A conjecture on the Hall topology for the free group - LaCIM
... A, and by F(A) the free group over A. The identity of F(A) is denoted by 1. The Hall (or profinite) topology for the free group was introduced by M. Hall in [6]. It is the coarsest topology on F(A) such that every group morphism from F(A) onto a finite discrete group is continuous. That is, a sequen ...
... A, and by F(A) the free group over A. The identity of F(A) is denoted by 1. The Hall (or profinite) topology for the free group was introduced by M. Hall in [6]. It is the coarsest topology on F(A) such that every group morphism from F(A) onto a finite discrete group is continuous. That is, a sequen ...
solutions to HW#8
... (c) Show that HxK and HyK are either the same set or are disjoint for all x, y ∈ G. Show that the set of HK double cosets partitions G. Every element g ∈ G must belong to an HK double coset because 1 ∈ H and 1 ∈ K, hence g = 1g1 ∈ HgK. Therefore the union of the HK double cosets is G. It remains to ...
... (c) Show that HxK and HyK are either the same set or are disjoint for all x, y ∈ G. Show that the set of HK double cosets partitions G. Every element g ∈ G must belong to an HK double coset because 1 ∈ H and 1 ∈ K, hence g = 1g1 ∈ HgK. Therefore the union of the HK double cosets is G. It remains to ...
The number of conjugacy classes of elements of the Cremona group
... Proof. — Let us denote by g the automorphism of odd order of the conic bundle induced by π : S → P1 . Recall that the action of g on the fibres of π induces an automorphism g of P1 of odd order m, whose orbits have all the same size m, except for two fixed points. Suppose that one fibre F of π is si ...
... Proof. — Let us denote by g the automorphism of odd order of the conic bundle induced by π : S → P1 . Recall that the action of g on the fibres of π induces an automorphism g of P1 of odd order m, whose orbits have all the same size m, except for two fixed points. Suppose that one fibre F of π is si ...
DIALGEBRAS Jean-Louis LODAY There is a notion of
... appears naturally in the computation of the Leibniz homology of dialgebras of matrices (cf. [F1]). (Co)homology of dialgebras with non-trivial coefficients is treated by Alessandra Frabetti in [F4]. Section 4 is devoted to the relationship between Leibniz algebras and dialgebras. The functor which a ...
... appears naturally in the computation of the Leibniz homology of dialgebras of matrices (cf. [F1]). (Co)homology of dialgebras with non-trivial coefficients is treated by Alessandra Frabetti in [F4]. Section 4 is devoted to the relationship between Leibniz algebras and dialgebras. The functor which a ...
ENDOMORPHISMS OF ELLIPTIC CURVES 0.1. Endomorphisms
... 1.1. Rational and complex representations. The fact that every complex elliptic curve admits a uniformization C/Λ is very helpful in analyzing the endomorphism ring. Indeed, an endomorphism of E is given by a C-linear map α – i.e., multiplication by some nonzero complex number – such that α(Λ) ⊂ Λ. ...
... 1.1. Rational and complex representations. The fact that every complex elliptic curve admits a uniformization C/Λ is very helpful in analyzing the endomorphism ring. Indeed, an endomorphism of E is given by a C-linear map α – i.e., multiplication by some nonzero complex number – such that α(Λ) ⊂ Λ. ...
A non-archimedean Ax-Lindemann theorem - IMJ-PRG
... and H(g −1 ) H(g)c for every g ∈ SL(d, Q). When d = 2, one even has H(g −1 ) = H(g). By abuse of language, if G is a linear algebraic Q-group, we implicitely choose an embedding in some linear group, which furnishes a height function on G(Q). The actual choice of this height function depends on th ...
... and H(g −1 ) H(g)c for every g ∈ SL(d, Q). When d = 2, one even has H(g −1 ) = H(g). By abuse of language, if G is a linear algebraic Q-group, we implicitely choose an embedding in some linear group, which furnishes a height function on G(Q). The actual choice of this height function depends on th ...
Topological modules over strictly minimal topological
... Comment.Math.Univ.Carolinae 44,3 (2003)461–467 ...
... Comment.Math.Univ.Carolinae 44,3 (2003)461–467 ...
11-4 PPT
... equivalent. If so, tell what property is applied. If not, explain why. 15 + ( 5 + 8) and (15 + 5) + 8 To determine if expressions are equal, perform the operations using the order of operations, then compare the answers. ...
... equivalent. If so, tell what property is applied. If not, explain why. 15 + ( 5 + 8) and (15 + 5) + 8 To determine if expressions are equal, perform the operations using the order of operations, then compare the answers. ...
Whitney forms of higher degree
... The forms we (resp., wf , wv ) are indexed over the set of these couples (resp., triplets, quadruplets), thus we use e (resp., f , v) also as a label since it points to the same object in both cases. When a metric (i.e., a scalar product) is introduced on the ambient affine space, differential forms ar ...
... The forms we (resp., wf , wv ) are indexed over the set of these couples (resp., triplets, quadruplets), thus we use e (resp., f , v) also as a label since it points to the same object in both cases. When a metric (i.e., a scalar product) is introduced on the ambient affine space, differential forms ar ...
INFINITESIMAL BIALGEBRAS, PRE
... Examples 2.2. Let (A, µ, ∆) be an ǫ-bialgebra. (1) A itself is an ǫ-Hopf module via µ and ∆, precisely by definition of ǫ-bialgebra. (2) More generally, for any space V , A⊗V is an ǫ-Hopf module via µ⊗id : A⊗A⊗V → A⊗V and ∆⊗id : A⊗V → A⊗A⊗V . (3) A more interesting example follows. Assume that the c ...
... Examples 2.2. Let (A, µ, ∆) be an ǫ-bialgebra. (1) A itself is an ǫ-Hopf module via µ and ∆, precisely by definition of ǫ-bialgebra. (2) More generally, for any space V , A⊗V is an ǫ-Hopf module via µ⊗id : A⊗A⊗V → A⊗V and ∆⊗id : A⊗V → A⊗A⊗V . (3) A more interesting example follows. Assume that the c ...
Abelian Varieties - Harvard Math Department
... a symplectic pairing H1 (X, Z) × H1 (X, Z) → Z such that the corresponding form on H 0 (X, ΩX )∗ is positive definite. (This pairing is given by the intersection pairing). We will later see that the addition law on J(X) is a morphism of projective varieties. An abelian variety over C is a projective ...
... a symplectic pairing H1 (X, Z) × H1 (X, Z) → Z such that the corresponding form on H 0 (X, ΩX )∗ is positive definite. (This pairing is given by the intersection pairing). We will later see that the addition law on J(X) is a morphism of projective varieties. An abelian variety over C is a projective ...
Lie groups - IME-USP
... Let G be a Lie group. A Lie subgroup of G is an immersed submanifold (H, ϕ) of G such that H is a Lie group and ϕ : H → G is a homomorphism. 3.3.3 Remark Similarly as in the case of immersed submanifolds (Problem 19 in Chapter 1), we consider two Lie subgroups (H1 , ϕ1 ) and (H2 , ϕ2 ) of G equivale ...
... Let G be a Lie group. A Lie subgroup of G is an immersed submanifold (H, ϕ) of G such that H is a Lie group and ϕ : H → G is a homomorphism. 3.3.3 Remark Similarly as in the case of immersed submanifolds (Problem 19 in Chapter 1), we consider two Lie subgroups (H1 , ϕ1 ) and (H2 , ϕ2 ) of G equivale ...
EQUIVALENCE CLOSURE IN THE TWO
... 2.1. Logics, structures and types. We employ standard terminology and notation from model theory throughout this paper (see, e.g., [3]). In particular, we refer to structures using Gothic capital letters, and their domains using the corresponding Roman capitals. We denote by GF2 the guarded two-vari ...
... 2.1. Logics, structures and types. We employ standard terminology and notation from model theory throughout this paper (see, e.g., [3]). In particular, we refer to structures using Gothic capital letters, and their domains using the corresponding Roman capitals. We denote by GF2 the guarded two-vari ...
On Schwartz groups - Instytut Matematyczny PAN
... We start Section 3 by defining Schwartz groups. Then we obtain permanence properties for this class with respect to subgroups, Hausdorff quotients, products and local isomorphisms, and show that bounded subsets of locally quasi-convex Schwartz groups are precompact. We finish this section by proving ...
... We start Section 3 by defining Schwartz groups. Then we obtain permanence properties for this class with respect to subgroups, Hausdorff quotients, products and local isomorphisms, and show that bounded subsets of locally quasi-convex Schwartz groups are precompact. We finish this section by proving ...
UNDECIDABILITY OF LINEAR INEQUALITIES IN GRAPH
... holds for all graphs G. The positive semi-definiteness characterization shows that it suffices to verify the validity of α1 f (H1 ) + . . . + αk f (Hk ) ≥ 0, for all graph parameters f satisfying Conditions (i) and (ii). Note that if the F × F matrix in Condition (ii) was finite, then there would exist ...
... holds for all graphs G. The positive semi-definiteness characterization shows that it suffices to verify the validity of α1 f (H1 ) + . . . + αk f (Hk ) ≥ 0, for all graph parameters f satisfying Conditions (i) and (ii). Note that if the F × F matrix in Condition (ii) was finite, then there would exist ...