The local structure of twisted covariance algebras
... The fundamental problem in investigating the unitary representation theory of a separable locally compact group G is to determine its space G^ of (equivalence classes of) irreducible representations. I t is known that when G is not type I, G^, with the Mackey Borel structure, is not standard, or eve ...
... The fundamental problem in investigating the unitary representation theory of a separable locally compact group G is to determine its space G^ of (equivalence classes of) irreducible representations. I t is known that when G is not type I, G^, with the Mackey Borel structure, is not standard, or eve ...
The congruence subgroup problem
... We will now deal with k-tori. These are abelain but they need much more subtle handling than unipotent groups. One has the following: Theorem (Chevalley). C(S, G) = {1} if S is finite and G is a torus. This is false if S is infinite. However one knows the structure of T in sufficient detail to get c ...
... We will now deal with k-tori. These are abelain but they need much more subtle handling than unipotent groups. One has the following: Theorem (Chevalley). C(S, G) = {1} if S is finite and G is a torus. This is false if S is infinite. However one knows the structure of T in sufficient detail to get c ...
Determination of the Differentiably Simple Rings with a
... always assume that K is associative with a unit elementacting unitallyon the algebra. Jacobsonnoted(at least in a special case, see [16]) the followingclass of simple rings A which are not simple: A is the examples of differentiably groupring SG whereS is a simpleringof primecharacteristicp and G # ...
... always assume that K is associative with a unit elementacting unitallyon the algebra. Jacobsonnoted(at least in a special case, see [16]) the followingclass of simple rings A which are not simple: A is the examples of differentiably groupring SG whereS is a simpleringof primecharacteristicp and G # ...
LECTURE 2 1. Motivation and plans Why might one study
... It turns out that many interesting commutative rings are actually cluster algebras (i.e., there is some surface or B-matrix such that the cluster algebra associated to it, is isomorphic to the commutative ring of interest). And, in fact, one needn’t restrict to commutative rings here, as there are n ...
... It turns out that many interesting commutative rings are actually cluster algebras (i.e., there is some surface or B-matrix such that the cluster algebra associated to it, is isomorphic to the commutative ring of interest). And, in fact, one needn’t restrict to commutative rings here, as there are n ...
Full text
... Integer representations by forms are sources of a series of very interesting Diophantine equations. For instance, the cubic form x3 +y3+z3 represents 1 and 2 in an infinite number of ways, whereas only two representations (1,1,1) and (4,4, -5) are known for the number 3 and it is unknown whether the ...
... Integer representations by forms are sources of a series of very interesting Diophantine equations. For instance, the cubic form x3 +y3+z3 represents 1 and 2 in an infinite number of ways, whereas only two representations (1,1,1) and (4,4, -5) are known for the number 3 and it is unknown whether the ...
Lie Groups and Lie Algebras
... Thus, for a local transformation group, the map Φ is defined on an open subset {e} × M ⊂ V ⊂ G × M , and the conditions (2.2) are imposed wherever they make sense. Example 2.8. An obvious example is provided by the usual linear action of the general linear group GL(n, R), acting by matrix multiplica ...
... Thus, for a local transformation group, the map Φ is defined on an open subset {e} × M ⊂ V ⊂ G × M , and the conditions (2.2) are imposed wherever they make sense. Example 2.8. An obvious example is provided by the usual linear action of the general linear group GL(n, R), acting by matrix multiplica ...
Splittings of Bicommutative Hopf algebras - Mathematics
... Hence in the picture the Verschiebung coincides with the multiplication by p map (up to an isomorphism) whereas the Frobenius is an isomorphism. We now give some examples of short exact sequences of Hopf algebras which split both as algebras and as coalgebras but do not split as Hopf algebras. There ...
... Hence in the picture the Verschiebung coincides with the multiplication by p map (up to an isomorphism) whereas the Frobenius is an isomorphism. We now give some examples of short exact sequences of Hopf algebras which split both as algebras and as coalgebras but do not split as Hopf algebras. There ...
Classification of Semisimple Lie Algebras
... raised considerable doubts about the validity of the proof, since there is no way any one person could check the proof from beginning to end. Since then, considerable effort has been devoted to the simplification of the proof, and since 2004, a third–generation of this effort is underway (good refer ...
... raised considerable doubts about the validity of the proof, since there is no way any one person could check the proof from beginning to end. Since then, considerable effort has been devoted to the simplification of the proof, and since 2004, a third–generation of this effort is underway (good refer ...
Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
... Theorem 1.1 The classical Drinfeld–Sokolov reduction defined on glˆn admits a one-parameter deformation to the Hamiltonian reduction on glˆλ . As a Poisson manifold the result of the reduction coincides with the entire Poisson–Lie group of pseudodifferential operators equipped with the quadratic Gel ...
... Theorem 1.1 The classical Drinfeld–Sokolov reduction defined on glˆn admits a one-parameter deformation to the Hamiltonian reduction on glˆλ . As a Poisson manifold the result of the reduction coincides with the entire Poisson–Lie group of pseudodifferential operators equipped with the quadratic Gel ...
Universal enveloping algebras and some applications in physics
... A natural question arises: Does the converse of Proposition 6 hold? More accurately: Is it possible to canonically construct an associative algebra out of any given Lie algebra? The answer is positive and “universal”: the enveloping algebra does the job, which explains its usefulness. The commutator ...
... A natural question arises: Does the converse of Proposition 6 hold? More accurately: Is it possible to canonically construct an associative algebra out of any given Lie algebra? The answer is positive and “universal”: the enveloping algebra does the job, which explains its usefulness. The commutator ...
Hecke algebras and characters of parabolic type of finite
... standard basis element ^ corresponding to SeWj\W/Wj onto the standard basis element associated with S~1. Statement (vi) implies that this antiautomorphism is the identity on H^(W, Wj), and hence H^(W, Wj) is commutative. We next prove that (iv) and (v) imply (vi). For this purpose we use the fact th ...
... standard basis element ^ corresponding to SeWj\W/Wj onto the standard basis element associated with S~1. Statement (vi) implies that this antiautomorphism is the identity on H^(W, Wj), and hence H^(W, Wj) is commutative. We next prove that (iv) and (v) imply (vi). For this purpose we use the fact th ...
A primer of Hopf algebras
... 1.1. After the pioneer work of Connes and Kreimer1 , Hopf algebras have become an established tool in perturbative quantum field theory. The notion of Hopf algebra emerged slowly from the work of the topologists in the 1940’s dealing with the cohomology of compact Lie groups and their homogeneous sp ...
... 1.1. After the pioneer work of Connes and Kreimer1 , Hopf algebras have become an established tool in perturbative quantum field theory. The notion of Hopf algebra emerged slowly from the work of the topologists in the 1940’s dealing with the cohomology of compact Lie groups and their homogeneous sp ...
Lie algebra cohomology and Macdonald`s conjectures
... The representations of G on the spaces constructed from V also induce representations of g on these spaces. With the help of equation 1.1 it is easily verified that these g-representations are just the ones given by above formulas, with ρ substituted by dπ. The next proposition explains the definiti ...
... The representations of G on the spaces constructed from V also induce representations of g on these spaces. With the help of equation 1.1 it is easily verified that these g-representations are just the ones given by above formulas, with ρ substituted by dπ. The next proposition explains the definiti ...
here - Rutgers Physics
... boundary governed by solutions to the MC equation. (A and L are mathematical structures which play an important role in open and closed string field theory, respectively. ) ...
... boundary governed by solutions to the MC equation. (A and L are mathematical structures which play an important role in open and closed string field theory, respectively. ) ...
full text (.pdf)
... trace-based and relational Kleene algebras with tests (KAT) that arise naturally from /rst-order structures. Such algebras are de/ned in terms of a specialized class of Kripke frames called Tarskian frames. A Tarskian frame is a Kripke frame whose states are valuations of program variables and whose ...
... trace-based and relational Kleene algebras with tests (KAT) that arise naturally from /rst-order structures. Such algebras are de/ned in terms of a specialized class of Kripke frames called Tarskian frames. A Tarskian frame is a Kripke frame whose states are valuations of program variables and whose ...
Representation rings for fusion systems and
... H ≤ G, the fixed point subspace X H has mod-p homology of a sphere S n(H) . We define the dimension function of X to be the super class function DimP X : P → Z such that (DimP X)(H) = n(H) + 1 for every p-subgroup H ≤ G, over all primes dividing the order of G. We prove the following theorem. Theore ...
... H ≤ G, the fixed point subspace X H has mod-p homology of a sphere S n(H) . We define the dimension function of X to be the super class function DimP X : P → Z such that (DimP X)(H) = n(H) + 1 for every p-subgroup H ≤ G, over all primes dividing the order of G. We prove the following theorem. Theore ...
LINEAR REPRESENTATIONS OF SOLUBLE GROUPS OF FINITE
... groups has been focused on the classification of the infinite simple ones. It was independently conjectured by Cherlin and Zil’ber that these are linear algebraic groups over algebraically closed fields. This algebraicity conjecture suggests another important question, namely which groups of finite ...
... groups has been focused on the classification of the infinite simple ones. It was independently conjectured by Cherlin and Zil’ber that these are linear algebraic groups over algebraically closed fields. This algebraicity conjecture suggests another important question, namely which groups of finite ...
Cohomology of Categorical Self-Distributivity
... 3.1. Self-Distributive Maps for Coalgebras Constructed From Racks In this section we note that quandles and racks can be used to construct selfdistributive maps in CoComCoalg simply by using their elements as basis. Let X be a rack. Let V = kX be the vector space over a field k with the elements of ...
... 3.1. Self-Distributive Maps for Coalgebras Constructed From Racks In this section we note that quandles and racks can be used to construct selfdistributive maps in CoComCoalg simply by using their elements as basis. Let X be a rack. Let V = kX be the vector space over a field k with the elements of ...
GENERALIZED CAYLEY`S Ω-PROCESS 1. Introduction We assume
... If V is a finite dimensional k–space, we say that an action M × V → V is polynomial if the associated map M → Endk (V ) is a morphism of affine algebraic monoids, in particular in the case that M is endowed with 0 ∈ M , in accordance with our definitions 0 · v = v for all v ∈ V . The action map M × ...
... If V is a finite dimensional k–space, we say that an action M × V → V is polynomial if the associated map M → Endk (V ) is a morphism of affine algebraic monoids, in particular in the case that M is endowed with 0 ∈ M , in accordance with our definitions 0 · v = v for all v ∈ V . The action map M × ...
Affine Hecke Algebra Modules i
... For α ∈ Zn , define X α := X1α1 · · · Xnαn . Our goal for this section is to prove the following theorem. Theorem 3.1. The set B := {X α Tw : α ∈ Zn , w ∈ Sn } is an F -basis for HF,n . Proof. The above lemma shows us that F B is invariant under left multiplication by the generators Ti , Xi and is h ...
... For α ∈ Zn , define X α := X1α1 · · · Xnαn . Our goal for this section is to prove the following theorem. Theorem 3.1. The set B := {X α Tw : α ∈ Zn , w ∈ Sn } is an F -basis for HF,n . Proof. The above lemma shows us that F B is invariant under left multiplication by the generators Ti , Xi and is h ...
Projective p-adic representations of the K-rational geometric fundamental group (with G. Frey).
... Néron-Ogg-Shafarevich: A has to have good reduction everywhere, i.e. at all points of C. A second condition is that the fixed fields of (finite quotients of) the kernels of the ρ̃Vi ’s are regular extensions of F . Note that by using group theoretical methods it not difficult to find examples of ab ...
... Néron-Ogg-Shafarevich: A has to have good reduction everywhere, i.e. at all points of C. A second condition is that the fixed fields of (finite quotients of) the kernels of the ρ̃Vi ’s are regular extensions of F . Note that by using group theoretical methods it not difficult to find examples of ab ...
Lecture notes up to 08 Mar 2017
... space equipped with an inner product). Let O(E) ⊂ GL(E) be the subgroup of orthogonal transformations, and let S(E) ⊂ E be the subset of unit-length vectors. Then O(E) is a locally compact topological group, S(E) is a locally compact topological space, and we have the natural action of O(E) on S(E). ...
... space equipped with an inner product). Let O(E) ⊂ GL(E) be the subgroup of orthogonal transformations, and let S(E) ⊂ E be the subset of unit-length vectors. Then O(E) is a locally compact topological group, S(E) is a locally compact topological space, and we have the natural action of O(E) on S(E). ...
Lie groups - IME-USP
... (i) The left-invariant vector fields on Rn are precisely the constant vector fields, namely, the linear combinations of coordinate vector fields (in the canonical coordinate system) with constant coefficients. The bracket of two constant vector fields on Rn is zero. It follows that the Lie algebra o ...
... (i) The left-invariant vector fields on Rn are precisely the constant vector fields, namely, the linear combinations of coordinate vector fields (in the canonical coordinate system) with constant coefficients. The bracket of two constant vector fields on Rn is zero. It follows that the Lie algebra o ...
Third symmetric power L-functions for GL(2)
... of type G2 over a non-archimedean field F of characteristic zero. Let P be the parabolic subgroup of G generated by the long root of G. Write G = MN. Let A be the center of M. Then the short root a of G may be identified as the unique simple root of A in the Lie algebra of N. If e is half the sum of ...
... of type G2 over a non-archimedean field F of characteristic zero. Let P be the parabolic subgroup of G generated by the long root of G. Write G = MN. Let A be the center of M. Then the short root a of G may be identified as the unique simple root of A in the Lie algebra of N. If e is half the sum of ...