ON ALTERNATIVESUPERMATRIX REDUCTIONSteven Duplij �yzPhysics Department, University of Kaiserslautern,Postfach 3049, D-67653 KAISERSLAUTERN,Germany31 May, 1995AbstractWe consider a nonstandard odd reduction of supermatrices (ascompared with the standard even one) which arises in connectionwith possible extension of manifold structure group reductions. Thestudy was initiated by consideration of the generalized noninvertiblesuperconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated on a par. They can be uni�edinto some kind of "sandwich" semigroups. Also we de�ne a specialmodule over even- and odd-reduced supermatrix sets, and the gener-alized Cayley-Hamilton theorem is proved for them. It is shown thatthe odd-reduced supermatrices represent semigroup bands and Reesmatrix semigroups over a unit group.PACS numbers: 02.20.Mp; 11.30.Pb KL-TH-95/15�Alexander von Humboldt FellowyOn leave of absence from Theory Division, Nuclear Physics Laboratory, Kharkov StateUniversity, KHARKOV 310077, UkrainezE-mail: duplij@physik.uni-kl.de 1

1 IntroductionAccording to the general theory of G-structures [7, 23, 30] various geometriesare obtained by a reduction of a structure group of a manifold to some sub-group G of the tangent space endomorphisms. In the local approach usingcoordinate description this means that one should reduce a correspondingmatrix in a given representation to some reduced form as a matter of fact.In the most cases this form is triangle, because of the simple observation fromthe ordinary matrix theory that the triangle matrices preserve the shape andform a subgroup. In supersymmetric theories, despite of appearance of oddsubspaces and anticommuting variables, the choice of the reduction shaperemained the same[24, 33, 44, 46], and a ground reason of this was the fullyidentity of the supermatrix multiplication with the ordinary one, and conse-quently the shape of the matrices from a subgroup was the same. However in�ne search of nontrivial supersymmetric manifestations one can observe thatthe closure of multiplication can be also achieved for other shapes, but due toexistence of zero divisors in the Grassmann algebra or in the ring over whicha theory is de�ned. So the meaning of the reduction itself could be extendedprincipally. Evidently, that some "good" properties of the transformationscould be lost in this direction, but opening of new possibilities, beauty andinteresting and unusual features which are distinctive for supersymmetriccase only are the su�cient price for the surprises arisen and reason for themto investigate.This paper was initiated by the study of superconformal symmetry semi-group extensions [14, 13]. Indeed superconformal transformations [3, 10, 9]appear as a result of the reduction of the structure group matrix to the trian-gle form [20, 19]. Also, the transition functions on semirigid surfaces [11, 22](see [12]) occurred in the description of topological supergravity [21] have thesame shape. In [14] we considered an alternative version of the reduction.The superconformal-like transformations obtained in this way have many un-usual features, e. g. they are noninvertible and twist parity of the tangentspace in the supersymmetric basis1.Here we study the alternative reduction of supermatrices from a moreabstract viewpoint without connecting a special physical model.1This situation is di�erent from the case of Q-manifolds [1], where changing parity ofthe tangent space is done by hand from the �rst de�nitions.2

2 PreliminariesLet � be2 a commutative Banach Z2-graded superalgebra over a �eld K (whereK = R; C or Qp) with a decomposition into the direct sum: � = �0��1. Theelements a from �0 and �1 are homogeneous and have the �xed even and oddparity de�ned as jaj def= fi 2 f0; 1g = Z2j a 2 �ig. The even homomorphismmb : �! B , where B is a purely even algebra over K , is called a body map,if for any other purely even algebra A and any homomorphism ma : � ! Athere is an even homomorphism mab : B ! A such that ma = mab � mb. Thekernel of mb is S � kermb def= fa 2 �jm (a) = 0g and is called the soul sectorof �. If there are exists an embedding n : B ,! � such that m � n = id, then� admits the body and soul decomposition � = B � S, and a soul map canbe de�ned as ms : �! S. Usually the isomorphism B �= K is implied (whichis not necessary in general and can lead to very nontrivial behavior of thebody). This is the case when � is modeled with the Grassmann algebras^ (N) having N generators [42, 41, 49] or ^ (1) [43, 5, 29], or with the freegraded-commutative Banach algebras ^BE over Banach spaces [28, 36, 6].The soul S is obviously a proper two-sided ideal of � which is generatedby �1. In case � is a Banach algebra (with a norm jj�jj) soul elements arequasinilpotent [27], which means 8a 2 S; limn!1 jjajj1=n = 0. But in the in�nite-dimensional case quasinilpotency of the soul elements does not necessarilylead to their nilpotency (8a 2 S9n; an = 0) [37]. These facts allow us toconsider noninvertible morphisms on a par with invertible ones (in somesense), which gives, in proper conditions, many interesting and nontrivialresults (see [14, 15, 16]).The (pjq)-dimensional linear model superspace �pjq over � (in the senseof [42, 49, 29, 50]) is the even sector of the direct product �pjq = �p0 � �q1.The even morphisms Hom0 ��pjq;�mjn� between superlinear spaces �pjq !�mjn are described by means of (m + n) � (p+ q)-supermatrices (for de-tails see [4, 32]). In various physical applications supermatrices are reducedto some suitable form which is necessary for concrete consideration. For in-stance, in the theory of super Riemann surfaces [18, 45] the (1 + 1)�(1 + 1)-supermatrices describing holomorphic morphisms of the tangent bundle havea triangle shape [19, 20].2The standard notations can be found in [4, 32], and here we list some of them neededonly. 3

Here we consider a special alternative reduction of supermatrices andstudy its features. We note that the supermatrix theory per se has manyown problems [2, 17, 26] and unexpected conclusions (e.g. the lowering ofthe degree of characteristic polynomials comparing to the standard Cayley-Hamilton theorem[48, 47]).For transparency and clarity we con�ne ourselves to (1 + 1) � (1 + 1)-supermatrices3 , and generalization to the (m + n)� (p+ q) case is straight-forward and can be mostly done by means of simple changing of notations.3 Structure of Mat� (1j1)In the standard basis in �1j1 the elements from Hom0 ��1j1;�1j1� are de-scribed by the (1 + 1)� (1 + 1)-supermatrices [4]M � a �� b ! 2 Mat� (1j1) (1)where a; b 2 �0; �; � 2 �1 (in the following we use Latin letters for elementsfrom �0 and Greek letters for ones from �1). For sets of matrices we also usecorresponding bold symbols, e. g. M def= fM 2 Mat� (1j1)g. In this simple(1j1) case the supertrace de�ned as str : Mat� (1j1) ! �0 and Bereziniande�ned as Ber : Mat� (1j1) n fM jmb (b) = 0g ! �0 arestrM = a� b; (2)BerM = ab + ��b2 : (3)Now we de�ne two types of possible reductions of M on a par and studysome of their properties simultaneously.De�nition 1 Even-reduced supermatrices are elements fromMat� (1j1) hav-ing the form S � a �0 b ! 2 RMat even� (1j1) : (4)3which will allow us not to melt ideas by large formulas, and only this size will be usedin the following consideration 4

Odd-reduced supermatrices are elements from Mat� (1j1) having the formT � 0 �� b ! 2 RMat odd� (1j1) : (5)The name of the odd-reduced supermatrices follows naturally from BerT =��=b2 ) (BerT )2 = 0 andBerT 2 = Ber �� �b�b b2 � �� ! = 0: (6)The explanation of the ground of the notations S and T comes fromthe fact that the even-reduced supermatrices give superconformal trans-formations which describe morphisms of the tangent bundle over the su-per Riemann surfaces [19], while the odd-reduced supermatrices give thesuperconformal-like transformations twisting the parity of the (1j1) tangentsuperspace in the standard basis (see [14, 15]).Assertion 2 M is a direct sum of diagonal D and anti-diagonal (secondarydiagonal) A supermatrices (the even and odd ones in the notations of [4])M = D�A; (7)where D� a 00 b ! 2 D � MatDiag� (1j1) ; (8)A� 0 �� 0 ! 2 A � MatAdiag� (1j1) ; (9)and D � S and A � T.For the reduced supermatrices one �ndsS \T = 0 �0 b ! 6= ;: (10)Nevertheless, the following observation explains the fundamental role ofS and T. 5

Proposition 3 The Berezians of even- and odd-reduced supermatrices areadditive components of the full BerezinianBerM = BerS + BerT: (11)The �rst term in (11) covers all subgroups of even-reduced supermatri-ces from Mat� (1j1), and only it was considered in the applications. Butthe second term is dual to the �rst in some sense and corresponds to allsubsemigroups of odd-reduced supermatrices from Mat� (1j1)4.4 Invertibility and ideals of Mat� (1j1)Denote the set of invertible elements of M by M�, and I =M nM�. In [4]it was proved that M� = fM 2Mjmb (a) 6= 0 ^ mb (b) 6= 0g. Then similarlyS� = fS 2 Sjmb (a) 6= 0 ^ mb (b) 6= 0g and T� = ;, i. e. the odd-reducedmatrices are noninvertible and T � I. Consider the invertibility structure ofMat� (1j1) in more detail. Let us denoteM0 = fM 2Mjmb (a) 6= 0g ;M00 = fM 2Mjmb (b) 6= 0g ;I0 = fM 2Mjmb (a) = 0g ;I00 = fM 2Mjmb (b) = 0g : (12)ThenM =M0 [ I0 =M00 [ I00 andM0 \ I0 = ; ,M00 \ I00 = ;, thereforeM� =M0\M00 and T �M00. The Berezinian BerM is well-de�ned for the matricesfromM00 only and is invertible when M 2M�, but for the matrices fromM0the inverse (BerM)�1 is well-de�ned and is invertible when M 2M� too [4].Under the ordinary matrix multiplication the set M is a semigroup ofall (1j1) supermatrices [35, 39, 40], and the set M� is a subgroup of M.In the standard basis M� represents the general linear group GL� (1j1) [4].According to the general de�nitions [8] a subset I �M is called a right (left)ideal of the semigroup M, if I �M � I (M � I � I), where the point denotesthe standard matrix set multiplication: A �B def= fSABjA 2 A; B 2 Bg.4The relation (11) is a supersymmetric version of the obvious equality detM = detD+detA, when D and A from (7) and (8) are ordinary matrices. The problem is that for Abeing a supermatrix BerA is not de�ned at all.6

An isolated ideal satis�es the relation [8]M1M2 2 I)M1 2 I _M2 2 I; (13)and a �lter F of the semigroup M is de�ned byM1M2 2 F)M1 2 F^M2 2 F: (14)Proposition 4 1) The sets I, I0 and I0 are isolated ideals of M.2) The sets M�, M0 and M00 are �lters of the semigroup M.3) The sets M0 and M00 are not subgroups5, but subsemigroups of M,which are M0 =M� [ J0 and M00 =M� [ J00 with the isolated ideals J0 =M0 nM� =M0 \ I00 and J00 =M00 nM� =M00 \ I0 respectively.4) The ideal of the semigroup M isI = I0 [ J0 = I00 [ J00: (15)Proof. Let M3 =M1M2, then a3 = a1a2+�1�2 and b3 = b1b2+ �1�2. Takingthe body part we derive mb (a3) = mb (a1)mb (a2) ; (16)mb (b3) = mb (b1)mb (b2) : (17)1) The left-hand side of (16) and (17) vanishes i� the �rst or secondmultiplier of the right-hand side equals zero. Then use (13).2) The left-hand side of (16) and (17) does not vanish i� the �rst andsecond multiplier of the right-hand side does not equal zero. Then use (14).3) J0 consists of the matrices with mb (a) 6= 0, but mb (b) = 0, and J00consists of the matrices with mb (a) = 0, but mb (b) 6= 0.4) The ideal of M consists of the matrices with mb (a) = 0 or mb (b) = 0.Then use 3) and the de�nitions. 2Remark. Since the ideals J0 and J00 are isolated, i. e. J0 \M� = J0 \M� = ;,they cannot be represented as sequences of elements from the group M�5as it was incorrectly translated in [4], pp. 95, 103 (in the original Russian edi-tion, Moscow, 1983, pp. 89, 93, the sets M0 and M00, denoted as G0Mat (1; 1j�) andG00Mat (1; 1j�) respectively, are called semigroups).7

(viz. no one noninvertible element can be derived from a sequence of in-vertible ones, see, e. g. [27]), and so the statement "that any element ofG00Mat (p; qj�) (the semigroup M00 here, and so the notation G00::: misleads)is the limit of a sequence from GMat (p; qj�) (the group M� here)" holdsonly for invertible elements from M00, i.e. belonging M�, and it means thatelements from M� can be obtained from a sequence of elements from M�,which is simply a group action.Assertion 5 For the odd-reduced matrices from (12) it follows T � I0 andA \M0 = A \M00 = ;.5 Multiplication properties of odd-reduced su-permatricesIn general the odd-reduced matrices do not form a semigroup, sinceT1T2 = �1�2 �1b2b1�2 b1b2 + �1�2 ! 6= T: (18)But from (18) it follows thatT �T \T 6= ; ) �� = 0;T �T \ S 6= ; ) �b = 0; (19)which can take place, because of the existence of zero divisors in �.Proposition 6 1) The subset TSG � T of the odd-reduced matrices satisfy-ing �� = 0 form an odd-reduced subsemigroup of M.2) In the odd-reduced semigroup TSG the subset of matrices with � = 0 isa left ideal, and one with � = 0 is a right ideal, the matrices with b = 0 forma two-sided ideal.5.1 Semigroup band representationsLet Z� (t) = 0 �t� 1 ! 2 Z�� TSG; (20)8

i.e. Z� is a set of the odd-reduced matrices parameterized by the even pa-rameter t 2 �0. Then Z� is a semigroup under the matrix multiplication (�numbers the semigroups) which is isomorphic to a one parameter semigroupwith the multiplication ft1g �� ft2g = ft1g : (21)This semigroup is called a right zero semigroup ZR = fS ftg ; ��g and playsan important role (together with the left zero semigroup ZL de�ned in a dualmanner) in the general semigroup theory (e.g., see [8], Theorem 1.27, and[25]).Let B� (t; u) = 0 �t�u 1 ! 2 B�� TSG; (22)then B� is a matrix semigroup (numbered by �) which is isomorphic to atwo �0-parametric semigroup B = fS ft; ug ; ��g, where the multiplication isft1; u1g �� ft2; u2g = ft1; u2g : (23)Here every element is an idempotent (as in the previous case too), and sothis is a rectangular band multiplication [25, 38].Let C� (t; u; v) = 0 �t�u v ! 2 C�� TSG, then C� is a matrix semi-group isomorphic to a semigroup BG = fS ft; u; vg ; ��g where the multipli-cation is ft1; u1; v1g �� ft2; u2; v2g = ft1v2; u2v1; v1v2g : (24)The parameter v describes the di�erence of an element from an idempotent,since ft; u; vg2 � ft; u; vg = ft (v � 1) ; u (v � 1) ; v (v � 1)g.Assertion 7 The one and two parametric subsemigroups of the semigroup ofodd-reduced supermatrices TSG having vanishing Berezinian represent semi-group bands, viz. the left and right zero semigroups and rectangular bands.Theorem 8 The continuous supermatrix representation of the Rees matrixsemigroup over a unit group G = e (see [8, 25]) is given by formulas (20)and (22).9

5.2 "Square root" of even-reduced supermatricesConsider the second equation in (19).Proposition 9 The elements TpSfrom the subset TpS � T of the odd-reduced matrices satisfying �b = 0 can be interpreted as "square roots" ofthe even-reduced matrices S.Example. 1) Let TpS = 0 �� � ! 2 TpS, then �TpS�2 = �� 1 0 �1 ! 2S n S�.2) If = 0 in 1), then we obtain A2 = �� 1 00 �1 ! 2 D n D� (see(8), (9) and compare with D2 = a2 00 b2 !). This could be accepted as ade�nition of a square root of �� in some sense. Thus we haveD �D = D;A �A = D; (25)and the second relation could be formally considered as an "odd branch" ofthe root pD.6 Uni�cation of reduced supermatricesNow we try to unify the even- and odd-reduced matrices (4) and (5) into acommon abstract object. To begin with consider the multiplication table ofall introduced sets including the even-reduced matrices productsS � S = S;D �D = D;D � S = S;S �D = S; (26)10

and ones for the odd-reduced matricesA �T = S;A � S = T;T �A = Sst;S �A = T�;S �T = S [TT � S = T: (27)Here st : Mat� (1j1)! Mat� (1j1) is a supertranspose [4], i. e. a �� b !st = a ��� b !. Also we use the �-transpose [34] de�ned by �: Mat� (1j1) !Mat� (1j1) and a �� b !� = b �� a ! : (28)Note that the sets of matrices S and T are not closed under st and � oper-ations, but Sst \ S = D and T� \T = A.First we observe from the �rst two relations of (27) that A plays a role ofthe left type-changing operator A : S! T and A : T! S, whileD does notchange the type. Next from the �rst two relations of (26) it is obviously seenthat the sets S and D are subsemigroups. Unfortunately, due to the next tolast relation of (27) the set T has no clear abstract meaning. However, thelast relationT�S = T is important from another viewpoint: any odd-reducedmorphism �1j1 ! �1j1 corresponding to T can be represented as a productof odd- and even-reduced morphisms, such that

T@@RS-?T (29)is a commutative diagram. This decomposition is crucial in the applicationto the superconformal-like transformations construction (see [14]).11

6.1 Reduced supermatrix set semigroupTo unify the introduced sets (26) and (27) we consider the triple productsS �A �T = S;T �A �T = T;S �D � S = S;T �D � S = T: (30)Here we observe that the matrices A and D play the role of "sandwich"elements in a special S and T multiplication. Moreover, the sandwich el-ements are in one-to-one correspondence with the right sets on which theyact, and so they are "sensible from the right". Therefore, it is quite naturalto introduce the followingDe�nition 10 A sandwich right sensible product of the reduced supermatrixsets R = S;T is R1 �R2 def= ( R1 �D �R2; R2 = S;R1 �A �R2; R2 = T: (31)In terms of the sandwich product instead of (30) we obtainS�T = S;T�T = T;S� S = S;T� S = T: (32)Proposition 11 The �-multiplication is associative.Proof. Consider the relations:(T� S)�T = (T �D � S) �A �T = T �D � S �A �T;T� (S�T) = T �D � (S �A �T) = T �D � S �A �T; (33)where the last equalities follow from the associativity of the ordinary matrixmultiplication. Therefore, (T� S)�T = T� (S�T). Other associativityrelations can be proved in a similar way6. 26We stress here that the associativity does not follow from the associativity of thesupermatrix multiplication only, but is a consequence of the special and nontrivial setmultiplication table (32). 12

De�nition 12 The elements S and T form a semigroup under �-multiplication(31), which we call a reduced matrix set semigroup and denote RMSset.Comparing (21) and (32) we observe that the reduced matrix set semi-group can be viewed as a right zero semigroup having two elements.Assertion 13 The reduced matrix set semigroup is isomorphic to a specialright zero semigroup, i. e. RMSset �= ZR = fR = S;T;�g.6.2 Scalars, anti-scalars and generalized modulesNow we introduce the analog of �-multiplication for the reduced matricesper se (not for sets). First we de�ne the structure of generalized �-module inHom0 ��1j1;�1j1� in some alternative way, the even part of which is describedin [32] (in the ordinary matrix theory this is a trivial fact that the productof a matrix and a number is equal to a product of a matrix and a diagonalmatrix having this number on the diagonal).De�nition 14 In Mat� (1j1) a scalar (matrix) E (x) and anti-scalar (ma-trix) E (�) are de�ned byE (x) def= x 00 x ! 2 D = MatDiag� (1j1) ; x 2 �0;E (�) def= 0 �� 0 ! 2 A = MatAdiag� (1j1) ; � 2 �1: (34)Assertion 15 The Berezin's queer subalgebra Q� (1) � x �� x ! � Mat� (1j1)[4] is a direct sum of the scalar and anti-scalarQ� (1) = E (x)� E (�) : (35)Assertion 16 The anti-scalars anticommute E (�1) E (�2) + E (�2) E (�1) =0, and so they are nilpotent.Proposition 17 The structure of the generalized �0 � �1-module inHom0 ��1j1;�1j1� is de�ned by action of the scalars and anti-scalars (34).13

This means that everywhere we exchange the multiplication of superma-trices by even and odd elements from � with the multiplication by the scalarmatrices and anti-scalar ones (34). The relations containing the scalars arewell-known [32], but for the anti-scalars we obtain new dual ones. Considertheir action on elements M 2Mat� (1j1) in more detail. First we needDe�nition 18 Left P and right Q anti-transpose are Hom0 ��1j1;�1j1� !Hom1 ��1j1;�1j1� mappings acting on M 2M as a �� b !P = � ba � ! ; (36) a �� b !Q = � ab � ! : (37)Corollary 19 The anti-transpose is a square root of the parity changingoperator (28) in the following sensePQ = QP = �: (38)Assertion 20 The anti-transpose satisfy(E (�)M)P = �M(E (�)M)Q = �M�(ME (�))P = M��(ME (�))Q = M� (39)Thus the concrete realization of the right, left and two-sided generalized�0 � �1-modules in Hom0 ��1j1;�1j1� is determined by the actionsE (�)M = �MP ;ME (�) = MQ�;E (�1)ME (�2) = �1M��2; (40)together with the standard �-module structure [32]E (x)M = xM;ME (x) = Mx;E (x1)ME (x2) = x1Mx2: (41)14

Corollary 21 The generalized �0 � �1-module relations are(E (x)M)N = E (x) (MN)(ME (x))N = M (E (x)N)M (NE (x)) = (MN)E (x)(E (�)M)N = E (�) (MN)(ME (�))N = M (E (�)N)M (NE (�)) = (MN) E (�) (42)where M;N 2 Mat� (1j1).Proposition 22 The structure of the generalized �0 � �1-module inHom1 ��1j1;�1j1� is determined by the analogous actions of odd scalarE (�) def= � 00 �� ! 2 Hom1 ��1j1;�1j1� (43)and odd anti-scalarE (x) def= 0 xx 0 ! 2 Hom1 ��1j1;�1j1� (44)respectively7.6.3 Reduced supermatrix sandwich semigroupOne way to unify the even- (4) and odd-reduced (5) supermatrices into anobject analogous to a semigroup is consideration of the sandwich multipli-cation similar to (31), but on the level of matrices (not sets), by means ofthe scalars and anti-scalars as sandwich matrices. Indeed, the ordinary ma-trix product can be written as M1M2 = M1E (1)M2. But we cannot �ndan analog of this relation using anti-scalar, because among � 2 �1 there isno unity. Therefore, the only possibility to include E (�) into equal play isconsideration of sandwich elements (34) having arbitrary (or �xed by otherspecial conditions) both arguments x and �. Thus we naturally come to7>From (43) and (44) it is clear why we use the name "anti-" and not "odd" (as in [4])for the secondary diagonal matrices E (�). 15

De�nition 23 A sandwich right sensible �0 � �1-product of the reducedsupermatrices R = S;T isR1 ?X R2 def= ( R1E (x)R2; R2 = S;R1E (�)R2; R2 = T; (45)where X = fx; �g 2�0 � �1.The ?X -multiplication table coincides with (32). The associativity can beproved similar to (33). Therefore, we haveProposition 24 Under �0 � �1-multiplication the reduced matrices form asemigroup which we call a reduced matrix sandwich semigroup RMSS.Assertion 25 The reduced matrix sandwich semigroup is isomorphic to aspecial right zero semigroup, i. e. RMSS �= ZR = fR = SS ST ;?Xg.6.4 Direct sum of reduced supermatricesAnother way to unify the reduced supermatrices is consideration of the con-nection between them and the generalized �0 � �1-modules.De�nition 26 The reduced supermatrix direct space RMDS is a directsum of the even-reduced supermatrix space and the odd-reduced one.In terms of sets we have R� = S�T.Assertion 27 In RMDS the scalar is the Berezin's queer subalgebra Q� (1)(see (35)).Theorem 28 In RMDS the scalars play the same role for the even-reducedsupermatrices, as the anti-scalars for the odd-reduced ones.Corollary 29 The eigenvalues of even- (4) and odd-reduced (5) supermatri-ces should be found from di�erent equations, viz.SV = E (x)V;TV = E (�)V; (46)where V is a column vector, and they arex1 = a; x2 = b;�1 = �; �2 = �: (47)(see (4) and (5)). 16

De�nition 30 The characteristic functions for even- and odd-reduced su-permatrices are de�ned in RMDS byHevenS (x) = Ber (E (x)� S) ;HoddT (�) = Ber (E (�)� T ) : (48)Remark. In the standard �-module over Mat� (1j1) [4] one derives charac-teristic functions and eigenvalues for any matrix (and for odd-reduced too)from the �rst equations in (46) and (48) and obtains di�erent results (see, e.g. [31, 47]).Using (4), (5) we easily foundHevenS (x) = (x� a) (x� b)(x� b)2 ;HoddT (�) = (�� �) (�� �)b2 : (49)Here we observe the full symmetry between even- and odd-reduced super-matrices (for this purpose the cancellation in the �rst equation was avoided)and consistency with their �0 � �1-eigenvalues (47).The characteristic polynomial8 of a supermatrixM is de�ned by PM (M) =0 and in complicated cases is constructed from the parts of the characteristicfunction HM (x) according to a special algorithm [31, 47]. Due to existenceof zero divisors in � the degree of PM (x) can be less than n = p + q ,M 2 Mat� (pjq). But this algorithm is not applicable for the odd-reducedand secondary diagonal supermatrices. As before, we introduce two dualcharacteristic polynomials and, using (49), obtain the Cayley-Hamilton the-orem in RMDS.Theorem 31 (The generalized Cayley-Hamilton theorem) The char-acteristic polynomials in the reduced supermatrix direct space areP evenS (x) = (x� a) (x� b) ;P oddT (�) = (�� �) (�� �) : (50)and P evenS (S) = 0 for any S, but P oddT (T ) = 0 for nilpotent b only.8For a nonsupersymmetric matrixM it evidently coincides with the characteristic func-tion PM (x) = HM (x) � det (Ix�M), where I is a unity matrix.17

Proof. The even case is well-known, but for clarity we repeat it too, demon-strating the avoiding of multiplication of a matrix by a constant and usinginstead the scalars and anti-scalars (34), i. e. the introduced �0��1-modulestructure. Thus, considering simultaneously the even and odd cases we ob-tain P evenS (S) = (S � E (a)) (S � E (b)) = 0 �0 b ! a �0 0 ! = 0; (51)P oddT (T ) = (T � E (�)) (T � E (�)) = 0 0� � � b ! 0 �� �0 b ! = 0 00 b2 ! = 0: (52)27 ConclusionsWe conclude that almost all above constructions are universal and ideasmostly do not depend on size of the supermatrices under consideration. Inparticular case of superconformal-like transformations it would be interest-ing to use the alternative reduction introduced here in building the objectsanalogous to super Riemann or semirigid surfaces, which can also lead tonew topological-like models.References[1] M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zabrodsky, Thegeometry of the master equation and topological quantum �eld theory,UC Davis preprint, 1995, hep-th-9502010.[2] N. B. Backhouse and A. G. Fellouris, Grassmann analogs of classicalmatrix groups, J. Math. Phys. 26 (1985), 1146{1151.[3] M. A. Baranov, I. V. Frolov, and A. S. Schwarz, Geometry of supercon-formal �eld theories in two dimensions, Theor. Math. Phys. 70 (1987),92{103.[4] F. A. Berezin, Introduction to Superanalysis, Reidel, Dordrecht, 1987.18

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