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O.V.Radko, A.A.Vladimirov
ON THE ALGEBRAIC STRUCTURE OF DIFFERENTIAL CALCULUS ON QUANTUM GROUPS
Submitted to «Journal of Mathematical Physics»
1997 I. Introduction Non-com mutative differential calculus on quantum groups initiated and thoroughly worked out by Woronowicz [1] is up to now a subject of ac tive discussions and development. Though meeting some problems [2, 3, 4, 5, 6] with non-classical dimensionalities of spaces of higher-order dif ferential forms (which, in its turn, stimulated very interesting alternative approaches [7, 5, 8]), original Woronowicz ’s construction remains highly attractive due to both its rich algebraic structure and useful applications. Probably, the best known realization of this scheme is bicoVariant differ ential calculus on the GLq(N) quantum groups [9, 3, 10). Closely related but somewhat parallel to Woronowicz ’s construction is another project [3, 7, 11, 12] that, in particular, has produced a bicovariant algebra of four types of elements: functions on a quantum group, differ ential forms, Lie derivatives along vector fields, and inner derivations - by exact analogy with classical differential geometry. However, this scheme, as it is, does not seem to be fully motivated by the Hopf-algebraic nature of non-commutative differential calculus. In the present paper, we suggest an extension of Woronowicz ’s axiomat- ics which naturally involves Lie derivatives and inner derivations in a way that respects the Hopf algebra structure of the whole scheme. Actually, in the framework of Woronowicz ’s noncommutative differential calculus [1, 10] one deals with the differential complex
a -i+ r -i* r2 —>... , (i) where A is a Hopf algebra (of functions on a quantum group), F is its bicovariant bimodule, F2 = F A F is its second wedge power, and so on. Exterior differential map d : Fn —> Fn "hl is assumed to obey the Leibniz rule d(ab) = [da)b -f a db (2) and the nilpotency condition d o d — 0. Brzezinski [13] has shown that
rA = a ® r ® r2 ®... (3) also becomes a (graded) Hopf algebra with respect to (wedge) multiplica tion and natural definitions of coproduct and antipode. In what follows,
2 we want to demonstrate how this Hopf structure can be used to build an associative noncommutative bicovariant algebra containing functions, differential forms, Lie derivatives and inner derivations. Similar algebras have been introduced and studied by several authors [3, 11, 14, 12] (and the idea to a use cross-product for constructing bicovariant differential calculus is due to [7]). Probably, the closest to ours is the approach by P. Schupp [14]. However, some of our results and, especially, starting points appear to be different. So, we propose the construction described below as entirely Hopf-algebra motivated (and, we believe, natural) new approach to the problem. II. Cross-product of dual Hopf algebras Notions of mutually dual Hopf algebras and their cross-product will actively be used throughout this paper. Let us recall the corresponding terminology and basic definitions [15, 16, 17, 18, 7]. Let A be a Hopf algebra with associative multiplication, coassociative coproduct
A: A—» A 0 A, A(fl) = fl(i)®fl(2), A(a6) = A(o)A(6) (4)
(we will use the notation
<2(i) 0 a(2) 0 0(3) == (A 0 id) o A(o) = (id 0 A) o A(o) , (5) and so on, for multiple coproducts), a counit
e:A—> C , e{ab) = e(a)e(b ), e(a{l)) o (2) = a(1)e(a(2)) = a , (6) and an invertible antipode
S : A—► A, S(ab) — S(b)S(a ), A(5(o)) = S(a^ 2 )) 0 S(a{i)),
e(5'(o)) = e(a), 5(o (i))o (2) = a(i)5(o( 2)) = e(a). (7) Algebra A* is a Hopf dual of A with <•, •>: A*0 A —> C being a duality map, if
< x, 1 >= e(x),
3 One can define left and right covariant actions A* t> A and A « A* by
x > a = a(!)
As usual, left and right actions imply
(xy) > a = x t> (y > a), a<(rj/) = (a whereas the covariance (or generalized differential property) means x > (ab) = (x(1) ▻ a)(x(2) > b), (a6) < z = (a i.e., the A*-actions respect multiplicative structure of A, or, in other words, A is a left (right) A*-module algebra. One can use (e.g., left) action (9) of A* on A to define on their tensor product A 0 A* the cross-product algebra A x A* [15, 18, 7]. This is an associative algebra with the cross-multiplication rule given by xa = (z(i) > a)x (2) = (multiplication inside A and A* does not change). A cross-product is not a Hopf algebra but exhibits remarkable A*-module and A-comodule properties [7, 19). First, A x A* is covariant under the right A*-action of the following form: right A*-action: a < x — a^2 ) < z, «(i) >, y < x = y < x, 1 >= e(x) y , (131) to be extended on arbitrary products in Ax A* by the covariance condition {pq) Surely, this needs to be consistent with (12). Let us check it: (ya) «x = (y« x(1))(a < x(2)) = e(x{l))y < x{2) ,a {l) > a {2) = < x,a (i) > Va (2) =< x,a {i) >< t/(i), a(3) > a {2) y{2) = £(1(2)) < ®(i)»a(i) >< V(i)i a(3) > a_(2)2/(2) = < V(i)’a(2) > (<%(i) < x{1) )(y{2) < x{2) ) = (< %/(!),0(2) > 0(i)2/(2)) 4 It is known [18] that a. covariant right action F < H of a Hopf algebra H on an algebra F implies a covariant left coaction F —> H* Q F of the Hopf dual //* on F. The correspondence is defined by (16) where a coaction is assumed to be / —> /(i) Q /* 0) with h £ //, £ //*, /,/*°* € F. For coactions, ‘co variant ’ still means ‘respecting multiplica tion'. This is expressed by (// In our case, the left A-coact.ion dual to (13) is left A-coact.ion: a A(a) = «d) (v fl(2), y 1 0 y (18) The very last relation explains why the elements of A* are called left- invariant in this situation. Further, A x A* is covariant under a left Abaction and also under its dual right A-coaetion. Explicit form of the Ab action is taken to be the well-known Hopf adjoint. ad x > p = x(l)pS{x{2)), p £ A X A* , (19) which is evidently covariant: .r ao d {pq) = x{l)pqS{x(2 )) = x(1) pS{x[2) )x{3)qS{x(4)) = (,r(i) p)(x{2) ^ q). Moreover, for p = a £ A one shows [20, 7] that .r t> a = x{l)aS(x {2) ) = «(i) left A*-a.et ion: x > a = a^) < .r, «(2) > , x > y = x^yyS{x(2 )) ■ (21) The corresponding dual right A-coaction is deduced from the general rule [18] analogous to (16), which relates left action H t> F with right coaction F -» F 0 H*\ (22 ) 5 and is explicitly given by [7] right .4-coaction: a —► A (a) = ft(1) 0 a(2). t/ —> (e° >d y) Q en , (23) where {c '} are dual bases in .4 and .4". Note that in both (18) and (23) the coaction on the .4-part of .4x3.4" is just a coproduct. Being the covariant (co)actions, eqs. (13),(18),(21) and (23) charac terize A X3.4* as a left (right) (co)module algebra. It is in this sense that the cross-product algebra A xt A* may be called bicovariant [7, 19]. Of course, this bicovariance is merely a reflection of the underlying Hopf al gebra structure of A. III. Woronowicz ’s differential complex as a Hopf algebra Let us now recall the basic definitions of the Woronowicz noncommuta- tive differential calculus [1, 10]. First, a basis {a;1} of left-invariant 1-forms should be chosen in the bimodule F in (1). Any element p £ T can be uniquely represented as p = atuat £ A. Next, one specifies commuta tion relations between functions and differential forms, w'a = (fj & , (24) the coalgebra structure of F, A(iv‘) = 1 © a/ A ujJ 0 r‘ , (25) and a differential map d : A —> F: da — (x, o a)u>'. (26) Here a is arbitrary element of A, r* £ A, y, and /• belong to A*. The Hopf-algebra consistency (or bicovariance) conditions of the calculus are: -v <3 10 (A © id) o A = (id 0 A) o A --- X II (27) uj(ab) — (uja)b a (./]) = /;@/y (28) A(wa) = A(cv)A(a) =4> (f!»a Vk = ri(a 6 which are obtained from the properties of counit and antipode. Worono wicz’s theory asserts that every set of elements {rj,f-,Xi} obeying eqs. (27) —(32) gives us an example of a bicovariant differential calculus on the Hopf algebra A. For illustration, let us derive (31) (cf. [21]). A( a)u>*) =< X,",ti(3) > (ti(i) ® °(2)X^J O r*. + 1 © w’) = 0(1 )W^ © (Xi ▻ o (2))r‘ + a(1) © (xi ▻ a(2))w* , (33) da {i) © rt(2) + «(i) © do (2) = (Xi > 0(i))w' © a(2) + o (1) © (x,- ▻ 0(2))w' = 0(1) < Xii 0(2) > W* © 0(3) + 0(1) © (x, > 0(2))w' = 0(1)W' © (o (2) < Xi) + 0(1) © (Xt ▻ 0(2)V . (34) Independence of {tv1} yields O(i) © (xi ▻ o (2))r} = O(i) © (o( 2) < Xj) • (35) Acting on both sides of this equation by e © id, we come to (31). Consider now the graded Hopf algebra (differential complex) FA given by (1),(3) jointly with its dual (FA)*: a -A r -A r2 A ... t t t (36) a * 4- r r2* £- ... (vertical arrows indicate non-zero duality brackets implied by grading). Analogously to (12), an associative algebra Q = FA x (FA)* can be intro duced using the cross-product construction (here (FA)* = A* © T* © ...). We place Q in the center of our approach. It means that we assume the following guiding principle: All cross-commutation relations among functions, forms, Lie deriva tives, and inner derivations are to be chosen according to the rules (12) of a cross-product algebra. In other words, given Woronowicz’s calculus (and, hence, the Hopf algebra FAj, we then have to use only standard Hopf- algebra technique FA =4»(FA)* =>rAx(rA)* to construct the whole algebra of these four types of elements. The resulting algebra is bico variant by construction. Its bicovariance in the sense of Woronowicz ’s left and right covariance [1] is implied by 7 the Hopf-algebra nature of FA [13], whereas its bicovariance in the sense of Schupp, Watts and Zumino, expressed by eqs. (13),(18),(21) and (23), proves to be an inherent feature of the cross-product (see Sect. 2), and stems, at the very end, from the same Hopf structure of FA. IV. Explicit form of commutational relations It only remains to put all the relevant objects in the corresponding ‘boxes'. We already know that functions and 1-forms are situated in A and F, respectively. Owing to (18), one may consider A* (acting on A from the left) as an algebra of left-invariant (and A x A* - of general) vector fields on a quantum group A. It is generally accepted [3, 7, 11, 22] that Lie derivatives Ch along a (left-invariant) vector field h £ A* must be related with its action on arbitrary elements of Q\ Ch = h (37) which, due to (20), reduces to ordinary left action (9) h o p for p £ I A. It seems also natural to relate inner derivations with elements of F* [14]. We propose the following definition [23]. Let 7, £ F" be determined by fixing its duality bracket with a general element of F, <7 i,au>j >= e(a) , (38) and < 7 i, p >= 0 for p £ A, F2, F3,.... Then we can define a basis of inner derivations {z,} as follows: 1 1 = 7t > (39) (the same comment as for eq. (37) applies). Here we make no attempt to associate some ih £ F* with any h £ A*, for it looks unnatural in the context of our approach (see, however, [11, 24] for a discussion of such a possibility). The cross-product algebra we are seeking for, i.e., an algebra which includes four types of differential-geometric objects, a,uj\Ch and zt, is implicitly contained in the above definitions. In order to make it more transparent, we employ these definitions for obtaining a set of helpful re lations. To begin with, the dual differential map d* is introduced by 8 It commutes with elements of A*. dmoh = hod*, i.e., dm(h0) = hd*0 , h E A* , (41) and transforms 7, to \t: Xi = dm 7,. (42) Both formulas are derived via duality: < d*{h0).p > = < h.0,dp >—< h © 0, A(r/p) >—< h,pn) >< 0,dp(2 ) > = < /?./->(!) >< d*0,p( 2 ) > = < /? © d*#, A(p) > = < hd*0,p > (43) (we used < h,dp^ >= 0), and < a > — < ~u,da > = < 7.,. (\ , ▻ a )ivJ >— t(\, ▻ a) = v-(a(1)) < \»a(2) > = < a > . (44) Further, to verify that the coproduct of 7, is given by A(7,-) = 1 © 7, + 7j ® SI (45) it suffices to compute its bracket with a general element in A O F + F © A: < A(7i) - 1 0 7, - 7 j 0 // , o 0 bujk + cijjk 0 e > = < 7i,a6u ’A' -F cufce > — t(a) < 7 ,, 5lv a > — < 7 J,avA: >< //, e > =< 7„r(/> ()u,m > -5(c) < /*.c >= 5(c)£(/‘>e)-5(c) < /*,c >= 0, where a, 6. c. e E A. A comparison of (45) with (25) displays a 'left ap pearance ’ of A('),). Nevertheless, unlike the u/ - case (25), we prefer not to use the words ‘left invariance ’ here, to avoid confusion with the left invari ance under A - coaction (18) appropriate to any object in TA*. However, a similarity of (45) and (25) enables one to show in a way quite analo gous to [1] that any element 0 E T* is uniquely represented in the form 0 = Id7 t, /?' E A*. Now we are in a position to derive 10 commutation relations among a E A, u/ E T, \, E A* and 7, E F*. Three of them are already present in the original Woronowicz theory. They ar%: internal multiplication rule inside the algebra A, eq. (24), and the recipe how to multiply u/. The latter is unambiguously fixed in the framework of Woronowicz ’s scheme [1] 9 but generally cannot be written down in a closed form (see [2, 10, 12]). Another four. \ta — a\ t = ( \ j ▻ a)f- , (46) 7 ta - «7t = 0 , (47) \,wJ -uJJ\t = C3lkJfk , Cfk =< \k,rj > , (48) 7,^ + u>J7; = // . (49) are immediately obtained by the application of the cross-product rule (12) to FAx(FA)\ The remaining commutation relations require the use of the duality arguments. Let us first derive a formula 7 ih = (r- ▻ h)ij = < h(2),d > 6(l)7, . (50) We have < 7 ih . aujk >=< 7, 0 /i,«( i) 0 G(2)W^ + a(i)u;J ® «(2)rj > = < li -«(jK' >< h,a (2) rk >= ^t(fl(i)) < /i(i), 0(2) >< 6(2), r* > = < rk t> h,a >=< rJt ▻ 6 ,ci(1) > e(a {2) )Sk = h ,a(1) >< 7j, ct(2)u;^ > = < (rj > /i)7j , awfc > . (51) Using (41) and (42), we come to analogous formula for X«6 = (rj >6)xj =< 6(2),r/ > 6(i>Xj • (52) This can be also proved by a direct calculation: < \ih ,« > = < Xn a(i) >< 6, a(2) >=< 6 , a < x. >=< 6 , (x> > o)d > = < 6(i), yH>< 6(2), rj >=< 6(1),tt(1) >< Xj,«(2) >< 6(2), r3 > = < r3 > 6 ,A(i) >< Xj, 0(2) > = < irt t>6)xj,a > • (53) It is wort h mentioning that the same technique leads to a helpful formula fih = (rf >/, = < hw,S-'(r>m) > hmf? < A(3,0 > (54) which can be used, in conjunction with (52), to deduce the structure rela tions of bicovariant differential calculus in the form given in [10, 25]: X, X, • <=, (55) 10 XkfT = <**!/" V- • (57) c’L/r/t + /jx-c = + cjui. (58) Now we can list the remaining three commutational relations. One of them is (55), and the other two are as follows: 7iXj - tfxilk = C*7* , (59) <7,7, ,«tv"V>= e(a)(<7j" - -S”'^). (60) Eq. (59) stems from (50), whereas (60) is verified by a straightforward calculation. Thus, we have completed the explicit construction of the cross-product algebra generated by a,ux,Xi and 7 V. Cartan identity Remarkably, this quantum algebra exhibits some features which exactly correspond to the well-known classical relations. First, the Lie derivatives commute with exterior differentiation: Ch o d = d o Ch , h E A* . (61) Really, prove it for a 6 A: h> da = < A , «(2) >= d(A t> a). (62) Then, from h>(dbi ... dbi )... (h(n)>dbn) = d(/qi)>6i) ... d(A( n )t>£>„) (63) and the Leibniz rule it follows that h > (d(a dbi... dbn)) = d(h > (adbi ... dbn)), (64) which is exactly (61). Furthermore, the Cartan identity in the classical form can be shown to be valid: CXi = d o ii -f- ii o d. (65) 11 One needs to verify that \« > P = di'ji > p) + 7, 1> (dp), pe rA . (66) For p — a£ A eq. (66) is almost trivial and follows from 7, t> a = 0 , 7, > da == \i t> a . (67) Let p — adb (a, b E A). To show that Xi > (a db) — ^(7, > a db) + 7,- 1> (da d&) (68) we calculate each term separately, Xi >(adb) = a(i)d6 (1) < 1 © Xi + Xj © // , «(2) © 6(2) > = a(x« > <*6) + (z\j > a)(// > db), (69) d(7i ▻ (a db)) = da (%, ▻ 6) + a d(%, t> 6), (70) h > (dadb ) = ~da(xt>b) + (Xj > a)Ul t> <*&), (71) and then use (62). At last, consider the general case p — adb B , B = dc% ... dcn , where a, 6,, c, E A : Xi>(adbB) = a db (xi>B.)Aa(xj>db)(f->B) + (xk>a){fj>db)(f->B ), (72) 7i > (adbB ) = —a d6 (7, t> B) + a(%j ▻ &)(// ▻ B), (73) d(7, o (a db B)) = —da db (7; ▻ B) + a(%j ▻ c/6)(// ▻ B) q- adbd{xi > B) + da (xj > b)(f/ > B) + a(%^ ▻ 6)d(// ▻ B), (74) 7, t> (da db B) — da d6 (7, > B) — da (x~ > b)(f ■ > B) + (;u > a)(/*=> »)(//▻ B). (75) After summing this up, it remains to prove that a db (xi > B) = a dbd(xi > B), (76) or Xi ▻ (dci ... dcn) = d(7, ▻ (dci . . . dc n)), (77) 12 that is the same problem at a lower level. Thus, the proof is completed by induction. To conclude this section, we compare the duality < T”, F > used above ('vertical' duality in (36) between 1-forms and inner derivations) with a duality « .4", F »between vector fields E A* and differential 1-forms E F. The latter is a natural generalization of ordinary classical duality, and is assumed as a basis of an alternative construction of bicovariant differential calculus on the Hopf algebras in [22]. It is easily seen that the dual differential map d* establishes a direct relation between these two dualities in the following way: « d*e,p » = < o,p >, 9 e r\ pe r. (78) VI. Comparison with other approaches Now the above results (mostly, the commutation relations (46)-(49), (55), (59) and (60)) are to be compared with other approaches known in the literature [3, 11, 14, 12]. To achieve this, it is convenient to chose another set of generators for the rA*-part of our cross-product algebra. We switch from 7, to defined by da = u/(\; o a), (79) <7,,u/a>= e(a) au?J = > a) (81) and proceeding by complete analogy with Sect. 3 and 4, we obtain = 5,-1(//), 7, = aZk'hv'i ' X» = ¥x\j = d*li ' (82) A(y }) = » ^(7 .) = , A(\i) = \,®l+^0Xj. (83 ) As for commutational relations, in the {a, w} - sector they remain un changed, those between a,tv and x, 7 follow directly from (12), X,« - (v?- ▻ a)\j - \t>a, (84) 7, a - (f - > a)7; = 0, (85) 13 X, w - ?Tcrik'jl x>“Xl — ( ki u ^ (86) 7 1 + crfk u;fc07 = ■ , (87) where - At °jV = and those inside {\« 7} look like ~ ~ ~kl~ ~ z"iA‘ ~ 7* \.j - — **- k l /" 1 k " Xt\j - v}l\i\k = ( ij \k , (90) <7,7, ,wVa> = c(«)(-d- + - (91) Formulas (89),(90) are obtained with the use of h = {h #]2T,T2 = T27i/Zi2, A(T) = T0T, c(T) = l, (93) Rl2 LfLt = LfLfRl2 , RuLtL: = L^L+Rn , (94) A (L±) = i±@L±, £(Z,±) = 1, (95) < Tt, L* >= Rn , < Ti, L2 >= /?2i‘ 1 (96) < TuS(Ll) >= Rj , < TuS(L~2) >= Rn (97) (generators t\ 6 A and € A* form matrices T and L±, respectively), where R is a special numerical matrix related to GLq{N) [26] which obeys the Yang-Baxter R12R13R23 — R23R13R12 (98) 14 and Hecke % = (A = ?-,-■) (99) conditions. Let us also introduce a numerical matrix D by />;. - = = (ioo) and fix the differential map d : A F via dT = Tn (101) in terms of left-invariant Maurer-Cart an forms fl. Then the Woronowicz bicovariant differential calculus on GLq(N) is produced by the following choice [9, 10, 12] of the elements r, : ffk = r'ks(i*) , xL = (102) A which serve to define the Hopf and differential structure of the calculus as follows (note doubling the indices due to the matrix format used): A m = (/;!> onf , (104) < = (x?>C)^ = Ca^ (105) (the last equation implies < Xqi^n >= <^™)- From (82) and (102) we get <4 = /+'s-‘(r;), x'=l[( - (D-')lk) (106) and can now write down all commutational relations in the matrix form. If, before doing so, we perform one more redefinition, J = -7Z) , X = -*/), Y = 1-AX, (107) so that Yj = i+iS(/-J), (108) we end up with a complete set of commutation relations in terms of ma trices T, ft, Y and J: 15 #12TiT2 = T2Ti#i2, (109) niT2 = T2 R^nlR~l1 , (HO) nlR21 1 n2 R2 , — —R21 ^2 R12 1 (HI) Y\T2 — T2 R21 Y1 R12 1 (112) n1 R12 Y2 R2 i = R\2 Y2 R2 \ft\, (113) J1T2 = T2 R2 I J1 R\2 1 (114) R12 J2 R2 I + ^12^2^21^1 — "^(1 — 7^12^21 ) , (115) Y1 R12 Y2 R2 I = R12F2R21F1 , (116) J1R12V2R2I = R12 Y2 R21 J1 , (117) JlRl2J2R2l — ~ R2 I J2 R2 \J\ ■ (118) Several comments are in order. In this specific realization of the Woro- nowicz calculus, it proves possible to present multiplication relations for ft in a closed form (111). The commutation rule (116) for Y is often called the reflection equation [27, 28, 29], and the related formula for X X \R\2 X2 fi'2 1 — R12 X2 R21 X1 = X 1 (X\ R12 R21 — R\2 R2 \X\) (119) - the quantum Lie algebra [1, 30, 7, 31, 32], because it generalizes classical commutator in the Lie algebra of left-invariant vector fields. In terms of T and V , the left and right A-coactions in (18) and (23) take the form left: t) —> t[ ® t), y; —> 1 @ y; , (120) right: t) —> ® t), Yj —► Ytk ® . (121) This shows explicitly that the algebra A* proves to.be left-invariant and right-coadj oi nt-covariant . Algebra (109)—(118) is exactly the GLq{N) bicovariant differential al gebra found in [3] and discussed further in [12]. We have shown that it is produced just by application of the cross-product recipe to the original Woronowicz differential complex, whose Hopf-algebra properties account for bicovariance of the algebra. We are grateful to G. Arutyunov, T. Brzezinski, C. Burdik, R. Giachetti, A. Isaev, P. Pyatov, J. Riembelinski and M. Tarlini for discussions. This work was supported by Russian Foundation for Basic Research. 16 References [1] S. L. Woronowicz, Comm. Math. Phvs. 122, 125 (1989). [2] U. Carow-VVatamura, M. Schlieker, S. Watamura, and W. Welch, Comm. Math. Phys. 142, 605 (1991). [3] P. Schupp, P. Watts, and B. Zumino, Lett. Math. Phys. 25, 139 (1992). [4] A. P. Isaev and P. N. Pyatov, Phys. Lett. A 179, 81 (1993). [5] A. P. Isaev and P. N. Pyatov, J. Phys. A. 28, 2227 (1995). [6] G. E. Arutyunov, A. P. Isaev, and Z. Popowicz, J. Phys. A 28, 4349 (1995). [7] P. Schupp, P. Watts, and B. Zumino, Comm. Math. Phys. 157, 305 (1993). [8] L. D. Paddeev and P. N. Pyatov, Amer. Math. Soc. Transl. (2) 175, 35 (1996); hep-th/9402070. [9] B. Juvcd, Lett. Math. Phys. 22, 177 (1991). [10] P. Aschieri and L. Castellan!, hit. J. Mod. Phys. A 8, 1667 (1993). [11] P. Schupp, P. Watts, and B. Zumino, preprint NSF-ITP-93-75 (1993); hep-th/9306022. [12] A. P. Isaev, Phys. Part. Nucl. 28:3 (1997). [13] T. Brzezinski, Lett. Math. Phys. 27, 287 (1993). [14] P. Schupp, preprint LMU-TPW 94-8 (1994); hep-th/9408170. [15] M. E. Sweedler, Hopf algebras (Benjamin, New York, 1969). [16] E. Abe, Hopf algebras (Cambridge U.P., 1980). [17] V. Chari and A. Pressley, A Guide to Quantum Groups (Cambridge U.P., 1994). [18] S. Majid, Int. J. Mod. Phys. A 5, 1 (1990). 17 [19] P. Schitpp, Ph D. thesis, UC Berkeley, LBL-3494‘2 (1993); hep-th/9312075. [20] S. Majid, J. Algebra 130, 17 (1990). [21] F. Bonechi, R. Giachetti, R. Maciocco, E. Sorace, and M. Tarlini, Lett. Math. Pliys. 37, 405 (1996). [22] P. Aschieri and P. Schupp. Int. J. Mod. Phys. A 11, 1077 (1996). [23] A. A. Vladimirov, Czech. J. Phys. 47, 131 (1997). [24] P. Schupp and P. Watts, preprint LBL-33655 (1994); hep-th/9402134. [25] D. Bernard, Phys. Lett. B 260, 389 (1991). [26] L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Alg. Analiz 1:1. 178 (1989); English transl.: Leningrad. Math. J. 1:1, 193 (1990). [27] P. P. Ixulish and E. K. Sklyanin, J. Phys. A 25, 5963 (1992). [28] P. P. kulish and R. Sasaki, Progr. Theor. Phys. 89, 741 (1993). [29] S. Majid, J. Math. Phys. 34, 1176 (1993). [30] S. Majid, Comm. Math. Phys. 156, 607 (1993). [31] G. W. Delius and A. Huffmann, J. Phys. A 29, 1703 (1996). [32] V. Lyubashenko and A. Sudbery, q-alg/9510004 (1995). Received by Publishing Department on February 14, 1997. PaubKO O.B., BaanHMHpoe A.A. E2-97-45 06 ajire6panwecK0H cxpyxxype zmchc^epeHiwajibHoro hcmhcjiciihb Ha KBaHTOBbix rpynnax floxasano, hto cxpyxxypa anre6pbi Xont^a na flHcfjcjjepemiHanbHOM xoMiuiexce BopoHOBHMa iio 3Boa«er nocxpOHXb GMXOBapnaHXHyfo HeKOMMyraiMBHyro anreGpy, BKJlK)HaK)myK) XOOpAHHaXHbie cfjyHXLtHH, JUl(|xJ)epeHUHaJlbHbie cjDOpMbl, npOH3BOflHbie JIh h BHyrpeHHHe npoH 3BeaeHHfl, xax Kpocc-ripoH 3BeaenHe aayx BsaHMHO ayanbHbix rpaayHpoeaHHbix ajire6p Xont^a. Taxon noaxoa b noanori Mepe yuHTbiBaex xorufDOB- cxyio c rpyxxypy 6HxoBapnanTHoro flHcfx^epeHUManbHoro HCMHC/ieHHsi BopoHOBHua. O h no 3Bo/i 5ieT aaib npaMoe aoxaxaxenbcxBo xo>xaecxBa Kaprana h BbisecTH paa apyrwx noae3Hbix cooxHouieHHH. npoBoaHTca aeraabHoe cpaBHeime npeanaraeMOH xoHCipyxuHH c noaxoaaMM apyrwx aeropoB. Pa6oxa BbinoaneHa b Jla6opaTopHH xeopeTHuecxoH 4)H3hxh hm . H.H.Boroato- 6oea OH5IH. ripenpHHT 06T.enHnenHoro HHCTHryra auepHbix iicc/ieyoBannH flydna, 1997 Radko O.V., Vladimirov A.A. E2-97-45 On the Algebraic Structure of Differential Calculus on Quantum Groups Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups — coordinate functions, differential formSjLie derivatives, and inner derivations — as the cross-product algebra of two mutually dual graded Hopf algebras. This construction, properly taking into account Hopf-algebraic properties of Woronowicz ’s bicovariant calculus, provides a direct proof of the Cartan identity and of many other useful relations. A detailed comparison with other approaches is also given. The investigation has been performed at the Bogoliubov Laboratory of Theoretical Physics, JINR. Preprint of the Joint Institute for Nuclear Research Dubna, 1997 Mbkct T.E.FIoneKO rioflnHcaHO b newarb 11.03.97