Super-Hopf realizations of Lie Superalgebras: Braided Paraparticle extensions of the Jordan-Schwinger map
aa r X i v : . [ m a t h - ph ] A ug Super-Hopf realizations of Lie superalgebras:Braided Paraparticle extensions of theJordan-Schwinger map
K. Kanakoglou ∗ ,† , C. Daskaloyannis ∗∗ and A. Herrera-Aguilar ∗ ∗ Instituto de Física y Matemáticas ( I FM ), Universidad Michoacana de San Nicolas de Hidalgo( U MSNH ), Edificio C-3, Cd. Universitaria, CP 58040, Morelia, Michoacán, M EXICO † School of Physics, Nuclear and Elementary Particle Physics Department, Aristotle University ofThessaloniki ( A UTH ), 54124 Thessaloniki, G REECE ∗∗ School of Mathematics, Analysis Department, Aristotle University of Thessaloniki ( A UTH ),54124 Thessaloniki, G REECE
Abstract.
The mathematical structure of a mixed paraparticle system (combining both parabosonicand parafermionic degrees of freedom) commonly known as the Relative Parabose Set, will beinvestigated and a braided group structure will be described for it. A new family of realizationsof an arbitrary Lie superalgebra will be presented and it will be shown that these realizationspossess the valuable representation-theoretic property of transferring invariably the super-Hopfstructure. Finally two classes of virtual applications will be outlined: The first is of interest forboth mathematics and mathematical physics and deals with the representation theory of infinitedimensional Lie superalgebras, while the second is of interest in theoretical physics and has to dowith attempts to determine specific classes of solutions of the Skyrme model.
Keywords:
Paraparticles, color Lie algebras, braiding, grading, realizations, Skyrme model
PACS:
Published at: A IP Conf. Proc. v. , p.193-200, 2010
INTRODUCTION
The idea of realizing an algebra of operators in terms of an other algebra, roughly amounts inconstructing a suitable homomorphism, i.e. a linear map preserving the multiplicative relations,between our initial algebra and the “target” algebra. In this way, the relations of the initial algebracan be directly reproduced by the relations of the target algebra (which are either easier tohandle or more well known). Moreover, the representations of the target algebra give rise -ina straightforward way- to representations of the initial algebra, enabling us to gain informationabout the spectrum, the eigenstates and thus the expected values of various physical quantities.The dynamical variables of any physical theory frequently have the structure of the universalenveloping algebra (U EA ) of some Lie algebra and this is exactly what initiated the interest inrealizations of Lie algebras. Historically, one of the earliest applications of such a methodologywas the famous Jordan-Schwinger map [6] which was first introduced in 1935 by P. Jordan: x J B J B ( x ) ≡ (cid:229) ni , j = E i j ( x ) b + i b − j x J F J F ( x ) ≡ (cid:229) ni , j = E i j ( x ) f + i f − j J B : U ( L ) → B and J F : U ( L ) → F are (associative) algebra homomorphisms from the (initial)U EA U ( L ) of L to the (target) bosonic (CCR) B or fermionic (CAR) F algebras respectively. is any Lie algebra with a fin. dim. n -matrix representation E , U ( L ) is the U EA of L and x is any element of L (thus x is a generator of U ( L ) ). As is well known, the boson-fermionversions J B , J F of the Jordan-Schwinger map, led to the construction of the symmetric and theantisymmetric representations respectively of the Lie algebras which were followed by extendedapplications in various areas of physics. For enlightening reviews regarding the history andthe usefulness of the Jordan-Schwinger realization in physics problems see [1]. The successand the applicability of the J.S. map inspired the extension of the idea and the construction ofother realizations as well using bosonic or fermionic degrees of freedom as target algebras. Therelevant examples, of realizing the dynamical variables of some model via bosonic or fermionicdegrees of freedom, are numerous: In the review article [9] and the references therein one canfind a host of boson-fermion realizations and their applications in models of nuclear physics,while in [2] such techniques are applied to models from various areas of physics.The introduction of paraparticle algebras [5] and the study of their representations in con-nection to the usual boson-fermion Fock representations, gave the occasion for the study of realizations of Lie algebras with parabosons or parafermions [3, 7] which may be consideredas formal (paraparticle) generalizations of the J.S. map.Since the introduction of the idea of supersymmetry, Lie superalgebras and their U EA s havedramatically come into play. Nowadays the Lie superalgebraic structure can be found throughoutmodels of quantum field theory, elementary particle physics, solid state physics etc. AlthoughLie superalgebra realizations is a much more recent topic than Lie algebra realizations, thereis already a significant number of references on this subject. Far from trying to present anexhaustive list, we feel it’s worth underlining the readers attention to the general works [4] -some of which may be considered as formal (superalgebraic) generalizations of the J.S. map-and the references therein. In these works, and to the best of our knowledge in any similarreference in the literature, the target algebra is almost always some mixture of bosons andfermions (ordinary or deformed). In some of these references, algebras of differential operators-acting on suitable polynomial spaces- have also been used. We can thus speak about particlerealizations or differential realizations of Lie superalgebras.The novel thing in this article will be the use of a mixed paraparticle system as a target algebra.Consequently we will deal with paraparticle realizations of an arbitrary Lie superalgebra, a topicwhich is almost totally unexplored in the literature. More specifically, our target algebra will bea system combining both parabosonic and parafermionic degrees of freedom be the so-called Relative Parabose Set P BF , according to the terminology introduced by Greenberg in [5]. Let usclose this introduction by presenting the multiplicative structure of P BF . Multiplicative structure of the Relative Parabose Set P BF . The Relative ParaboseSet has been historically the only -together with the
Relative Parafermi Set P FB - attempt fora mixture of interacting parabosonic and parafermionic degrees of freedom. We present it herein terms of generators and relations, adopting a handy notation: P BF is generated -as an assoc.alg.- by the (infinite) generators B x i , F h j , for all values i , j = , , ... and x , h = ± . The relationssatisfied by the above generators are:The usual trilinear relations of the parabosonic and the parafermionic algebras which can becompactly summarized as (cid:2) { B x i , B h j } , B e k (cid:3) = ( e − h ) d jk B x i + ( e − x ) d ik B h j (cid:2) [ F x i , F h j ] , F e k (cid:3) = ( e − h ) d jk F x i − ( e − x ) d ik F h j (1)or all values i , j , k = , , ..., and x , h , e = ± , together with the mixed trilinear relations (cid:2) { B x k , B h l } , F e m (cid:3) = (cid:2) [ F x k , F h l ] , B e m (cid:3) = (cid:2) { F x k , B h l } , B e m (cid:3) = ( e − h ) d lm F x k , (cid:8) { B x k , F h l } , F e m (cid:9) = ( e − h ) d lm B x k (2)for all values k , l , m = , , ..., and x , h , e = ± , which represent a kind of algebraically estab-lished interaction between parabosonic and parafermionic degrees of freedom and characterizethe relative parabose set.One can easily observe that the relations (1) involve only the parabosonic and theparafermionic degrees of freedom separately while the “interaction” relations (2) mix theparabosonic with the parafermionic degrees of freedom according to the recipe proposed in[5]. In all the above and in what follows, we use the notation [ x , y ] (i.e.: the “commutator”) toimply the expression xy − yx and the notation { x , y } (i.e.: the “anticommutator”) to imply theexpression xy + yx , for x and y any elements of the algebra P BF .Finally we remark that all vector spaces, algebras and tensor products in this article will beconsidered over the field of complex numbers C and that the prefix “super” will always amountto Z -graded. ( G , q ) -LIE ALGEBRAS: COLORS AND BRAIDINGS In this section G will always be considered a finite, abelian group.A ( G , q ) -Lie algebra or a q -colored, G -graded Lie algebra [11] is a mathematical ideageneralizing the idea of Lie algebras and Lie superalgebras. It consists of a G -graded vectorspace L i.e. L = ⊕ g ∈ G L g (equiv.: a representation of the group Hopf algebra CG on the vectorspace L ), a non-associative multiplication h .., .. i : L × L → L on L respecting the gradation, in thesense that h L a , L b i ⊆ L a + b , ∀ a , b ∈ G and a function q : G × G → C ∗ taking non-zero complexvalues. The above data must be subject to the following set of axioms: • q - braided ( G -graded) antisymmetry: h x , y i = − q ( a , b ) h y , x i • q - braided ( G -graded) Jacobi id.: q ( c , a ) h x , h y , z ii + q ( b , c ) h z , h x , y ii + q ( a , b ) h y , h z , x ii = • q : G × G → C ∗ color function ! q ( a + b , c ) = q ( a , c ) q ( b , c ) , q ( a , b + c ) = q ( a , b ) q ( a , c ) , q ( a , b ) q ( b , a ) = x ∈ L a , y ∈ L b , z ∈ L c and ∀ a , b , c ∈ G .The q function is called a color function on G or a commutation factor on G . The axiomsimposed on it imply that it is a skew-symmetric bicharacter on G which has been shown tobe equivalent [10] to a triangular universal R-matrix on the group Hopf algebra CG . We thusconclude that by definition, the notion of q -colored G -graded Lie algebra implicitly presupposesthe existence of a triangular structure for the group Hopf algebra CG which finally entails a symmetric braiding in the monoidal category CG M of its representations. This last remark,fully justifies the use of the term braided , in both the antisymmetry property and the generalizedJacobi identity included in the defining axioms of the q -colored G -graded Lie algebra.The U EA of L , U ( L ) is a G -gr. assoc. algebra. It will be generated by any linearly inde-pendent set of homogeneous elements of L . Inside U ( L ) we will have the relations h x , y i = xy − q (cid:0) deg ( x ) , deg ( y ) (cid:1) yx , for any x , y homogeneous in L and deg ( x ) , deg ( y ) ∈ G .The G -graded v.s. U ( L ) ⊗ U ( L ) equipped with the multiplication ( x ⊗ y )( z ⊗ w ) = q ( a , b ) xz ⊗ yw is denoted U ( L ) ⊗ U ( L ) and called q -braided, G -graded tensor product algebra . ( L ) can be equipped with a “comultiplication” D : U ( L ) → U ( L ) ⊗ U ( L ) which is an ho-momorphism of G -gr. assoc. alg., uniquely defined by its values on the homog. elements of L D ( x ) = ⊗ x + x ⊗ S : U ( L ) → U ( L ) which is a twisted [10] or braidedantihomomorphism in the sense that: S ( xy ) = q (cid:0) deg ( x ) , deg ( y ) (cid:1) S ( y ) S ( x ) for any homogeneous x , y ∈ U ( L ) . S is uniquely defined by its values on the homogeneous elements of L : S ( x ) = − x .If the above are supplemented with e ( x ) = ∀ x ∈ U ( L ) x = e ( ) = G -graded Hopf algebras or ( G , q ) -graded Hopf algebras. According to the modern terminology[10], developed in the ’90’s and originating from Quantum Group theory, such a structure willbe called a q - braided group in the sense of the braiding induced in the CG M category by thecommutation factor q . It is also customary to speak of U ( L ) as a Hopf algebra in the symmetricmonoidal category CG M of representations of the group Hopf algebra CG . Lets proceed to acouple of examples. Lie superalgebras: . If we set G = Z and the commutation factor q : Z × Z → C ∗ given by q ( a , b ) = ( − ) a · b ( ∀ a , b ∈ Z , and the oper. in the exponent in the Z ring), we get a Z -graded Lie algebra or aLie superalgebra. Thus: L = L ⊕ L , and in U ( L ) , we have: h x , y i = xy − ( − ) deg ( x ) deg ( y ) yx ( ∀ x , y hom. in L ) which imply: L ∋ h L , L i = [ L , L ] commutator L ∋ h L , L i = [ L , L ] commutator L ∋ h L , L i = { L , L } anticommutator ( Z × Z ) -graded Lie algebras: . If we set G = Z × Z and the commutation factor q : (cid:0) Z × Z (cid:1) × (cid:0) Z × Z (cid:1) → C ∗ given by q ( a , b ) = ( − ) ( a · b + a · b ) ( ∀ a = ( a , a ) , b = ( b , b ) ∈ Z × Z , and the oper. in the exponent are in the Z ring), we get(one of the possible) examples of a ( Z × Z ) -graded Lie algebra. Thus: L = L ⊕ L ⊕ L ⊕ L , and in U ( L ) , we have: h x , y i = xy − q ( deg ( x ) , deg ( y )) yx ( ∀ x , y hom. in L ) which imply: L ∋ h L , L i = [ ., . ] commutator L ∋ h L , L i = [ ., . ] commutatorL ∋ h L , L i = { ., . } anticommutator L ∋ h L , L i = [ ., . ] commutatorL ∋ h L , L i = { ., . } anticommutator L ∋ h L , L i = [ ., . ] commutatorL ∋ h L , L i = [ ., . ] commutator L ∋ h L , L i = { ., . } anticommutatorL ∋ h L , L i = [ ., . ] commutator L ∋ h L , L i = { ., . } anticommutator THE RELATIVE PARABOSE SET AS A BRAIDED GROUP
In this section we review results presented in [8, 13] regarding the braided, graded algebraicstructure of the Relative Parabose Set P BF . Z × Z ) -Graded structure of P BF . The relative parabose set P BF is the universal en-veloping algebra UEA of a q -colored ( Z × Z )-graded Lie algebra L Z × Z . This implies that P BF is a ( Z × Z )-graded associative algebra P BF ∼ = U ( L Z × Z ) (3)Its generators are homogeneous elements in the above gradation, with the paraboson generators B + k , B − l , k , l = , , ... spanning the L subspace of L Z × Z , and the parafermion generators F + a , F − b , a , b = , , ... spanning the L subspace of L Z × Z , thus their grades are given as follows deg ( B e k ) = ( , ) deg ( F ha ) = ( , ) (4)where e , h = ± . At the same time the polynomials { B e k , B h l } and [ F ea , F hb ] ∀ k , l , a , b = , , ... and ∀ e , h = ± span the subspace L of L Z × Z , and the polynomials { F ea , B h k } ∀ k , a = , , ... and ∀ e , h = ± span the subspace L of L Z × Z . Consequently their grades are given as follows deg ( { B e k , B h l } ) = deg ([ F ea , F hb ]) = ( , ) deg ( { F ea , B h k } ) = ( , ) (5)Finally the color function used in the above construction is given q : (cid:0) Z × Z (cid:1) × (cid:0) Z × Z (cid:1) → C ∗ q ( a , b ) = ( − ) ( a b + a b ) (6)for all a = ( a , a ) , b = ( b , b ) ∈ Z × Z , and the operations in the exponent are considered inthe Z ring. Braided Group structure of P BF . The relative parabose set P BF has the structure of a q -braided group where the commutation factor q : (cid:0) Z × Z (cid:1) × (cid:0) Z × Z (cid:1) → C ∗ is given by (6).The relations can be given explicitly as D ( B ± i ) = ⊗ B ± i + B ± i ⊗ S ( B ± i ) = − B ± i D ( F ± j ) = ⊗ F ± j + F ± j ⊗ S ( F ± j ) = − F ± j D ( { B e k , B h l } ) = ⊗ { B e k , B h l } + { B e k , B h l } ⊗ S ( { B e k , B h l } ) = −{ B e k , B h l } D ([ F ea , F hb ]) = ⊗ [ F ea , F hb ] + [ F ea , F hb ] ⊗ S ([ F ea , F hb ]) = − [ F ea , F hb ] D ( { F ea , B h k } ) = ⊗ { F ea , B h k } + { F ea , B h k } ⊗ S ( { F ea , B h k } ) = −{ F ea , B h k } (7)for all i , j , k , l , a , b = , , ... and for all e , h = ± . We also have e ( x ) = ∀ x ∈ P BF . On subalgebras of P BF . The linear subspace of the Relative Parabose set P BF (or ofthe ( Z × Z , q )-Lie algebra L Z × Z ) spanned by the elements of the form { B e k , B h l } , [ F ea , F hb ] and { F ea , B h k } for all k , l , a , b = , , ... and for all e , h = ± is a Z -graded Lie algebra (orequivalently a Lie superalgebra). The UEA of this Lie superalgebra is a subalgebra of P BF .Let us see this last statement in a little more detail: According to the first statement of thissection, the elements { B e k , B h l } , [ F ea , F hb ] and { F ea , B h k } ∀ k , l , a , b = , , ... and ∀ e , h = ± spanhe L ⊕ L subspace of the L Z × Z = L ⊕ L ⊕ L ⊕ L . Now it suffices to notice that thesubset { ( , ) , ( , ) } of the Z × Z group is a subgroup isomorphic to the Z group as we cansee from the multiplication tables of the corresponding groups (written in the additive notation): Z × Z ∼ = Z ⊕ Z group : + ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , ) ! { ( , ) , ( , ) } is a subgroupo f the Z × Z groupisomorphic to the Z = { , } group : + 0 10 and that the restriction of the commutation factor (6) on { ( , ) , ( , ) } coincides (as a function)with the usual commutation factor (on Z ) of Lie superalgebras. PARAPARTICLE REALIZATIONS OF LIE SUPERALGEBRAS
For detailed computations and proofs of the results presented in this section one can see [8].Let L = L ⊕ L be any complex Lie superalgebra of either finite or infinite dimension andlet V = V ⊕ V be a finite dimensional, complex, super-vector space i.e. dim C V = m and dim C V = n . If V is the carrier space for a super-representation (or: a Z -graded representation)of L , this is equivalent to the existence of an homomorphism P : U ( L ) → E nd gr ( V ) of assoc.superalgebras, from U ( L ) to the algebra E nd gr ( V ) of Z -graded linear maps on V . For anyhomogeneous element z ∈ L the image P ( z ) will be a ( m + n ) × ( m + n ) matrix of the form P ( z ) = A ( z ) B ( z ) C ( z ) D ( z ) ! րց P ( X ) = (cid:18) A ( X ) D ( X ) (cid:19) ( i f z = X even ) P ( Y ) = (cid:18) B ( Y ) C ( Y ) (cid:19) ( i f z = Y odd ) (8)where the complex submatrices A m × m , B m × n , C n × m , D n × n , of P ( m + n ) × ( m + n ) constitute the parti-tioning ( Z -grading) of the representation.The linear map J P BF : L → P BF defined by X i J P BF ( X i ) = m (cid:229) k , l = A kl ( X i ) { B + k , B − l } + n (cid:229) a , b = D ab ( X i )[ F + a , F − b ] (9)for any even element ( Z = X i ) of an homogeneous basis of L and by Y j J P BF ( Y j ) = m (cid:229) k = n (cid:229) a = (cid:16) B k , a ( Y j ) { B + k , F − a } + C a , k ( Y j ) { F + a , B − k } (cid:17) (10)for any odd element ( Z = Y j ) of an homogeneous basis of L , can be extended to an homomor-phism of associative algebras J P BF : U ( L ) → U ( L ⊕ L ) ⊂ P BF (11)etween the universal enveloping algebra U ( L ) of the Lie superalgebra L and the relativeparabose set P BF , in other words it constitutes a realization of L with paraparticles . Thestatement is fully justified by the fact that the linear map J P BF preserves (see [8]) for all values i , j , p , q = , , ... the Lie superalgebra brackets: J P BF ( (cid:2) X i , X j (cid:3) ) = (cid:2) J P BF ( X i ) , J P BF ( X j ) (cid:3) J P BF ( (cid:2) X i , Y p (cid:3) ) = (cid:2) J P BF ( X i ) , J P BF ( Y p ) (cid:3) J P BF ( { Y p , Y q } ) = { J P BF ( Y p ) , J P BF ( Y q ) } (12)Furthermore, the above constructed homomorphism of associative algebras J P BF : U ( L ) → P BF , isan homomorphism of super-Hopf algebras (equivalently an homomorphism of Z -graded Hopfalgebras) between U ( L ) and the U ( L ⊕ L ) Z -graded subalgebra of P BF . To see this it isenough to compute (see [8]) the rhs and the lhs of the following D ◦ J P BF = ( J P BF ⊗ J P BF ) ◦ D L e ◦ J P BF = e L S ◦ J P BF = J P BF ◦ S L (13)where D L , e L , S L are the Lie superalgebra super Hopf structure maps and D , e , S are the braidedgroup structure maps (7) of the Relative Parabose Set P BF . CONCLUDING REMARKS - FUTURE PROSPECTS
The Lie superalgebra paraparticle realizations presented in this article constitute direct general-izations of the Jordan-Schwinger map and of realizations presented in works such as [3, 4, 7]as well. Our results, generalize these previous works in various aspects: We use Lie superal-gebras (instead of Lie algebras), the target algebra consists of the mixed paraparticle system P BF (instead of using simply bosons or fermions or some supersymmetric mixture of them), ourformal expressions in (9), (10) make use of paraparticle number operators (instead of particlenumber operators) and finally our results are applicable for both finite and infinite dimensionalLie superalgebras as well.Similarly to the J.S. map which initiated developments in the representation theory of Liealgebras (construction of the symmetric and the antisymmetric representations of LA), theparaparticle realizations presented in this article can be utilized for pure mathematical purposesas well: Given representations of P BF , (9), (10) can be the starting point for the constructionof possibly new representations of an arbitrary Lie superalgebra, a topic which -at least in theinfinite dimensional case- is virtually unexplored. Of course this presupposes the constructionof at least the Fock-like representations of P BF which constitutes a hard task on its own.Finally let us close this work, by outlining a virtual application of the paraparticle realizations(9), (10), which -partially- provided us with the motivation for developing them. It has to do withattempts of the first and the third author of this paper to determine a specific class of solutions ofthe Skyrme model [12]: The Euler-Lagrange equations of the Skyrme model read: ( ~ L m : Maurer-Cartan covariant vectors) ¶ m (cid:0) F p ~ L m + e (cid:2) ~ L n , [ ~ L m ,~ L n ] (cid:3)(cid:1) = f smnr ∈ R or C ) (cid:2) [ ~ L m ,~ L n (cid:3) ,~ L r ] = f smnr ~ L s etween the ~ L m , enforcing them to be generators of a parafermion-like algebra, then realizingthe Maurer-Cartan covariant vectors ~ L m in terms of some paraparticle algebra, may prove a help-ful intermediate step in determining the constants f smnr . The next step will be the computationof the restrictions imposed on the forms of the fields U ( x ) , implied by the above mentioned“parafermionic” conditions. Finally, if the produced forms will prove acceptable, we will pro-ceed in computing the representations of the parafermion-like generated algebra and the physicsproduced on these representations.The above constitute part of projects we are already working in and hopefully we will reportprogress in the near future. ACKNOWLEDGMENTS
KK would like to thank the whole staff of I FM , U MSNH for providing encouragement and a stim-ulating atmosphere while this research was in progress. He also acknowledges the postdoctoralgrant C
ONACYT /No. J60060 which partially supported his stay in Michoacan. The research ofAHA was supported by grants C IC ONACYT /No. J60060; he is also grateful to S NI . REFERENCES
1. L.C. Biedenharn, M.A. Lohe,
Quantum group theory and q-tensor algebras , World Scientific, Sin-gapore, 1995, §2 .
3; L.C. Biedenharn, J.D. Louck, “Angular momentum in quantum Physics: Theoryand applications”,
Encyclopedia of Mathematics and its Applications , , Cambridge University Press,Cambridge, 1989, ch.52. W.J. Caspers, Spin systems , World scientific, Singapore, 1989; F. Iachello, “Lie algebras and applica-tions”, in
Lecture Notes in Physics v. , Springer, Berlin, 20063. C. Daskaloyannis, K. Kanakoglou, I. Tsohantjis, J. Math. Phys. v. , 2, pp.652-660, (2000) // e-print:arXiv:math-ph/9902005v24. Hong-Chen Fu, J. Math. Phys. v. , 3, pp.767-775, (1991); T.D. Palev, J. Math. Phys. v. , 10,pp.2127-2131, (1981); Chang-Pu Sun, J. Phys. A:Math. Gen. v. , pp.5823-5829, (1987); D.S Tang, J. Math. Phys. v. , 10, (1984); L. Frappat, P. Sorba, A. Sciarrino, Dictionary on Lie Algebras andSuperalgebras , Academic Press, New York, 2000 // eprint: hep-th/96071615. H.S. Green, Phys. Rev., v. , 2, pp.270, (1953); O.W. Greenberg, A.M.L. Messiah, Phys. Rev. v. ,5B, pp.1155-1167, (1965); D. V. Volkov, Sov. Phys.-JETP , pp.1107-1111, (1959); Sov. Phys.-JETP , pp.375-378, (1960)6. P. Jordan, Z. Physik , v. , pp.531-535, (1935); J. Schwinger, 1952 unpublished work reprinted in: Selected papers in quantum theory of angular momentum , edited by L.C. Biedenharn, H. van Dam,Academic Press, New York, 19657. K. Kademova et al,
Nucl. Phys. B v. , pp.350-354, (1970); Intern. J. Theor. Phys. v. , 2, pp.109-114,(1970); Intern. J. Theor. Phys. v. , 2, pp.115-118, (1970); Intern. J. Theor. Phys. v. , 3, pp.185-189,(1970); Intern. J. Theor. Phys. v. , 2, pp.159-170, (1971); Intern. J. Theor. Phys. v. , pp.337, (1970)8. K. Kanakoglou, C. Daskaloyannis, A. Herrera-Aguilar, arXiv:0912.1070v1 [math-ph]9. A. Klein, E.R. Marshalek, Rev. of Mod. Phys. v. , 2, pp.375-558, (1991)10. S. Montgomery, Hopf Algebras and their Actions on Rings , Regional Conference Series in Mathemat-ics , AMS, NSF-CBMS, De Paul university, Chicago, 1992; M. Scheunert, arXiv:q-alg/9508016v1;S. Majid, Foundations of Quantum Groups Theory , Cambridge University Press, Cambridge, 199511. M. Scheunert,
J. Math. Phys. v. , 4, pp.712-720, (1979); “The theory of Lie superalgebras", LectureNotes in Mathematics v. , Springer, Berlin 1978, pp.1-27012. T.H.R. Skyrme, Nucl. Phys. v. , pp.556-569, (1962)13. W. Yang, Sicong Jing, Commun. in theor. phys. v. , 6, pp.647-650, (2001) // e-print:arXiv:math-ph/0212009v1; Science in China (Series A) v.44