S. Montgomery, \Hopf Algebras and their actions on rings," 82, Regional Conference Series in Mathematics, AMS, Providenc, RI, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a polynomial can have nonnegative coefficients as an ab polynomial but not as a cd polynomial (for example, a b = c d) An important tool in studying the cd index is that the cd index is a coalgebra homomorphism. We give a short explanation here; for basic notions of coalgebras, see [24,32]. For more information on the coalgebra discussed here, we refer the reader to [17] We extend the ring d# to a coalgebra; that is, we enrich the ring with a coproduct #, which is a linear map # : d# #Z#c,d## d#. We will use the Sweedler notation for the coproduct; hence for the element ....

Montgomery, S.: Hopf Algebras and Their Actions on Rings (CBMS, Regional Conference Series in Mathematics, Number 82) Providence: American Mathematical Society 1993

....extensions. We provide a combinatorial description for the Hopf kernel of the map SSym QSym. These results are expanded on and proven in the manuscript [2] of the same name. For a background on quasi symmetric functions, see [22, x7.19] for Hopf Algebras, we recommend the book of Montgomery [18]. We also recommend the papers [20] of Poirier and Reutenauer and of [7] of Duchamp, Thibon, and Hivert, who studied this same Hopf algebra of permutations from a di erent perspective, the latter under the name free quasi symmetric functions . We thank Swapneel Mahajan, Nantel Bergeron, and the ....

Susan Montgomery, Hopf algebras and their actions on rings, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1993. MR 94i:16019

....a simpli cation to the proof of Theorem 3.1, Nantel Bergeron, one of whose questions motivated the results of Section 8, and the referees of an abridged version for helpful comments. 1. Basic definitions and results We use only elementary properties of Hopf algebras, as given in the book [26]. Our Hopf algebras H will be graded connected Hopf algebras over . Thus the Q algebra H is the direct sum L fH n j n = 0; 1; g of its homogeneous components H n , with H 0 = the product and coproduct respect the grading, and the counit is projection onto H 0 . Throughout, n is a ....

Susan Montgomery, Hopf algebras and their actions on rings, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1993. MR 94i:16019

....considered recently by R. Wood [31] as a framework for integral lifts of Steenrod operations. N. Ray [25] showed that the algebra of differential operators is isomorphic to the Landweber Novikov algebra in complex cobordism. Here, we postpone any reference to topology to x3. We refer the reader to [22] for all information concerning Hopf algebras. All rings and algebras we consider in this paper are assumed graded by complex dimension, so that products commute without signs. We start by recalling some concepts and notation from [31] The Weyl algebra W is the associative algebra with unit ....

S. Montgomery. Hopf Algebras and Their Action on Rings, volume 82 of Regional Conf. Ser. in Math. Amer. Math. Soc., Providence, RI, 1993.

....weak order on the symmetric group, with the label of a cover ul v the integer i where (i; i 1) vu . Then R( 1; w] is the set of reduced decompositions of w. The expressions relating f I (P ) to d I (P ) and F I;n to M ff give f I(ff) P )M ff : 3) 3. Incidence Hopf algebras. See [9, 13] for more on Hopf algebras. Let P be a class of graded posets closed under taking subintervals and products. The (reduced) incidence coalgebra [7, 10] IP of P is the graded free abelian group generated by isomorphism classes of posets in P with grading induced by the rank of a poset and coproduct ....

S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS 82, AMS, 1993.

....are: I. Hopf algebras and braided tensor categories II. Lie algebras and enveloping algebras III. Deformations of Hopf algebras IV. Perverse Sheaves The following book contains a very nice up to date account of the theory of Hopf algebras, and it also includes some useful things on quantum groups. [Mo] S. Montgomery, Hopf Algebras and their Actions on Rings , Regional Conference Series in Mathematics 82, American Mathematical Society, 1992. The book by Chari and Pressley [CP] contains a nice introduction to monoidal categories and braided monoidal categories. The following little book is a ....

....very helpful proofreading. Let k be a field. Unless otherwise specified all maps between vector spaces over k are assumed to be k linear and, if V is a vector space over k, then id V : V V denotes the identity map from V to V . The proofs of most of the statements in this chapter can be found in [Mo]. The proof that the antipode is an antihomomorphism (2.1) is given in [Sw] 4.0.1. The statement of Theorem (5.3) giving the construction of the quantum double, is given explicitly in [D1] x13, and the proof can be found in [Ma] p. 287 289. A statement similar to Proposition (5.5) is in [Ta] ....

S. Montgomery, "Hopf Algebras and their Actions on Rings", Regional Conference Series in Mathematics 82, American Mathematical Society, 1992.

....all h 2 H and r; s 2 R. Given an invertible map oe : H Omega H R, define R# oe H to be the algebra with the additive structure of R Omega H and multiplication (r Omega h) s Omega k) X r(h 1 Delta s)oe(h 2 ; k 1 ) Omega h 3 k 2 (1) for all h; k 2 H and all r; s 2 R. It is known (see [6]) that the crossed product is associative with identity 1 Omega 1 if and only if oe is a normalized Hopf cocycle and R satisfies a twisted H module condition. We interpret the generalities above in the specific case H = u(L) The comultiplication in u(L) is of course given by Delta(x) x Omega ....

Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series, vol. 82, American Mathematical Society, 1993.

....If H is nite dimensional, then this is equivalent to A being an H module and R the algebra of invariants for H . The notation E = R; A) will be used for H extensions. A general reference for all unproved assertions about Hopf algebras which cannot be found in [MSch] is the book [Mon93]. PRIMITIVE IDEALS IN HOPF ALGEBRA EXTENSIONS 3 2. Faithfully flat Galois extensions In this section we consider the fundamentals of faithfully at Galois extensions. The results in this section allow for great simpli cation of proofs later on, and will be used extensively. The H extension E = ....

S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, 1993.

....not divide dim P . Then the trace map tr P= splits via oe : End (P ) k 7 (dim P ) Gamma1 k Delta Id P . Therefore, is isomorphic to a direct summand of End (P ) Inasmuch as End (P ) P Omega P is projective, by the Fundamental Theorem of Hopf Modules ([Mo], Theorem 1.9.4. we conclude that is projective as well. This forces H to be semisimple, contrary to our assumption. Remarks. The semisimplicity hypothesis in (a) is definitely necessary. Indeed, the restricted enveloping algebra H = u(g) of the restricted p Lie algebra g = sl(2; ....

....subalgebras L of H such that the augmentation ideal L = Ker L is nilpotent. 8 MARTIN LORENZ Lemma. Let L be a local Hopf subalgebra of H. Then dim L divides the dimension of every projective H module. Proof. Since H is free as (left and right) L module, by the Nichols Zoeller Theorem ([Mo], Theorem 3.1.5) every projective H module is projective, and hence free, over L. Remarks and Examples. If H = G is the group algebra of a finite group G then the local Hopf subalgebras of H are the group subalgebras P , where P is a p subgroup of G (0 subgroup means h1i) The local ....

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S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Series in Math. No. 82, Amer. Math. Soc., Providence, 1993.

....or upper intervals. 5.1 Duality and the associated coalgebras We discuss first the connection between the algebras studied here to the coalgebras of [11] We will see that the algebra A can be viewed as the graded dual to the coalgebra Qha; bi. For the basic definitions of coalgebras we refer to [16] or [21] That A appeared to be the dual of some coalgebra was first pointed out to us by Moss Sweedler. The discussion in this subsection was suggested to us by Richard Ehrenborg and is included here for completeness. Let C be a coalgebra with coassociative coproduct Delta : C Gamma C Omega ....

S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, Number 82, American Mathematical Society, Providence, RI, 1993. 15

....we shall consider a K algebra H such that (H1) H is a K affine K Hopf algebra; H2) H satisfies a polynomial identity; H3) H has finite injective dimension as a (right and left) H module; H4) H is (right and left) Noetherian. Our standard references for Hopf algebraic results will be [41], 54] We ll denote the comultiplication, counit and antipode of a Hopf algebra by Delta, ffl and S respectively, and follow the convention of writing Delta(h) P h 1 Omega h 2 . The antipode will be assumed to be a bijective map. Note in passing that the existence of a bijective ....

S. Montgomery. Hopf Algebras and their Actions on Rings, volume 82 of CBMS Regional Conf. Series. Amer.Math.Soc., Providence, R.I., 1993.

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S. Montgomery, \Hopf Algebras and their actions on rings," 82, Regional Conference Series in Mathematics, AMS, Providenc, RI, 1993.

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S. Montgomery, \Hopf Algebras and their actions on rings," 82, Regional Conference Series in Mathematics, AMS, Providenc, RI, 1993.

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Montgomery S., Hopf algebras and their actions on rings, Regional Conference Series in Mathematics, V.82, Amer. Math. Soc., Providence, Rhode Island, 1993.

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S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conf. Series in Math. 82, A.M.S., 1993.

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S. Montgomery, "Hopf algebras and their actions on rings," CBMS number 82, AMS, 1993.

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S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conf. Series in Math. 82, A.M.S., 1993.

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S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conf. Series in Math. 82, A.M.S., 1993.

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S. Montgomery, Hopf algebras and their action on rings, CBMS Lecture Notes 82 (1993), Amer. Math. Soc..

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Susan Montgomery. Hopf Algebras and Their Actions on Rings, volume 82 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1993.

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S. Montgomery, Hopf algebras and their actions on rings, American Mathematical Society, Providence, 1993.

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S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series, no. 82, American Mathematical Society, Providence, R. I., 1993.

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S. Montgomery, Hopf algebras and their actions on rings, AMS (1993), CBMS 82.

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S. Montgomery. Hopf algebras and their actions on rings. Number 82 in C. B. M. S. American Mathematical Society. Providence, Rhode Island, 1993.

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Susan Montgomery, Hopf Algebras and their Action on Rings, Regional Conference Series in Mathematics 82, Amer. Math. Soc., Providence, Rhode Island 1993.

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