acta Informatca 7, pp , 1977 A Schönhage and V Strassen Schnelle Multplkaton großer Zahlen Computng 7, pp , 1971 J H M Wedderburn On hypercomplex numbers Proc London Math Soc 6(2), pp , 1907 Wayne Eberly s an Assocate Professor n the Computer Scence Department at the Unversty of Calgary Prof Eberly s the author of several papers n the areas of parallel algorthms and computatonal algebra For more nformaton, see eberly Mark Gesbrecht s an Assstant Professor n the Department of Computer Scence at the Unversty of Mantoba Prof Gesbrecht obtaned hs PhD n Computer Scence from the Unversty of Toronto He s the author of a number of papers on computer algebra, algebrac complexty and compler optmzaton and automatc parallelzaton More nformaton can be found on hs WWW homepage: mwg 178