Quillen, D., On the associated graded ring of a group ring, J. Algebra, vol. 10 (  Home/Search   Document Not in Database   Summary   Related Articles   Check

....) P i 1 P d i k=1 i i;k , and all the basis elements of degree i form a basis of I i =I i 1 ( 22] This theorem gives a close connection between restricted Lie algebras and nite p groups. It is an important relation. It was generalized by Quillen to all groups not just p groups, see [33]. 2.4 Polynomial Group Laws Its not hard to see that any group G of order p n is isomorphic to a group of the form (IF n p ; where : IF n p IF n p IF n p is the group law and i th component i (a; b) is given by a polynomial in the coordinates (a 1 ; a n ; b 1 ; ....

D. Quillen. On the associated graded ring of a group ring. J. Alg., 10:411-418, 1968.

....= P i 1 P d i k=1 i i;k , and all the basis elements of degree i form a basis of I i =I i 1 ( 25] This theorem gives a close connection between restricted Lie algebras and nite p groups. It is an important relation. It was generalized by Quillen to all groups not just p groups, see [36]. 2.4 Polynomial Group Laws It is not hard to see that any group G of order p n is isomorphic to a group of the form (IF n p ; where the group law : IF n p IF n p IF n p is a polynomial function. That is, the i th component function i (a; b) is given by a polynomial in ....

D. Quillen. On the associated graded ring of a group ring. J. Algebra, 10:411{ 418, 1968.

....for more details) If = Q , consider the following graded Lie algebras over : L(G) # M n=1 G n G n 1# Z Q , L(G) # M n=1 # n (G) # n 1 (G)# Z Q . If = F p , consider the restricted Lie F p algebra L p (G) # M n=1 G n G n 1 . Then Quillen s Theorem [Qui68] asserts that A(G) is the universal enveloping algebra of L(G) in characteristic 0 and is the universal p enveloping algebra of L p (G) in positive characteristic. Let us introduce the following numbers: a n (G) dim (# n # n 1 ) b n (G) rank(G n G n 1 ) Here by the rank of ....

Daniel G. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411--418.

....where one considers the cohomology with the constant coefficients Z=l or Q , respectively. In the second case the group Gamma should be better finitely generated. One can reduce the last two statements to the one about the cohomology of an augmented algebra using the results of Quillen s paper [11], or prove all the three assertions independently just in the same way as it was done for coalgebras above. 5. Morphisms of Graded Algebras and Koszulity The results of this and the next sections were found in an attempt to generalize and clarify some of the statements from the paper of J. ....

D. Quillen. On the associated graded ring of a group ring. Journ. Algebra 10, #4, p. 411--418, 1968.

....elements in the dual coalgebra, so k n is equal to the dimension of the subspace of primitive elements in An (P k ) Now, denote by Gfng the abelian group fl n G=fl n 1 G. The graded group L Gfng is a Lie algebra with the Lie bracket induced by the group commutator. A theorem of Quillen [16] says that the algebra L A Q n (G) is the universal enveloping algebra of of the Lie algebra L Gfng Omega Q. However, the space of primitive elements of a universal enveloping algebra is naturally isomorphic to the original Lie algebra. This means that k n is equal to the rank of the ....

D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411--418.

....(n) This follows from the fact that g m Gamma 1 2 (IG) n for some m, and from the formula: g m Gamma 1 = P m i=1 i m i j (g Gamma 1) i . The point of considering the rational FINITE TYPE INVARIANTS, THE MAPPING CLASS GROUP AND BLINKS 15 closures is the theorem of Jennings (see [Qu]) which says that the converse is true: if g Gamma 1 2 (IG) n , then g 2 G (n) This proves Remark 1.7 (b) 2.2. Proof of proposition 1.4. Let h be as in the statement of proposition 1.4. A Mayer Vietoris argument shows that H 1 (M h ) H 1 ( Sigma) L Gamma h (L ) Thus it suffices ....

D. Quillen, On the associated graded ring of a group ring, Journal of Algebra 10 (1968) 411-418.

....d 0 ) ae J 2 F and thus d 1 is well defined. Moreover, its image is precisely D J 2 F , and (ii) follows from this. We now relate the computed modules JG=J 2 G ; J 2 G =J 3 G with the sought ones Gamma 1 = Gamma 2 , Gamma 2 = Gamma 3 G Omega R applying a theorem by D. Quillen ([Q2]) Theorem. Let G be a group, k a field of characteristic zero, kG the group algebra and j : Phi Gamma n= Gamma n 1G Omega k PhiJ n G =J n 1 G given by g 7 g Gamma 1 over the homogeneous components. Then j induces an isomorphism of algebras U( Phi Gamma n = Gamma n 1G Omega R) ....

D. Quillen, On the Associated Graded Ring of a Group Ring, J. Algebra 10 (1968) 411-418.

....Let GQG be the associated graded algebra, i.e. G n QG = I n =I n 1 . Note that QG is a Hopf algebra with comultiplication defined by Delta(g) g Omega g for g 2 G. Then the maps of equation (24) induce a map: U(GG Omega Q) GQG (25) This map was shown by Jennings (see Quillen [Qu]) to be a Hopf algebra isomorphism. In particular, the primitive elements of GQG are isomorphic to the Lie algebra GG Omega Q. We end the section with the following lemma: Lemma 2.8. For x i 2 G we have the following identity in the graded quotient I n =I n 1 : 1 Gamma [x 1 ; x ....

....5 The case of m = 1. We now describe explicitly the map D L Sigma ;1 . Assume that we are given an admissible Heegaard genus g surface f . Recall first that GmT g Omega Q is a finite dimensional, stable with respect to the genus, representation of Sp(H) It follows by a theorem of Quillen [Qu] (see also [Ha1] that it is a rational representation of Sp(HQ ) It is a very interesting question to analyze the structure of the above representation. Motivated by the above question Morita [Mo5] developed a theory of higher Johnson homomorphisms, known to form a Lie algebra. The structure of ....

D. Quillen, On the associated graded ring of a group ring, Journal of Algebra 10 (1968) 411--418.

....1; n) modulo the relations (N) and 2[g] 0; if n = 1; N) R) S) and (T 0 ) if n 2: For general groups G, the consecutive factors J m (G) J m 1 (G) of the filtration : J m (G) J m 1 (G) have been studied much more intensively than the filtration itself. In [7], the consecutive factors have been calculated when the ground ring is a field. In [9] a general homological procedure for calculating these factors with integral coefficients has been proposed. Finally, in the case of an elementary abelian p group G = G(m) of rank m, these factors are completely ....

Quillen D., On the associated graded ring of a group ring, J. Algebra 10 (1968), 411--418.

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Quillen, D., On the associated graded ring of a group ring, J. Algebra, vol. 10 (

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D. Quillen. On the associated graded ring of a group ring. J. Algebra, 10:411{ 418, 1968.

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D. Quillen. On the associated graded ring of a group ring. J. Algebra, 10:411{ 418, 1968.

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Quillen, D., On the associated graded ring of a group ring, J. Algebra, vol. 10 (

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D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411-418.

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D. G. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411--418.

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D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411418. MR 38 #245

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D. Quillen, On the associated graded ring of a group ring, Journal of Algebra 10 (1968) 411--418.

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D. Quillen, On the associated graded ring of a group ring, Journal of Algebra 10 (1968) 411--418.

No context found.

D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411418. MR 38 #245

No context found.

D. Quillen, On the Associated Graded Ring of a Group Ring, Journal of Algebra 10, 1968, 411-418.

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