INFINITESIMAL BIALGEBRAS, PRE
... In Appendix B we study certain special features of counital ǫ-bialgebras. We construct another monoidal category of algebras and show that comonoid objects in this category are precisely counital ǫ-bialgebras (Proposition B.5). The relation to the constructions of Appendix A is explained. We also de ...
... In Appendix B we study certain special features of counital ǫ-bialgebras. We construct another monoidal category of algebras and show that comonoid objects in this category are precisely counital ǫ-bialgebras (Proposition B.5). The relation to the constructions of Appendix A is explained. We also de ...
Quantum Groups - International Mathematical Union
... Finally define a quantum group to be the spectrum of a (not necessarily commutative) Hopf algebra. So the notions of Hopf algebra and quantum group are in fact equivalent, but the second one has some geometric flavor. Let me make some general remarks on the definition of Hopf algebra. First of all, ...
... Finally define a quantum group to be the spectrum of a (not necessarily commutative) Hopf algebra. So the notions of Hopf algebra and quantum group are in fact equivalent, but the second one has some geometric flavor. Let me make some general remarks on the definition of Hopf algebra. First of all, ...
Algebra 1
... the white fur gene (W) is recessive. This means that a guinea pig with at least one dominant gene (BB or BW) will have black fur. A guinea pig with two recessive genes (WW) will have white fur. The Punnett square below models the possible combinations of color genes that parents who carry both genes ...
... the white fur gene (W) is recessive. This means that a guinea pig with at least one dominant gene (BB or BW) will have black fur. A guinea pig with two recessive genes (WW) will have white fur. The Punnett square below models the possible combinations of color genes that parents who carry both genes ...
Weighted semigroup measure algebra as a WAP-algebra H.R. Ebrahimi Vishki, B. Khodsiani, A. Rejali
... W AP (A) = A∗ . Further information for the Arens regularity of Banach algebras can be found in [5, 6]. WAP-algebras, as a generalization of the Arens regular algebras, have been introduced and intensively studied in [9]. A Banach algebra A for which the natural embedding x 7→ x̂ of A into W AP (A)∗ ...
... W AP (A) = A∗ . Further information for the Arens regularity of Banach algebras can be found in [5, 6]. WAP-algebras, as a generalization of the Arens regular algebras, have been introduced and intensively studied in [9]. A Banach algebra A for which the natural embedding x 7→ x̂ of A into W AP (A)∗ ...
Introduction to the Lorentz algebra
... which is free as a Z-module and its Z-basis is B. This allows us to define the Lorentz type algebra LK := LZ ⊗Z K over any field K . In this talk we study the ideal structure of Lorentz type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground ...
... which is free as a Z-module and its Z-basis is B. This allows us to define the Lorentz type algebra LK := LZ ⊗Z K over any field K . In this talk we study the ideal structure of Lorentz type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground ...
skew-primitive elements of quantum groups and braided lie algebras
... algebras. They often have the following form. Let K = kG be the group algebra of a commutative group. Let H be a Hopf algebra in the category of Yetter-Drinfel'd modules over K . Then the biproduct K ? H is a Hopf algebra [R, Ma, FM], which in general is neither commutative nor cocommutative. More g ...
... algebras. They often have the following form. Let K = kG be the group algebra of a commutative group. Let H be a Hopf algebra in the category of Yetter-Drinfel'd modules over K . Then the biproduct K ? H is a Hopf algebra [R, Ma, FM], which in general is neither commutative nor cocommutative. More g ...
Full Text (PDF format)
... i+j=n Ci ⊗ Cj and ε(C>0 ) = 0). A graded algebra (coalgebra) structure induces an augmented algebra (coalgebra) structure in an evident way. The homology coalgebra (cohomology algebra) of a graded algebra (coalgebra) is equipped with a natural second grading, as it can be seen from the explicit reso ...
... i+j=n Ci ⊗ Cj and ε(C>0 ) = 0). A graded algebra (coalgebra) structure induces an augmented algebra (coalgebra) structure in an evident way. The homology coalgebra (cohomology algebra) of a graded algebra (coalgebra) is equipped with a natural second grading, as it can be seen from the explicit reso ...
A brief introduction to pre
... and Sokolov(1994), Etingof and Soloviev(1999), Golubschik and Sokolov(2000), · · · Poisson brackets and infinite-dimensional Lie algebras: Gel’fand and Dorfman(1979), Dubrovin and Novikov(1984), Balinskii and Novikov(1985), · · · Quantum field theory and noncommutative geometry: Connes and Kreimer(1 ...
... and Sokolov(1994), Etingof and Soloviev(1999), Golubschik and Sokolov(2000), · · · Poisson brackets and infinite-dimensional Lie algebras: Gel’fand and Dorfman(1979), Dubrovin and Novikov(1984), Balinskii and Novikov(1985), · · · Quantum field theory and noncommutative geometry: Connes and Kreimer(1 ...
On bimeasurings
... correspond to bialgebra maps from C to B(B, A) as well as bialgebra maps from B to B(C, A). In fact Bialg(C, B(B, A)) Bimeas(C ⊗ B, A) Bialg(B, B(C, A)) and hence the functor B( _, A) on the category of bialgebras is adjoint to itself. In the special case A = k, this gives a new proof that the fi ...
... correspond to bialgebra maps from C to B(B, A) as well as bialgebra maps from B to B(C, A). In fact Bialg(C, B(B, A)) Bimeas(C ⊗ B, A) Bialg(B, B(C, A)) and hence the functor B( _, A) on the category of bialgebras is adjoint to itself. In the special case A = k, this gives a new proof that the fi ...
Profinite Orthomodular Lattices
... condition 12, 81 is that the intersection of all closed and open (= clopen) congruences on L is trivial. The following question has interested many authors [2, 81 and has still remained as open: what are necessary and sufficient conditions for a zero-dimensional compact universal algebra to be profi ...
... condition 12, 81 is that the intersection of all closed and open (= clopen) congruences on L is trivial. The following question has interested many authors [2, 81 and has still remained as open: what are necessary and sufficient conditions for a zero-dimensional compact universal algebra to be profi ...
Recognisable Languages over Monads
... Running Example 2. Consider the following ∞-language L = {an1 ban2 b · · · : the sequence ni is unbounded, i.e. lim sup ni = ∞.} One can show that this language does not have a syntactic morphism, not even if the target algebra is allowed to have infinite universe. The idea is that for every n, ther ...
... Running Example 2. Consider the following ∞-language L = {an1 ban2 b · · · : the sequence ni is unbounded, i.e. lim sup ni = ∞.} One can show that this language does not have a syntactic morphism, not even if the target algebra is allowed to have infinite universe. The idea is that for every n, ther ...
STRONGLY REPRESENTABLE ATOM STRUCTURES OF
... an atom structure is weakly representable if at least one atomic representable relation algebra has that atom structure, and it is strongly representable if every atomic relation algebra with that atom structure is representable. Since representability is preserved under subalgebras, and every atomi ...
... an atom structure is weakly representable if at least one atomic representable relation algebra has that atom structure, and it is strongly representable if every atomic relation algebra with that atom structure is representable. Since representability is preserved under subalgebras, and every atomi ...
CONVERGENCE THEOREMS FOR PSEUDO
... Definitions 2.1 and 2.2 below are found in Allan [1]. Definition 2.1. Let E be a locally convex algebra and let B1 denote the collection of all subsets B of E satisfying (i) B is absolutely convex and B 2 ⊂ B, (ii) B is bounded and closed. If E has an identity 1, we take 1 ∈ B. For every B ∈ B1 , E( ...
... Definitions 2.1 and 2.2 below are found in Allan [1]. Definition 2.1. Let E be a locally convex algebra and let B1 denote the collection of all subsets B of E satisfying (i) B is absolutely convex and B 2 ⊂ B, (ii) B is bounded and closed. If E has an identity 1, we take 1 ∈ B. For every B ∈ B1 , E( ...
Hochschild cohomology: some methods for computations
... that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about computations for particular classes of finite–dimensional algebras, since the computations of these ...
... that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about computations for particular classes of finite–dimensional algebras, since the computations of these ...
Notes
... Example 3. Let g be complex simple Lie algebra, and choose a Borel subalgebra b. b can be given the structure of a Lie bialgebra [D, Example 3.2]. The double D(b) is not quite the original algebra g, but it surjects onto g as a Lie algebra with kernel a Lie bialgebra ideal. Thus, g inherits a quasit ...
... Example 3. Let g be complex simple Lie algebra, and choose a Borel subalgebra b. b can be given the structure of a Lie bialgebra [D, Example 3.2]. The double D(b) is not quite the original algebra g, but it surjects onto g as a Lie algebra with kernel a Lie bialgebra ideal. Thus, g inherits a quasit ...
Applying Universal Algebra to Lambda Calculus
... semantics, and by Bastonero and Gouy for Berry’s stable semantics [6]. The proofs of the above results are syntactical and very difficult. In [65, 66] it was shown the equational incompleteness of all semantics of lambda calculus that involve monotonicity with respect to some partial order and have ...
... semantics, and by Bastonero and Gouy for Berry’s stable semantics [6]. The proofs of the above results are syntactical and very difficult. In [65, 66] it was shown the equational incompleteness of all semantics of lambda calculus that involve monotonicity with respect to some partial order and have ...
A family of simple Lie algebras in characteristic two
... should be satisfied for every values of 2 F? , f g, which cannot occur. Then the only possible non-zero values can be assumed for (e(0; ) ; e(1; ) ): the sole ...
... should be satisfied for every values of 2 F? , f g, which cannot occur. Then the only possible non-zero values can be assumed for (e(0; ) ; e(1; ) ): the sole ...
slides
... Draw two diagrams, one with bounded RL at the top, the other with RL at the top, and the variety of one-element algebras at the bottom Then ll in all the varieties from these slides to show how they are related to each other ...
... Draw two diagrams, one with bounded RL at the top, the other with RL at the top, and the variety of one-element algebras at the bottom Then ll in all the varieties from these slides to show how they are related to each other ...
Fredrik Dahlqvist and Alexander Kurz. Positive coalgebraic logic
... will be of fundamental importance in what follows: the adjunction S0 a P0 : Posop → DL which is the backbone of diagram (2) is in fact Pos-enriched; that is to say DL and Pos are Pos-enriched categories and S0 , P0 are Pos-enriched functors ([16]). Clearly, it would be a shame not to use this extra ...
... will be of fundamental importance in what follows: the adjunction S0 a P0 : Posop → DL which is the backbone of diagram (2) is in fact Pos-enriched; that is to say DL and Pos are Pos-enriched categories and S0 , P0 are Pos-enriched functors ([16]). Clearly, it would be a shame not to use this extra ...
ON QUANTIC CONUCLEI IN ORTHOMODULAR LATTICES
... adjoint to b ! ; for any b 2 L. An immediate result from this is that the operator & distributes over arbitrary joins (on the left), i.e., it is \left distributive". Moreover & can be thought of as a noncommutative nonassociative meet operator. In fact, if & is either associative or commutative then ...
... adjoint to b ! ; for any b 2 L. An immediate result from this is that the operator & distributes over arbitrary joins (on the left), i.e., it is \left distributive". Moreover & can be thought of as a noncommutative nonassociative meet operator. In fact, if & is either associative or commutative then ...
ON QUANTIC CONUCLEI IN ORTHOMODULAR LATTICES
... hypothesis, and this of course is equal to the RHS of the last inequality. Therefore φ = uz where z = φ(0) = j(1)⊥ ; i.e., if a is an arbitrary element of L we have φ(a) = a ∨ j(1)⊥ but φ(a) = j(a⊥ )⊥ hence j(a⊥ )⊥ = a ∨ j(1)⊥ and j(a) = a ∧ j(1) = mj(1) . The converse is of course trivial. We want ...
... hypothesis, and this of course is equal to the RHS of the last inequality. Therefore φ = uz where z = φ(0) = j(1)⊥ ; i.e., if a is an arbitrary element of L we have φ(a) = a ∨ j(1)⊥ but φ(a) = j(a⊥ )⊥ hence j(a⊥ )⊥ = a ∨ j(1)⊥ and j(a) = a ∧ j(1) = mj(1) . The converse is of course trivial. We want ...
Composition algebras of degree two
... problem. However, if we impose certain general identities on these algebras then such a classification can be afforded. Symmetric composition algebras have been studied in [20, 21]. In [19] Okubo shows that, over fields of characteristic not 2, any finite dimensional composition algebra verifying th ...
... problem. However, if we impose certain general identities on these algebras then such a classification can be afforded. Symmetric composition algebras have been studied in [20, 21]. In [19] Okubo shows that, over fields of characteristic not 2, any finite dimensional composition algebra verifying th ...
Tense Operators on Basic Algebras - Phoenix
... Let us note that if a basic algebra A is commutative then the assigned lattice L(A) is distributive (see [7], Theorem 8.5.9). The propositional logic corresponding to a commutative basic algebra was already described (see [3]). Our aim is to introduce tense operators G, H , F , P on any basic algebr ...
... Let us note that if a basic algebra A is commutative then the assigned lattice L(A) is distributive (see [7], Theorem 8.5.9). The propositional logic corresponding to a commutative basic algebra was already described (see [3]). Our aim is to introduce tense operators G, H , F , P on any basic algebr ...
Connections between relation algebras and cylindric algebras
... a representation h : A → ℘(U 2 ) of a (finite simple) relation algebra A. Recall from formula (1) in section 2.1 that a subset of the base set U can be viewed as an A-network. We can make this a little tighter by considering maps instead of subsets. Definition 1. Let N = (N1 , N2 ) be an A-network, ...
... a representation h : A → ℘(U 2 ) of a (finite simple) relation algebra A. Recall from formula (1) in section 2.1 that a subset of the base set U can be viewed as an A-network. We can make this a little tighter by considering maps instead of subsets. Definition 1. Let N = (N1 , N2 ) be an A-network, ...
Slides
... For a crutched set (S, f ) and C*-relations R ⊆ AS,f on (S, f ), let JR be the two-sided, norm-closed ideal generated by R in AS,f . Then, the unital C*-algebra presented on (S, f ) subject to R is hS, f |Ri1C∗ := AS,f /JR . C∗ hSi ...
... For a crutched set (S, f ) and C*-relations R ⊆ AS,f on (S, f ), let JR be the two-sided, norm-closed ideal generated by R in AS,f . Then, the unital C*-algebra presented on (S, f ) subject to R is hS, f |Ri1C∗ := AS,f /JR . C∗ hSi ...