Constructing quantales and their modules from monoidal
... P(M) arose in Lambek’s [10] formal language theory and Girard’s [6] linear logic. The left P(M)-modules P(X) can be found in the labeled transition systems of Ambra.msky and Vickers [1], but there the a.ction was induced by a ...
... P(M) arose in Lambek’s [10] formal language theory and Girard’s [6] linear logic. The left P(M)-modules P(X) can be found in the labeled transition systems of Ambra.msky and Vickers [1], but there the a.ction was induced by a ...
http://www.math.cornell.edu/~irena/papers/ci.pdf
... elements. In section 1.3, we outline our main results. Throughout the book, we use the notation introduced in section 1.4. ...
... elements. In section 1.3, we outline our main results. Throughout the book, we use the notation introduced in section 1.4. ...
TRACES IN SYMMETRIC MONOIDAL CATEGORIES Contents
... Thus, we can’t expect to have a fixed-point operator acting on the whole category, since there is no way to specify a fixed point for a map which has no fixed points. Instead, we would like to know, given a map, does it have any fixed points, and if so, how many and what are they? Thus, we need an o ...
... Thus, we can’t expect to have a fixed-point operator acting on the whole category, since there is no way to specify a fixed point for a map which has no fixed points. Instead, we would like to know, given a map, does it have any fixed points, and if so, how many and what are they? Thus, we need an o ...
lecture notes
... Whenever a binary operation ∗ has an identity e, we can introduce the notion of inverse element. The definition is as follows: Definition 1.1.3. Given a binary operation ∗ with identity e, an element x ∈ X is called invertible if there exists y ∈ X such that x ∗ y = y ∗ x = e. In this case, we call ...
... Whenever a binary operation ∗ has an identity e, we can introduce the notion of inverse element. The definition is as follows: Definition 1.1.3. Given a binary operation ∗ with identity e, an element x ∈ X is called invertible if there exists y ∈ X such that x ∗ y = y ∗ x = e. In this case, we call ...
o deliteljima nule, invertibilnosti i rangu matrica nad komutativnim
... over commutative semirings. For such matrices, there already exists a number of rank functions, generalizing the rank function for matrices over fields. In this thesis, a new rank function is proposed, which is based on the permanent, which is possible to define for semirings, unlike the determinan ...
... over commutative semirings. For such matrices, there already exists a number of rank functions, generalizing the rank function for matrices over fields. In this thesis, a new rank function is proposed, which is based on the permanent, which is possible to define for semirings, unlike the determinan ...
Families of ordinary abelian varieties
... despite the best effort; this paper is no exception. As a feeble substitute for a proof, we offer some evidence for the conjectures. We prove in 8.6, using the main theorem of de Jong in [10], that every Tate-linear subvariety of the ordinary locus of a Hilbert modular variety, attached to a totally ...
... despite the best effort; this paper is no exception. As a feeble substitute for a proof, we offer some evidence for the conjectures. We prove in 8.6, using the main theorem of de Jong in [10], that every Tate-linear subvariety of the ordinary locus of a Hilbert modular variety, attached to a totally ...
Lecture Notes for Math 614, Fall, 2015
... equations? No algorithm is known for settling questions of this sort, and many are open, even for relatively small specific examples. In the example considered here, it turns out that 3 equations are needed. I do not know an elementary proof of this fact — perhaps you can find one! One of the themes ...
... equations? No algorithm is known for settling questions of this sort, and many are open, even for relatively small specific examples. In the example considered here, it turns out that 3 equations are needed. I do not know an elementary proof of this fact — perhaps you can find one! One of the themes ...
EQUIVARIANT SYMMETRIC MONOIDAL STRUCTURES 1
... Much of this formalism arose from our attempt to understand the transfer. One approach in the finite group case is via the Wurtmuller map: G+ ∧H X −→ FH (G+ , X) for any H-spectrum X. This is a natural map arising from the the fact that G+ ∧H − is left adjoint to the forgetful functor from G-spectra ...
... Much of this formalism arose from our attempt to understand the transfer. One approach in the finite group case is via the Wurtmuller map: G+ ∧H X −→ FH (G+ , X) for any H-spectrum X. This is a natural map arising from the the fact that G+ ∧H − is left adjoint to the forgetful functor from G-spectra ...
MATH 113: PRACTICE FINAL SOLUTIONS Note: The final is in
... Similarly, F + G is bilinear if F and G are because (F + G)♯w = Fw♯ + G♯w , Fw♯ and G♯w are linear, with analogous formulae for (F + G)♭v . Now, if U is any set (and Z is a vector space as above), the set M(U ; Z) of all maps from U to Z is a vector space with the usual addition and multiplication b ...
... Similarly, F + G is bilinear if F and G are because (F + G)♯w = Fw♯ + G♯w , Fw♯ and G♯w are linear, with analogous formulae for (F + G)♭v . Now, if U is any set (and Z is a vector space as above), the set M(U ; Z) of all maps from U to Z is a vector space with the usual addition and multiplication b ...
REPRESENTATION THEORY Tammo tom Dieck
... with the properties v(hg) = (vh)g and ve = v, and K-linear right translations rg : v 7→ vg. These will be called right representations as opposed to left representations defined above. The map rg : v 7→ vg is then the right translation by g. Note that now rg ◦ rh = rhg (contravariance). If V is a ri ...
... with the properties v(hg) = (vh)g and ve = v, and K-linear right translations rg : v 7→ vg. These will be called right representations as opposed to left representations defined above. The map rg : v 7→ vg is then the right translation by g. Note that now rg ◦ rh = rhg (contravariance). If V is a ri ...
n-ARY LIE AND ASSOCIATIVE ALGEBRAS Peter W. Michor
... of [7]. Related (but not graded) concepts are discussed in [4] in terms of operads and their Koszul duality. The recent preprints [2] and [5] propose dynamical models which correspond to the not graded case with even n in our construction. 2. Review of binary algebras and bimodules In this section w ...
... of [7]. Related (but not graded) concepts are discussed in [4] in terms of operads and their Koszul duality. The recent preprints [2] and [5] propose dynamical models which correspond to the not graded case with even n in our construction. 2. Review of binary algebras and bimodules In this section w ...
DRINFELD ASSOCIATORS, BRAID GROUPS AND EXPLICIT
... the last relation taking place in the group exp(t̂k4 ), where tk4 is the k-Lie algebra with generators tij , 1 ≤ i 6= j ≤ 4 and relations tji = tij and [tij , tik + tjk ] = [tij , tkl ] = 0 for i, j, k, l distinct; t̂k4 is its degree completion, where the generators tij have degree 1; and if a is a ...
... the last relation taking place in the group exp(t̂k4 ), where tk4 is the k-Lie algebra with generators tij , 1 ≤ i 6= j ≤ 4 and relations tji = tij and [tij , tik + tjk ] = [tij , tkl ] = 0 for i, j, k, l distinct; t̂k4 is its degree completion, where the generators tij have degree 1; and if a is a ...
Algebraic K-Theory and Topological Spaces
... Let X be an admissible space, and let A = C(X). The algebra A is a Fréchet algebra. Indeed, if K0 ⊂ K1 ⊂ K2 ⊂ . . . is a countable sequence of compact subsets such that X = ∪K i , define a countable family of multiplicative semi-norms p i by pi (f ) = sup{|f (x)| : x ∈ Ki }. The collection {pi } of ...
... Let X be an admissible space, and let A = C(X). The algebra A is a Fréchet algebra. Indeed, if K0 ⊂ K1 ⊂ K2 ⊂ . . . is a countable sequence of compact subsets such that X = ∪K i , define a countable family of multiplicative semi-norms p i by pi (f ) = sup{|f (x)| : x ∈ Ki }. The collection {pi } of ...
NOETHERIAN MODULES 1. Introduction In a finite
... In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely g ...
... In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely g ...
Symmetric tensors and symmetric tensor rank
... is convenient to assume that the summation is always done on the second matrix index. For instance, the multilinear transform in (2.1) may be expressed as A′ = A•1 L•2 M •3 N . An alternative notation for (2.1) from the theory of group actions is A′ = (L, M, N ) · A, which may be viewed as multiplyi ...
... is convenient to assume that the summation is always done on the second matrix index. For instance, the multilinear transform in (2.1) may be expressed as A′ = A•1 L•2 M •3 N . An alternative notation for (2.1) from the theory of group actions is A′ = (L, M, N ) · A, which may be viewed as multiplyi ...
A note on actions of a monoidal category
... the corresponding f : V−→ [V, V] has (f X)Y = X ⊗ Y . For another simple example of an action, take V to be the cartesian closed category Set and A to be any category admitting small multiples (also called copowers—for X ∈ Set and A ∈ A, the multiple X · A is the coproduct of X copies of A); clearly ...
... the corresponding f : V−→ [V, V] has (f X)Y = X ⊗ Y . For another simple example of an action, take V to be the cartesian closed category Set and A to be any category admitting small multiples (also called copowers—for X ∈ Set and A ∈ A, the multiple X · A is the coproduct of X copies of A); clearly ...
Representation Theory Of Algebras Related To The Partition Algebra
... Representation theory is concerned with the study of how various algebraic structures (such as groups, monoids, algebras) act on vector spaces while respecting the operations on these algebraic structures. In group theory, the idea of representation is to find a group of permutations or linear trans ...
... Representation theory is concerned with the study of how various algebraic structures (such as groups, monoids, algebras) act on vector spaces while respecting the operations on these algebraic structures. In group theory, the idea of representation is to find a group of permutations or linear trans ...
![[math.QA] 23 Feb 2004 Quantum groupoids and](http://s1.studyres.com/store/data/015053496_1-11323fe54a6fa6579b01cd3020b05b55-300x300.png)
[math.QA] 23 Feb 2004 Quantum groupoids and
... of certain L-bialgebroids, were L is a base algebra over a Hopf algebra H in the sense of Definition 2.1. The simplest bialgebroid of this kind, namely the smash product L ⋊ H, was introduced in [Lu]. It is interesting to note that bialgebroids of [Lu] were considered over exactly the same class of ...
... of certain L-bialgebroids, were L is a base algebra over a Hopf algebra H in the sense of Definition 2.1. The simplest bialgebroid of this kind, namely the smash product L ⋊ H, was introduced in [Lu]. It is interesting to note that bialgebroids of [Lu] were considered over exactly the same class of ...
A course on finite flat group schemes and p
... 2.3.1. Algebraic affine group schemes. An algebraic affine group scheme over R is an affine group scheme such that its Hopf algebra is finitely generated as an R-algebra, which means that ‘we have only finitely many coordinates’. 2.3.2. Translation action. For g ∈ G(R) we have the left translation λ ...
... 2.3.1. Algebraic affine group schemes. An algebraic affine group scheme over R is an affine group scheme such that its Hopf algebra is finitely generated as an R-algebra, which means that ‘we have only finitely many coordinates’. 2.3.2. Translation action. For g ∈ G(R) we have the left translation λ ...
INTRODUCTION TO FINITE GROUP SCHEMES Contents 1. Tate`s
... the category of affine R-schemes, the dual category. Let F be a covariant functor C → Grps (the category of groups) and F ∨ : C∨ → Grps the corresponding contravariant functor. Example. If S is an R-algebra, we can let F (S) = S × , for if we have a map f : S → T , then we have an induced map F (S) ...
... the category of affine R-schemes, the dual category. Let F be a covariant functor C → Grps (the category of groups) and F ∨ : C∨ → Grps the corresponding contravariant functor. Example. If S is an R-algebra, we can let F (S) = S × , for if we have a map f : S → T , then we have an induced map F (S) ...
3 - UCI Math
... 4. R3 with cross/vector product ×. 5. {Functions f : V → V }, where V is any set and ∗ = ◦ (composition of functions). 6. C with ∗ the Hermitian product: z ∗ w = zw. 7. N = Z+ with ∗ the least common multiple: e.g. 4 ∗ 5 = 20. 8. N with ∗ the highest common prime factor: e.g. 65 ∗ 20 = 5. 9. The set ...
... 4. R3 with cross/vector product ×. 5. {Functions f : V → V }, where V is any set and ∗ = ◦ (composition of functions). 6. C with ∗ the Hermitian product: z ∗ w = zw. 7. N = Z+ with ∗ the least common multiple: e.g. 4 ∗ 5 = 20. 8. N with ∗ the highest common prime factor: e.g. 65 ∗ 20 = 5. 9. The set ...
Solution 1 - D-MATH
... m(a)p−k for a ∈ Zp . To check that m is an isomorphism, we show that it is both injective and surjective. If there is a nonzero element a in the kernel of m, we can assume without loss of generality that it lies in Zp and by definition this implies that an ≡ 0 modulo pn and therefore that a = 0. For ...
... m(a)p−k for a ∈ Zp . To check that m is an isomorphism, we show that it is both injective and surjective. If there is a nonzero element a in the kernel of m, we can assume without loss of generality that it lies in Zp and by definition this implies that an ≡ 0 modulo pn and therefore that a = 0. For ...
MONADS AND ALGEBRAIC STRUCTURES Contents 1
... We now look to define the general notion of an algebraic systems of a certain ‘type’, so that we can obtain a category corresponding to a variety of algebras. For the way in which we will need this concept for this paper, which is for now just to obtain a “free-forgetful” type adjunction between Set ...
... We now look to define the general notion of an algebraic systems of a certain ‘type’, so that we can obtain a category corresponding to a variety of algebras. For the way in which we will need this concept for this paper, which is for now just to obtain a “free-forgetful” type adjunction between Set ...
as a PDF
... can be completed to a commutative diagram only by automorphisms of F. If (a) is satisfied (and perhaps not (b)), then k: F , M is called a flat precover of M. If a flat cover exists, then it is unique up to isomorphism. If a ring R is left perfect, then every left R-module has a projective cover. Ov ...
... can be completed to a commutative diagram only by automorphisms of F. If (a) is satisfied (and perhaps not (b)), then k: F , M is called a flat precover of M. If a flat cover exists, then it is unique up to isomorphism. If a ring R is left perfect, then every left R-module has a projective cover. Ov ...
The Theory of Polynomial Functors
... be needed so as to properly understand the functors), to see how they fit into Professor Roby’s framework of strict polynomial maps (Chapter 5). 3:o. To conduct a survey of numerical rings (in order to understand the maps). This has, admittedly, been done before, in a somewhat diverent guise, but ou ...
... be needed so as to properly understand the functors), to see how they fit into Professor Roby’s framework of strict polynomial maps (Chapter 5). 3:o. To conduct a survey of numerical rings (in order to understand the maps). This has, admittedly, been done before, in a somewhat diverent guise, but ou ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.