E. Abe. Hopf algebras. Cambridge University Press, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the cyclicity condition of C and the fact that it is compatible with the symmetry follows easily. 4. Entropic Hopf algebras 4.1. Hopf algebras and representations. In this section we give a quick summary of the necessary background in bialgebras and Hopf algebras. For suitable introductions, see [29, 1]. A Hopf algebra is a k vector space, H, equipped with an algebra structure, a compatible coalgebra structure ( bialgebra) and an antipode satisfying the appropriate equations. Table 1 summarizes the necessary structure [29] We say a Hopf algebra is (co)commutative if the (co)multiplication is ....

E. Abe. Hopf algebras. Cambridge University Press, 1980.

....46 128, 1994. 2 algebras, but called them differently and also developed the theory in a different direction. I thank Mathijs Dijkhuizen for commenting on a preliminary version of this paper. 1. Generalities about Hopf algebras Standard references about Hopf algebras are the books by Abe [1] and Sweedler [6] see also Hazewinkel [5, x37.1] Below we will assume ground field C . By the tensor product V Omega W of two linear spaces V and W we will always mean the algebraic tensor product. Thus the elements of V Omega W are finite sums of elements v i Omega w i (v i 2 V , w i 2 W ....

E. Abe, Hopf algebras, Cambridge University Press, 1980.

....u of A is called irreducible if u is not equivalent to a corepresentation v of the form v = 0 ; i.e. if not for some m, 1 m n Gamma 1, we have v ij = 0 for all (i; j) such that m 1 i n, 1 j m. 2.4. NOTES There are two textbooks on Hopf algebras: Sweedler [42] and Abe [1]. A concise introduction to Hopf algebras is given in Hazewinkel [15, x37.1] An informal account of some basic facts and examples can be found in Bergman [8] Manin [26] discusses Hopf algebras in connection with quantum groups. We have introduced notation for Hopf algebra operations as in ....

....and examples can be found in Bergman [8] Manin [26] discusses Hopf algebras in connection with quantum groups. We have introduced notation for Hopf algebra operations as in Woronowicz [49] In fact, the notation Delta for comultiplication, for counit and S for antipode is more common, cf. [1], 11] This last notation will be used in x7 for dual Hopf algebras, in particular for (quantized) universal enveloping algebras. 3. Quantum Groups In this section we will introduce our main examples of quantum groups: the quantum analogues of the groups SL(2; C ) and SU(2) We will also give ....

E. Abe, Hopf Algebras, Cambridge University Press, 1980.

....time to construct a autonomous category, these techniques branch in two broad classes: those that have the involution of linear negation already embedded in the monoid like structure, and those that do not. An important subset of the rst method is the one that uses Hopf algebras. Hopf algebras [19, 1] are well known to provide fundamental examples of monoidal categories, and have been used in [5, 6, 7] to construct various models of (multiplicative) LL. Linear negation is provided by the interaction of the take the inverse operation in a group 1 (the monoid like structure one starts with) ....

....on A, which can be related to the structure map of an object in the Chu category using the entropy. 4 Mixed Hopf algebras 4.1 Hopf algebras and representations In this section we give a quick summary of the necessary background in bialgebras and Hopf algebras. For suitable introductions, see [19, 1]. A Hopf algebra is a k vector space 3 , H , equipped with an algebra structure, a compatible coalgebra structure ( bialgebra) and an antipode satisfying the appropriate equations. Table 2 summarizes the necessary structure [19] We say a Hopf algebra is (co)commutative if the ....

E. Abe. Hopf algebras. Cambridge University Press, 1980.

....space time. Solving models of 1 1 dimensional quantum field theory, Sklyanin, Takhtadzhyan and Faddeev revealed a new algebraic structure that was called quantum algebra [20, 21] In the more axiomatic treatment of Drinfel d and Jimbo [18, 46] it appeared as a special class of Hopf algebras [1] and consequently as a generalization of groups. The new name quantum group was used henceforth. To cut a long story short we just mention that signs of quantum group symmetries have been found in many quantum mechanical models such as integrable spin chains (e.g. 69, 75] rational conformal ....

E. Abe, Hopf algebras, Cambridge University Press, Cambridge, 1980

....quantities, which concern both the objects that define a quantum reference frame and the objects described by the theory, have to be invariant under the action of the group G. Various authors [9, 10, 11, 12] have suggested that the quantum aspects of spacetime can be described by a quantum group [13, 14] obtained from a deformation of the commutative Hopf algebra of the functions defined on the group G. From the point of view of the quantum frames, even in the absence of gravitation, there are two problems: 1. A quantum group cannot describe the internal degrees of freedom of the quantum ....

....frames discussed in ref. 2] 3 Classical reference frames. By considering the classical limit of the formalism developed in the preceding section, we clarify the meaning of various concepts and we can understand why the relations between quantum frames cannot be described by a quantum group [13, 14] obtained by a quantization of the group G. In a classical theory the algebras A 0 ; A n Gamma1 and A (n) are commutative and can be considered as algebras of L 1 functions (essentially bounded measurable functions) on the phase spaces Gamma 0 ; Gamma n Gamma1 and Gamma ....

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E. Abe: Hopf Algebras. Cambridge University Press (1977).

....from sequentialization for Yetter s nets, cf. 31] and the planarity condition described above. 2 5 Hopf algebras and Representations 5.1 Algebras and Coalgebras In this section we give a quick summary of the necessary background in bialgebras and Hopf algebras. For suitable introductions, see [1, 35, 21]. Definition 5.1 A Hopf algebra is a k vector space y , H, equipped with an algebra structure and a compatible coalgebra structure ( bialgebra) and an antipode satisfying the appropriate equations [21, 35] The following chart summarizes the necessary structure. All maps shown are linear. y ....

....notation, f(hv) hf(v) for all h 2 H; v 2 V . We thus obtain a category MOD(H) The above definition is a straightforward generalization from group representations; indeed, the latter arises as the special case H = k[G] A similar remark applies to the Hopf algebra associated to a Lie algebra [1]. If U and V are modules, then U Omega V has a natural module structure given by: H Omega U Omega V Delta Omega id H Omega H Omega U Omega V c 23 H Omega U Omega H Omega V ae Omega ae U Omega V Denote this module as U Omega H V . We will frequently drop the subscript if there ....

K. Ab'e, Hopf Algebras, Cambridge University Press, (1977).

.... 1) j 3 i 1 (t) and as a special case of a construction of Cherednik, one can associate with the maximal permutation n of S n the element Pi n = 3 1 i (3 2 Gamma f 13 )3 1 j : i (3 n Gamma1 Gamma f 1n ) 3 2 Gamma f 13 ) 3 1 j of H n (q) Omega C (t) where f 1k = [1] t;q [1] n Gamma1 t;q Gamma 1 [1] n t;q Gamma 1 with [1] t;q = 1 t(q Gamma 1) It would be interesting to clarify the relation between 3 n and V n (t) These elements are clearly related in the two extreme cases t = 1 and t = Gamma1=q but we do not know if such relations still exist ....

.... 3 i 1 (t) and as a special case of a construction of Cherednik, one can associate with the maximal permutation n of S n the element Pi n = 3 1 i (3 2 Gamma f 13 )3 1 j : i (3 n Gamma1 Gamma f 1n ) 3 2 Gamma f 13 ) 3 1 j of H n (q) Omega C (t) where f 1k = 1] t;q [1] n Gamma1 t;q Gamma 1 [1] n t;q Gamma 1 with [1] t;q = 1 t(q Gamma 1) It would be interesting to clarify the relation between 3 n and V n (t) These elements are clearly related in the two extreme cases t = 1 and t = Gamma1=q but we do not know if such relations still exist in the ....

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E. Abe , Hopf algebras, Cambridge University Press, 1977.

....scenario we deliberately present these arguments in rather elementary steps. 2 Coactions and crossed products To fix our conventions and notations we start with shortly reviewing some basic notions on Hopf module actions, coactions and crossed products. For full textbook treatments see e.g. [A,M3,Sw]. Let G and G be a dual pair of finite dimensional Hopf algebras. We denote elements of G by Roman letters a; b; c; and elements of G by Greek letters ; The units are denoted by 1 2 G and 1 2 G. Identifying G = G, the dual pairing G Omega G C is written as ....

E. Abe, Hopf Algebras, Cambridge University Press, Cambridge, 1980.

.... non commutative numbers one should completely get rid of the field K from the outset. As yet we have no concrete proposal for this scenario. Section 3 is devoted to the study of integrals in weak Hopf algebras. Using the notion of weak Hopf modules which is a generalization of the Hopf modules [1, 19] we show that non zero integrals exist. A weak Hopf version of Maschke s Theorem characterizes semisimple WHAs as those possessing normalized integrals. An other important class of WHAs are those which are Frobenius algebras. They are characterized by possessing non degenerate left integrals. This ....

.... yield the left and right subalgebras, respectively: Inv A A = A L ; Inv AA = A R : Investigating the structure of the mixed modules A A A and A A A , that incorporate the whole bialgebra structure of A, one arrives to a weak generalization of the notion of Hopf modules [1, 19]. Definition 3.6 A right weak Hopf module (right WHM) over A is a right A module M which is also a right A comodule such that the compatibility relation (m Delta x) 0 Omega (m Delta x) 1 = m 0 Delta x (1) Omega m 1 x (2) 3.12) holds for x 2 A; m 2 M . Lemma 3.7 Let M be a right WHM ....

E. Abe, Hopf Algebras, Cambridge University Press, Cambridge, 1980

....of time on these Sections, and s he can move on to Section 4.3 and, if still interested, come back later for the mathematical details. 2 Deformed Gauge Theories In order to generalize the concepts involved in discussing a deformed gauge theory, one must use the language of Hopf algebras (HAs) [12, 13, 14], as is usually the case when talking about QGs. As a result, this section will be rather abstract and mathematical. Let M be a unital associative algebra and A a Hopf algebra (both over a field k) which (left) coacts on M [6] i.e. there exists a linear algebra map A Delta : M A Omega M, ....

E. Abe, Hopf Algebras (Cambridge University Press, 1977)

....of Hopf algebras provides a natural generalization of that of groups and may ultimately allow us to generalize the previous results to the noncommutative, braided and cyclic settings. We briefly review the basic theory before stating our conservativity result. For a more complete discussion, see [1] or [45] 11.1 Definition and Categorical Structure Definition 11.1 A Hopf algebra is a vector space, H, equipped with an algebra structure, a compatible coalgebra structure and an antipode. These must satisfy equations as outlined in [45] The following chart summarizes the necessary ....

....= g Omega g ffl: H k ffl(g) 1 k Antipode S: H H S(g) g Gamma1 One obtains a Hopf algebra from a group G by taking the vector space generated by the elements of the group. The previous chart also illustrates this associated Hopf structure. Many other examples are discussed in [45] and [1]. We now discuss the representation theory of Hopf algebras. In the following definition V can either be a vector space, an object of T VEC or RT VEC. In the latter cases, we topologize the Hopf algebra discretely, and use the appropriately topologized tensor product. Definition 1 A (left) ....

K. Ab'e, Hopf Algebras, Cambridge University Press, (1977).

....bialgebras and Hopf algebras that we consider are over k.# and Hom will mean# k and Hom k . For a coalgebra C, we will use Sweedler s # notation, that is, #(c) c (2) #)#(c) c (3) etc. We will also use the Sweedler notation for left and right C comodules: #M (m) m [1] for any m in a right C comodule M,and#N (n) n [0] for any n in a left C comodule N . If V and W are two vector spaces, # : V# W W# V will denote the switch map, that is, # (v# w) w# v for all v V and w W . Let H be a Hopf algebra. For an element R H,weusethenotationR = We ....

....(2) y) for all x, y, z H. As a consequence of the above conditions we have that: BB4) #(x, 1H ) #(1H,x) #(x) for all x H. We say that a Hopf algebra H coacts weakly on a coalgebra C if there exists a k linear map c [0] such that the following conditions hold: #(c [ 1] )c [0] c C . Suppose that H coacts weakly on C,andlet# : H : c # 1 (c)# # 2 (c)beak linear map for which the following three conditions hold for all (CC) # 1 (c (1) 0] # 2 (c (2) # 2 (c (1) 0] # 2 (c (2) 2) # 2 (c (2) cocycle condition) NC) #)# I)# = ....

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E. Abe, Hopf Algebras, Cambridge University Press, Cambridge, 1977.

....formula for 0n o Un(q) To state precisely this result, we need to introduce the elements n(q) of Z(q) Sn] defined according to the parity of n by 2n(q) qi 1 [2 2i 4 1]q (12. 2n i 2n 2n i4.1 . 2n 1) i 1 1]q n l(q) qn i [2i 1]q (12 . n i 2n n i 1 . 2n 1) [2 1]q [21 2i 4. 2]q (12 . 2n i4.1 2n4.1 2n i4.2 . 2n) 2i]q (12 n i 2n4.1 n i4.1 2n) As a consequence of Proposition 5.4, we can give a factorization formula for 0n o Un(q) Corollary 5.6 For all n ) 2, On o Un(q) 1 q) 1 q2) 1 qn 1) 2(q) 3(q) Tn(q) 33) Example ....

.... type relations i t(q 1) i t(q 1) i(t) i 1(t) 2 t(q 1) i(t) i 1(t) i(t) 2 t(q 1) and as a special case of a construction of Cherednik, one can associate with the maximal permutation of the element On 1 ( 2 f13)l) n 1 fin) 2 f13)1) of H(q) C(t) where [11n 1 fXk [1]t,q L Jt,q 1 i n with [1]t,q = 1 t(q 1) It would be interesting to clarify the relation between and V(t) These elements are clearly related in the two extreme cases t = 1 and t = 1 q but we do not know if such relations still exist in the general case. 44 Deformations of the ....

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E. ABE , Hopf algebras, Cambridge University Press, 1977.

....is not only to introduce the concepts needed in the manipulations of these mathematical objects, but to also establish much of the notation which will appear throughout this work. For the interested reader, much more information about Hopf algebras and their properties is readily available in [5, 6, 7]. 2.1 Basic Definitions An algebra is a vector space A over a field k such that the algebra multiplication m : A Omega A A is a bilinear map satisfying m(a Omega (b c) m(a Omega b) m(a Omega c) m( a b) Omega c) m(a Omega c) m(b Omega c) 2.1) for all a; b; c 2 A. In ....

E. Abe, Hopf Algebras, Cambridge University Press, 1977

.... G (the symmetry algebra ) with unit e 2 G, a one dimensional representation ffl : G 7 C (the co unit ) a homomorphism Delta : G 7 G Omega G (the co product ) and an anti automorphism S : G 7 G (the antipode ) These objects obey a set of basic axioms which can be found, e.g. in [1, 47]. The Hopf algebra (G; ffl; Delta; S) is called quasi triangular if there is an invertible element R 2 G Omega G such that R Delta( Delta 0 ( R for all 2 G ; id Omega Delta) R) R 13 R 12 ; Delta Omega id) R) R 13 R 23 : Here Delta 0 ( P Delta( P , with P being ....

E.Abe, Hopf algebras, Cambridge University Press, Cambridge, 1980.

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E. Abe. Hopf algebras. Cambridge University Press, 1980.

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E. Abe, Hopf algebras, Cambridge University Press, Cambridge, 1977.

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