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PlanetMath: bialgebra A bialgebra is a vector space that is both a unital algebra and a coalgebra, such that the comultiplication and counit are unital algebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. This is version 3 of bialgebra, born on 2002-10-18, modified 2004-12-29. planetmath.org /encyclopedia/Bialgebra.html   (95 words)

Bialgebra -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-29) In formulas, the bialgebra compatibility conditions look as follows (using the sumless (additional info and facts about Sweedler notation) Sweedler notation): Here we wrote the algebra multiplication as simple juxtaposition, and 1 is the multiplicative identity. For examples of bialgebras, refer to the articles on (additional info and facts about coalgebra) coalgebras and (additional info and facts about Hopf algebra) Hopf algebras. www.absoluteastronomy.com /encyclopedia/b/bi/bialgebra.htm   (174 words)

Bialgebra - TheBestLinks.com - Associative algebra, Field (mathematics), Mathematics, TheBestLinks.com:Find or fix a ...   (Site not responding. Last check: 2007-10-29) In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. Equivalently, one may require that the multiplication and the unit of the algebra both be coalgebra morphisms. For examples of bialgebras, refer to the articles on coalgebras and Hopf algebras. www.thebestlinks.com /Bialgebra.html   (250 words)

Encyclopedia: Quantum mechanics Noncommutative quantum field theory In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes. In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. www.nationmaster.com /encyclopedia/Quantum-mechanics   (10771 words)

[No title] Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a $q$-deformed universal enveloping algebra of a certain simple Lie algebra. In~\cite{Lu} a method is proposed how to add such relations to (\ref{1}) without destroying the bialgebra (the Hopf algebra structure also survives). Let us see now that, quite analogously, the bialgebra structure of $sl_q(2)$ is unambiguously determined if we require the $q$-oscillator algebra to be its covariant comodule. thsun1.jinr.ru /~alvladim/pap/czjp96.txt   (1142 words)

Bialgebra structures on a real semisimple Lie algebra. (ResearchIndex) Bialgebra structures on a real semisimple Lie algebra. We then determine all the bialgebra structures on a real semisimple Lie algebra for the nonzero standard modified Yang-Baxter equation. Finally we consider the case of a real simple Lie algebra the complexification of which is not simple and we give some partial results about... citeseer.ist.psu.edu /chloup95bialgebra.html   (281 words)

[No title] Note also that we call K a bialgebra, and not a Hopf algebra, since, like the Dyer-Lashof algebra, it is not connected and thus has no conjugation. In section 4 we study actions of K on itself and on K*, arising from the Nishida re- lations, including a description which may shed light on the A-algebra structure of the Dickson algebras, and we present a formula analogous to the Thom isomorphism theorem. K be the Verschiebung, the bialgebra map dual to the Frobenius (squaring) map on K*. hopf.math.purdue.edu /Pengelley-Williams/newlowerops.txt   (8078 words)

[No title] Consider two bialgebras $ Y_+ $ and $ Y_- $ with generators $ \{ u_{j}^{i}, F^i \}, $ $ \{ t_{j}^{i}, E_i \}, $ respectively, which form matrices $T,U$, a row $E$ and a column $F$. The bialgebras defined above form a matched pair of bialgebras, when a left action of $ Y_+ $ on $ Y_- $ and a right action of $ Y_- $ on $ Y_+ $ are suitably defined. Then we build bialgebras (\ref{1}),(\ref{2}) (which may, under suitable conditions, be extended to Hopf algebras \cite{Vl1}) being paired via (\ref{3}),(\ref{6}). thsun1.jinr.ru /~alvladim/pap/lue-vlad.txt   (1749 words)

PlanetMath: almost cocommutative bialgebra The significance of the almost cocommutative condition is that Cross-references: identity, category, representations, natural isomorphism, tensor product, map, comultiplication, opposite, unit, cocommutative, bialgebra This is version 2 of almost cocommutative bialgebra, born on 2003-03-24, modified 2003-03-24. planetmath.org /encyclopedia/AlmostCocommutativeBialgebra.html   (121 words)

On the Bialgebra of Functional Graphs and Differential Algebras (ResearchIndex) On the bialgebra of functional graphs and differential algebras 1 we consider a bialgebra structure on G and three interesting subalgebras: T the set of labeled forests; S the set of permutation graphs; and L the set of well labeled forests,... 20 Coalgebras and bialgebras in combinatorics (context) - Joni, Rota - 1979 citeseer.ist.psu.edu /25841.html   (404 words)

[No title] Furthermore, K is a bialgebra with comultiplication defined on gen- erators by the (Cartan) formula Xi OE(Di) = Dk Di-k. Definition and discussion of M and K(1) In this section we define and discuss the extended Milnor bialgebras M and M+ from [BJ ] and the stabilized algebra K(1) from [PW ]. 4 T. The positive extended Milnor bialgebra is M+ = F2[0; 1; 2; : :]: with comultiplication X p (i) = 2i-p p: The extended Milnor Hopf algebra is M = F2[0 ; 1; 2; : :]:, the result of adjoining -10to the bialgebra M+. hopf.math.purdue.edu /Bisson-Pengelley-Williams/stablops.txt   (2755 words)

Representation Theory Seminar   (Site not responding. Last check: 2007-10-29) Let A is an associative algebra and B a bialgebra. A B-module algebra structure on A is an action of B on A such that the comultiplication on bialgebra B is compatible with the multiplication on algebra A. In this case, for example, group-like elements of the bialgebra act on A as algebra homomorphisms, and primitive elements of B act on A as derivations. In this lecture, we consider bialgebras that give rise to B-module algebra structures on polynomial, Weyl, and quiver algebras. styx.math.neu.edu /%7Ealexmart/rtrt/20032004/vlassov.html   (83 words)

FreeScience.info The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. Also, application of linear bialgebra to bicodes is given. freescience.info   (571 words)

Amazon.ca: Books: Quantum Groups   (Site not responding. Last check: 2007-10-29) The notion of a bialgebra is also discussed, which is essentially a vector space equipped with both an algebra structure and a coalgebra structure. A Hopf algebra is then a bialgebra that has a special endomorphism of the underlying vector space. The author introduces the concept of a braided bialgebra, which contain a "universal" R-matrix which induces a solution of the Yang-Baxter equation on all of their modules, and thus giving a systematic method for constructing solutions of the Yang-Baxter equation. www.amazon.ca /exec/obidos/ASIN/0387943706   (1322 words)

Bialgebra Cohomology, Deformations, and Quantum Groups -- Gerstenhaber and Schack 87 (1): 478 -- Proceedings of the ... We introduce cohomology and deformation theories for a bialgebra A (over a commutative unital ring k) such that the second cohomology group is the space of infinitesimal deformations. Our theory gives a natural identification between the underlying k-modules of the original and the deformed bialgebra. Certain explicit deformation formulas are given for the construction of quantum groups--i.e., Hopf algebras that are neither commutative nor cocommutative (whether or not they arise from quantum Yang-Baxter operators). www.pnas.org /cgi/content/abstract/87/1/478   (165 words)

Abstract for "A triple construction for Lie bialgebras"   (Site not responding. Last check: 2007-10-29) We study the triple of a quasitriangular Lie bialgebra as a natural extension of the Drinfel'd double. The triple is itself a quasitriangular Lie bialgebra. We prove several results about the algebraic structure of the triple, analogous to known results for the double. www.maths.qmul.ac.uk /~jeg/triple.abstract.html   (80 words)

Quantization, deformations, and new homological and categorical methods in mathematical physics   (Site not responding. Last check: 2007-10-29) Graded manifolds with additional structures are naturally connected with Lie (bi)algebras and (bi)algebroids. A Lie superalgebra g is an odd Lie bialgebra if the dual space with the opposite parity Πg* is also a Lie superalgebra, and these two brackets are compatible in a certain way. An "odd double" of an odd Lie bialgebra is again an odd Lie bialgebra. www.ma.umist.ac.uk /tv/LMS/ted.html   (149 words)

Manchester Geometry Seminar   (Site not responding. Last check: 2007-10-29) A Lie bialgebra is a Lie algebra such that the dual space is also a Lie algebra, with a natural compatibility condition.) The relation will be explained between Lie (bi)algebras and such structures on supermanifolds as homological fields, Poisson and Schouten (Gerstenhaber) brackets. This is a far-going generalization of Drinfeld's bialgebras and their doubles, and at the same time provides a very natural simple framework for the original Drinfeld's construction, in terms of super Poisson geometry. www.ma.umist.ac.uk /tv/Seminar/2000-2001/ted.html   (186 words)

OhioLINK ETD: Kaygun, Atabey   (Site not responding. Last check: 2007-10-29) We show that one can extend the definition of Hopf cyclic homology with coefficients such that one can use bialgebras and a larger class of coefficient module/comodules. With the help of this extension, we calculate the bialgebra cyclic homology of the quantum deformation of an arbitrary semi-simple Lie algebra and the Hopf algebra of foliations of codimension N, with several coefficient modules. Hopf Algebra; Foliations; Cyclic Homology; Bialgebras; Yetter-Drinfeld; Connes-Moscovici www.ohiolink.edu /etd/view.cgi?osu1107564231   (78 words)

THE BIALGEBRA OF CURVES WITH SYMMETRIES The notion of bialgebra was introduced recently by Gizatullin One of the main problems is the structure of the bialgebras, especially the question of semi-simplicity. The goal of this project is to determine the structure of the bialgebras www.conicyt.cl /bases/fondecyt/proyectos/01/2001/1010432.html   (243 words)

AMCA: Biquantization of Lie bialgebras of the $B_n$ type by Valentina Golubeva   (Site not responding. Last check: 2007-10-29) The notion of the Lie bialgebra was introduced by V. Drinfeld (1982, 1987) in the framework of his algebraic treatment of the quantum inverse scattering method. The Lie bialgebra is the Lie algebra g provided with a Lie cobracket, related to the Lie bracket by some compatibility relations. The quantization problem for g consists in finding a bialgebra structure on the module of formal power series F=U(g)[[h]] which induces the given bialgebra structure and the Poisson cobracket on F/h. at.yorku.ca /c/a/h/z/34.htm   (281 words)

[No title]   (Site not responding. Last check: 2007-10-29) A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. They are explicitly used to generate new non-semisimple quantum algebras as it is the case for the Euclidean, Poincar\'e and Galilean algebras. www.cs.odu.edu /~dlibug/ups/rdf/xxx/hep-th/9412083.rdf   (97 words)

DMTCS vol 1 no 1 (1997), pp. 229-237   (Site not responding. Last check: 2007-10-29) We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of {1, 2,...n} in itself as in the frame of the discrete dynamical systems. Maurice Ginocchio (1997), On the bialgebra of functional graphs and differential algebras, Discrete Mathematics and Theoretical Computer Science 1, pp. dmtcs.loria.fr /volumes/abstracts/dm010114.abs.html   (205 words)

IngentaConnect Lie bialgebra quantizations of the oscillator algebra and their u...   (Site not responding. Last check: 2007-10-29) IngentaConnect Lie bialgebra quantizations of the oscillator algebra and their u... All coboundary Lie bialgebras and their corresponding Poisson - Lie structures are constructed for the oscillator algebra generated by. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal R-matrices. api.ingentaconnect.com /content/iop/jphysa/1996/00000029/00000015/art00006   (136 words)

A Natural Differential Calculus on Lie Bialgebras with Dual of Triangular Type (ResearchIndex)   (Site not responding. Last check: 2007-10-29) Abstract: We prove that for a specific class of Lie bialgebras, there exists a natural differential calculus. This class consists of the Lie bialgebras for which the dual Lie bialgebra is of triangular type. The differential calculus is explicitly constructed with the help of the R- matrix from the dual. citeseer.lcs.mit.edu /vandenhijligenberg95natural.html   (276 words)

IHES PREPRINT M/00/66   (Site not responding. Last check: 2007-10-29) In particular the bialgebra of chord diagrams arises as some subspace of this homology (in this case $d=3$). We provide also a simplification for the calculation of the Vassiliev spectral sequence in the first term. Mots-clé, Keywords : discriminant of the knot space, bialgebra of chord diagrams, Hochschild complex, Poisson (resp. www.ihes.fr /PREPRINTS/M00/Resu/resu-M00-66.html   (123 words)

Publications and Preprints   (Site not responding. Last check: 2007-10-29) The dual Lie bialgebra of a certain quasitriangular Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson-Lie group G. Motivated by the data at the Poisson (classical) level, we then construct on A its locally compact quantum group structures: comultiplication, counit, antipode and Haar weight, as well as its associated multiplicative unitary operator. By working with several specific Poisson-Lie groups arising from Heisenberg Lie bialgebras and by carrying out their quantizations, a case is made for a useful but simple method of constructing locally compact quantum groups. equinox.unr.edu /homepage/bjkahng/research/publist.html   (1023 words)

EULER Record Details   (Site not responding. Last check: 2007-10-29) Bergman, George M. On the growth of algebras with bialgebra action. The author studies the growth rate of $A$ under its algebra structure together with the action of $B$. In particular, he proves that if $B$ is a coassociative bialgebra which is finitely generated as a $k$-algebra and its length growth rate is less than exponential and $A$ is commutative then the depth growth rate of $A$ is less than doubly exponential. www.emis.de /projects/EULER/detail?ide=1996berggrowalgewith&matchno=11&matchtotal=127&q=cr%3AG%2A+Bergman+   (179 words)

Hopf algebras For a monoid G, we have seen that the monoid algebra R(G) is a bialgebra. If G is a group then the group algebra R(G) becomes a Hopf algebra with antipode The matrix bialgebra M!(n) (earlier example) is not a Hopf algebra. www-texdev.ics.mq.edu.au /Quantum/Sect9/Sect9.html   (426 words)

FRT-Construction   (Site not responding. Last check: 2007-10-29) Epimorphic images of this bialgebra are called matric bialgebras (cf. According to the proposition we may assign a graded matric bialgebra to each subset This bialgebra will be called the FRT-construction corresponding to the subset www.mathematik.uni-stuttgart.de /mathB/abt/oehms/frt/node5.html   (300 words)

[No title]   (Site not responding. Last check: 2007-10-29) I am interested in classifying Hopf or bialgebras up to cocycle deformation rather than up to isomorphism. For this, it seems natural to extend the framework to coquasi-bialgebras, the dual notion of quasi bialgebras due to Drinfeld. In terms of extensions, I will give some method of constructing coquasi-bialgebras and classifying them up to coquasi-isomorphism. www.fuw.edu.pl /~pmh/conf/pfiles/masuoka.html   (55 words)

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