migroup Forum Vol. 54 (1997) 253-260 1997 Springer-Verlag New York Inc.

RESEARCH ARTICLE

Some Abstract Properties of Semigroups Appearing in Superconformal Theories

S t e v e n Duplij*

Communicated by John M. Howie

Abstract A new type of semigroup which appears while dealing with N ---- 1 superconformal symmetry in superstring theories is considered. The ideal series having unusual ab- stract properties is constructed. Various idealisers are introduced and studied. The ideal quasicharacter is defined. Green's relations are found and their connection with the ideal quasicharacter is established.

1. I n t r o d u c t i o n

Mathematical objects with new properties often appear from concrete physical con- siderations and models. The discovery of supersymmetry [36, 37] gave many new mathematical features, but its influence on the general abstract properties of the the- ory, in spite of the fact that among principal objects there were noninvertible ones and zero divisors [16], needs to be emphasized. The latter leaded to the conclusion that the abstract ground of supersymmetric theory should have a semigroup nature [8]. It was also realised that the noninvertible transformations and semigroups appearing in that way have many new nontrivial properties [7, 10]. In particular, it would be interesting to work out the general abstract structure of the N = 1 superconfor- real semigroup, which is important in the consistent construction of the superstring unified theories [17, 12]. In this paper we provide a consideration of the superconfor- mal semigroups from the abstract-algebraic point of view and present their abstract properties without proofs which will appear elsewhere.

2. P r e l i m i n a r i e s

The semigroup of N = 1 superconformal transformations of C 1'1 complex superspace with the coordinates (z, 0) valued in the Grassmann algebra [2], where z E 6 1'0 and 0 E C °A, is isomorphic to the semigroup S of the even C 1'° -+ C 1'° and odd C ~'° --+ C °'1 functions satisfying some multiplication law (for details see [7, 9]). The even part of the law (1) s3= s~*s2, s ie S,

in terms of the even functions g(z) can be presented as

(2) g3(:) = [g,(~) + hl(~)], g~(z), where ~ is some shifting and hi(z) is some even nilpotent function of second degree, i.e. h~(z) =_ hi(z), h~(z) = 0. We stress that, because of the shifting z -+ ~ and

*: Alexander yon Humboldt Fellow

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the second term in the brackets (2), S differs from the semigroups of functions with point by point multiplication [6], and also from the semigroups of functions [24, 25]. This leads to new unusual abstract properties of S considered below. Further we note that to study these properties it is sufficient to know the formal expression (2) only. This parametrisation of N = 1 superconformal transformations was given in [7, 9] (where one can also find the exact formulas and the concrete background). For other considerations we refer to [1, 27, 28, 15].

Here we do not consider the physical interpretations of g(z) (see [1, 5]) and stress only that g(z) controls invertibility of the superconformal transformations [8]. Therefore, the index of g(z) which is defined by

(3) indg(z) d~S {n • g I g"(z) = O,g'~-l(z) ¢ 0}

plays a crucial part in the following. We mention here that in (2) and (3) the multi- plication is a point by point one in the Grassmann algebra [2] (for clarity sometimes we use a point for it), but the star in (1) denotes the semigroup multiplication.

So the semigroup S can be divided into two disjoint parts S = G U T, G Cl T = O, where

(4) a eAs {s • S I indg(z) = e c ) ,

(5) T e&s {s • S I indg(z) < oo}.

Here G is a group corresponding to the invertible transformations. From the multi- plication law (2) it follows that T is a two-sided ideal. The unity element e • S has g(z) = 1, h(z) = 0, and the zero element has g(z) = O, h(z) = 0 (for other details see [8, 9]). From (2) and the relation indh(z) = 2 it follows that T is a nilsemi- group [21, 13, 11, 34], i.e. Vt • T 3n • Z, t ' n = z (here the multiplication in the power expression is implied as the semigroup one (1)). So every element from T is nilpotent without bound on its index and of finite order, but every element from G is of infinite order.

The superconformal transformations corresponding to G were studied earlier in [1, 5, 29]. Therefore we concentrate our attention on the ideal T , which gives the evidence of some unusual abstract properties of the parametrised superconformal semigroup S.

3. I d e a l Ser ies

To classify the elements from the ideal part T we take the n- th power of the equation (2) in the Grassmann algebra and, using the relation ind h(z) = 2, obtain

(6) 9 (z) = +

We see that the natural classification can be done by means of the index of g(z) (see (3)). Let us define the following sets

(7) A~ d~_i {s • S I indg(z) = n ) .

(s) U k<n

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Then we notice that T is a disjoint union of the sets An, because T = U , A , A t3An_ 1 = O. From (6) it follows that In-1 C In and I n \ I n - I = A , . Therefore we obtain the following infinite chain of the sets In

(9) z C I1 c 12 c . - - c I , c .-- c T.

To understand the meaning of In we use (6) and obtain

(10) S * I , C_ I , ,

(11) In* S C_C_ I~+l,

(12) S * I ~ * S C_ I~+1,

From these relations we can easily observe that the sets In are left ideals of the semigroup S, but not right ideals, because of (11). Moreover, the appearance of n + 1 in the right side of (11) and (12) is very unusual, and so is natural to call these strange sets I~ "jumping ideals". Therefore I~-a <~lI~ and the chain (9) is a left and "jumping ideal" series. Then In are quasiideals [33, 4] since they satisfy S * In r) In * S C I~ . Simultaneously, the sets I~ are biideals, because In * S * In C_ In [3, 20]. It is exciting that in our case the regularity is not necessary for the coincidence of quasiideals and biideals in superconformal semigroup (as distinct from [20]). Because of the inclusion In <1U => In <3 S, VU <1S the semigroup S is a filial semigroup [18]. The indices in (9) form a well ordered set for which n is an ordinal. Because of In-t <ll In the chain (9) can be called a left ascending ideal series of S . From (11) and (12) we derive

(13) S * In U In * S C I ,+l ,

This condition is opposite to that for which the chain (9) is an ascending annihilator series of S [14, 31]. So we call it an ascending antiannihilator series of S .

The multiplication law for the sets I~ and Am is

I,,* In+k C_ In+l, In+k-1 * I . C_ In, A ~ * A~+ k C In+l, A ~ + k - l * A n C In, I~* An+k C_ I~+l,

(14) In+k-1 * An C_ In, An*ln+k C In+l, An+k-I * I . C_ I . , I ,~*G C_ In+l, G * I , , C_ In, A ~ * G C_ In+l, G , A n C A~.

where k > 0. It follows that the set In is a subsemigroup of S , because from (14) we have I ~ * I n C I~ but the set A~ is not a subsemigroup, since A ~ , A C I,~. This is a consequence of the fact that our semigroup is defined over the Grassmann algebra [2] which contains nilpotents and zero divisors, and the latter fact should be taken into account properly [16].

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From the last two relations of (14) and (12) we can obtain

(15) G * A ,GC_I ,+ I ,

i.e. some of the elements from A are conjugated by the subgroup G with the elements of the next set A,+I. By analogy with [35, 23, 22] we define G-normal subsets A, BC__S as follows: g - a , A , g C _ B , g G G . Then from (15) we make a conclusion that any two sets An contain G-normal elements and one can reach any A,~ using the subgroup action only. Further general abstract properties of such elements can be found in [23, 30].

4. Idealisers

The left (right, two-sided) idealiser Ii (U) ( L (U), I (U)) of the subset U C S can be defined as the largest subsemigroup of S within which U is a left (right, two-sided) ideal, i.e.

(16) Ii (U) a,__/{s C_ S I s * U C U},

(17) /~ (O) aL-/{s C S I U , s C U ) ,

(18) I ( u ) d e = I { s C _ S I s * U C U , U * s C _ U } .

The set I (U) is a subsemigroup, since from

U*sC_ U, s , U C_U, U * t C U a n d t * U C U

we may deduce that

U * s * t C U * t C U a n d s * t * U C s * U C U .

Also, if V is a subsemigroup of U and U <1V , then v * U C_ U, U * v C_ U for all U in V and so V C _ I ( U ) .

Let us consider the idealisers of the various introduced subsets of S. First the left idealiser for I , is S, as follows directly from (10), i.e.

(19) Is (I~) = S.

From the last relation in (14) we find

(20) ~t ( A ) = G.

For the right idealisers of IN the situation is more complicated.Using (11) we divide S into two disjoint parts S = SI. U San, where SI. N S a . = O, and they satisfy the relations (21) I , * S l , C_ I , .

(22) I , * S a , C h , + 1

By definition (17) Sx, is the right idealiser for I , , i.e.

(23) I, (I.) = SI..

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Obviously I~ C SI~, since In * I~ C I~. Therefore Si~ = I~ U SHn. From (6) it follows that for the elements from SIt~ the second term in the brackets should disappear, and therefore we find

(24) SII n -~ {S e T \ I n I g~'-' (~)" g~' (z) = 0, h I (z ) . g~ (z) -- 0}.

Then the "jumping" set SA. from (22) is equal to SAn -- (S \ In) \ SXl.. Another way to vanish the second term in (6) is by considering the special

superconformal transformations (they are called Ann-transformations in [9]) for which the relation gn-, (z)- h (z) = 0 is valid (see (2) and (6)). Let us divide I , in two

disjoint parts In = IA, U I#A~, where Inn d~! {s G In I g " - ' (z)" h (z) = 0}. It was shown in [9] that the Ann-property is preserved from the right only, and so we obtain Inn * S C Inn , which means that IAn is a right ideal in S, then

(25) L (IAn) = S.

For the sets A a m = IAn \ IAn_l we find AAn * G C_ A A , , and therefore

(26) Ir (AAn) = G.

We note here that by means of the right group action we can reach a set In with any large n, because the relation ACAn * G C z~¢An+~ (see also (15)).

5. Ideal Quasieharacter

Let us define (27) X (s) de=] {n E N I ind g (z) = n}.

Using (10) and (11) we obtain

~" x ( t ) , X(s) _> x ( t ) (28) m a x x ( s * t ) X ( s ) + I , X(s) < k ( t ) .

In particular,

(29) )¢ ( g - s ) = X(s) , s # z . X(s*g) = ~ ( s ) + l ,

From (28) it follows that n8 = IX (s * t) - X (s) - X (t)l is bounded. This value nz shows how much the mapping s ~ X (s) differs from a homomorphism [19]. The boundedness of n~ allows us to conclude that X (s) is a quasicharacter [32] which can be called an ideal quasicharacter. The elements of S having finite ideal quasicharacter are nilpotent and belong to the ideal T , and X (g) = c~, g E G . Another description of the ideal quasicharacter can be written as follows X (s) = n <==~ s E ~ , . Since A ,~ ~ ~,~ = O, n # m , we conclude that X (s) indeed separates the elements having different indices, and the relation 7r defined as STr$ ¢==~ )¢ (s) = X (t) is an equivalence relation in S.

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6. G r e e n ' s R e l a t i o n s

In our notations the Green's L and T~ relations are

(30) s £ t ¢ ~ 3u, v E S , u , s = t , v * t = s , s T ~ t -u-4- 3 u , v E S , s . u = t , t , v = s .

Let us find L and T£ equivalent elements in the superconformal semigroup S. Using (10) and (28) we find that s / : t =:~ X(s) < )~(t) A x ( t ) < X(s) =~ X (s) = X (t). Therefore £ --- rr, and L-equivalent elements have the same ideal quasicharacter, (31) s z: t ~ x (s) = x ( t ) ,

and they belong to the same set A~. By analogy from (11) for the 7~-equivalent elements we derive s T i t =~ X(s) < x ( t ) + l A x ( t ) < ) ; ( s ) + 1. Then the ideal quasicharacters of the Tg-equivalent elements can differ only by 1 or coincide, i.e.

(32) s T¢ t ~ IX (s) - X (t)] < 1.

Since 7-/= L fl 7~, the sets A~ consist also of H-equivalent elements. Consider the L-equivalent elements. Let s ~ t, s ~ z, t ~ z. From (30) we de-

rive t h a t s = v * ( u * s ) = ( v * u ) * s = ( v * u ) * k * s f o r a n y k E N . I f v E T V u E T , then (v * u) *k E T , since T is an ideal in S. Because of T is a nilsemigroup 3n E N such that (v * u) *n = z. Through the arbitrariness of k we choose k = n and obtain s = (v * u) *n * s = z * s = z or s = t , which contradicts the initial assumptions. The same is valid for other Green's relations. Therefore v E G A u E G, i.e. nontrivial L and T~ equivalences can be constructed with regard to the invertible elements of S only. In fact the principal left and right ideals generated by Y t E S and defined by L (t) de_=/S * t and R (t) 4¢1 t * S are analogous to the left and right cosets of G in S introduced in [26, 30].

A c k n o w l e d g e m e n t s

The author is grateful to Prof. J. M. Howie for fruitful conversations and remarks and the kind hospitality at the University of St. Andrews, where the work was begun. Also the discussions with D. A. Arinkin, V. G. Drinfeld, J. Kupsch, M. V. Lawson, B. V. Novikov, W. Rfihl, S. D. Sinelshchikov and J. Wess are greatly acknowledged.

R e f e r e n c e s

[1] Baranov, M. A., I. V. Frolov, and A. S. Schwarz, Geometry ofsuperconfo:'mal field theories in two dimensions, Theor. Math. Phys. 70 (1987), 92-103.

[2] Berezin, F. A., "Introduction to Superanalysis", Reidel, Dordrecht, 1987.

[3] Catino, F., On bi-ideals in eventually regular semigroups, Riv. Mat. Pure Appl. 4 (1989), 89-92.

[4] Clifford, A. H., Remarks on O-minimal quasi-ideals in semigroups, Semigroup Forum 16 (1978), 183-196.

[5] Crane, L., and J. M. Rabin, Super Riemann surfaces: uniformization and Teichmiiller theory, Commun. Math. Phys. 113 (1988), 601-623.

258

DUPLIJ

[6] Csdszdr, A., and E. Th/immel, Multiplieative semigroups of continuous map- pings, Acta Math. Hung. 56 (1990), 189-204.

[7] Duplij, S., On N =- 4 super Riemann surfaces and superconformal semi- group, J. Phys. A24 (1991), 3167-3179.

[8] , On semigroup nature of superconformal symmetry, J. Math. Phys. 32 (1991), 2959-2965.

[9] , Ideal structure of superconformal semigroups, University of Kaiserslautern, preprint KL-TH-95/4, CERN-SCAN-9503193, 1995 (to ap- pear in Theor. Math. Phys.).

[10] ~, On alternative supermatrix reduction, University of Kaiser- slautern, preprint KL-TH-95/15, alg-geom-9506004, 1995 (to appear in Lett. Math. Phys.).

[11] Garba, G. U., Nifpotents in semigroups of partial one-to-one transforma- tions, Semigroup Forum 48 (1994), 37-49.

[12] Green, M. B., J. H. Schwarz, and E. Witten, "Superstring Theory", Vol. 1,2, Cambridge Univ. Press, Cambridge, 1987.

[13] Grillet, P. A., A construction of finite commutative nilsemigroups, Commun. Algebra 19 (1991), 3145-3172.

[14] Hmelnitsky, I. L., On semigroups with the idealizer condition, Semigroup Forum 32 (1985), 135-144.

[15] Hodgkin, L., On metrics and super-Riemann surfaces, Lett. Math. Phys. 14 (1987), 177-184.

[16] Huckaba, J. A., "Commutative Rings with Zero Divisors", Dekker, New York, 1988.

[17] Kaku, M., "Introduction to Superstrings", Springer-Verlag, Berlin, 1988.

[18] Kandasamy, W. B. Vasantha, Filial semigroups and semigroup rings, Liber- tas Math. 12 (1992), 35-37.

[19] Kazhdan, D., On ~ - representations, Israel J. Math. 43 (1982), 315-323.

[20] Lajos, S., Bi-ideals in semigroups. I. A survey, Pure Math. Appl. A2 (1992), 215-237.

[21] Lallement, G., On nilpotency in semigroups, Pacific J. Math. 42 (1972), 693-700.

[22] Levi, I., Green's relations on G-normal semigroups, Louisville preprint, 1992.

[23] Levi, l., and W. Williams, Normal semigroups of partial one.to-one trans- formatwns, 2, Semigroup Forum 43 (1991), 344-356.

[24] Magill, K. D., Semigroups of functions on topological spaces, Proc. London Math. Sac. 16 (1966), 507-518.

[25] , A survey of semigroups of continuous selfmap.~, Semigroup Forum 11 (1975), 1-189.

[26] McAlister, D. B., Embedding inverse semigroups in easel semigroups, Semi- group Forum 19 (1980), 255-267.

259

DUPLIJ

[271

[28]

[29]

[30]

[31]

[32]

[331

[34]

[351

[361

[371

Rabin, a. M., Status of the algebraic approach to super Riemann surfaces, in "Physics and Geometry", Chau, L.-L., and W. Nahm, eds., Plenum Press, New York, 1991, pp. 653-668. Rogers, A., Super Riemann surfaces, in "The Interface of Mathematics and Particle Physics", Quillen, D. G., G. B. Segal, and S. T. Tson, eds., Clarendon Press, New York, 1990, pp. 87-96. Rosly, A. A., A. S. Schwarz, and A. A. Voronov, Geometry ofsuperconformal manifolds, Commun. Math. Phys. 119 (1988), 129-152. Schein, B. M., Cosets in groups and semigroups, in "Semigroups With Applications", Howie, J. M., W. D. Munn, and H. J. Weinert, eds., World Sci., River Edge, 1992, pp. 205-221. Shevrin, L. N., On two longstanding problems concerning nilsemigroups, in "Semigroups With Applications", Howie, J. M., W. D. Munn, and H. a. Weinert, eds., World Sci., River Edge, 1992, pp. 222-235. Shtern, A. I., Quasirepresentations and pseudorepresentations, Func. Anal. Appl. 25 (1991), 140-143. Steinfeld, O., "Quasi-ideals in Rings and Semigroups", Akad4miai Kiado, Budapest, 1978. Sullivan, R. P., Semigroups generated by nilpotent transformations, J. Alge- bra 110 (1987), 324-343. Symons, J. S.V., Normal transformation semigroups, J. Austr. Math. Soc. A22 (1976), 385-390. Volkov, D. V., and V. P. Akulov, On the possible universal neutrino inter- action, JETP Lett 16 (1972), 621-624. Wess, J., and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys. B70 (1974), 39.

Theoretical Physics Department, University of Kaiserslautern, 67653 Kaiserslautern, Germany and Theory Division, Nuclear Physics Laboratory, Kharkov State University, Kharkov 310077, Ukraine. E-maih daplij@physik.uni-kl .de.

Received May 18, 1995 and in final form October 19, 1995

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