DOCSLIB.ORG

RESEARCH SYNOPSIS Miodrag Cristian Iovanov

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RESEARCH SYNOPSIS Miodrag Cristian Iovanov Research Synopsis M.C.Iovanov RESEARCH SYNOPSIS Miodrag Cristian Iovanov (In this text, all references are made to the papers in my publication list) My research interests are in representation theory, Hopf Algebras and quantum groups, tensor categories, category theory and homological algebra, algebraic topology. My PhD thesis was in 2009 from SUNY Buffalo, under the advisement of the late Samuel D. Schack. I. Representation theory of algebras and coalgebras [4],[11],[16],[17],[19],[20],[21],[22] Let A be a K-algebra. In the studies of the properties of A, one is often lead to considering (the category of) finite dimensional representations of A. In general, this category does not have ”enough” injective or projective objects, and one is sometimes lead to considering the category Ind(Rep(A)) whose objects are colimits of finite dimensional A-representations. Such A-modules which are sums of their finite dimensional submodules are sometimes called rational modules. This leads one naturally to the introduction of coalgebras and their comodules: the category of rational left A-modules is equivalent to the category of comodules over the representative coalgebra R(A), also called the finite dual of A. This coalgebra consists of all linear maps f : A → K such that ker(f) contains an ideal of finite codimension, n equivalently, there are finite families (gi)i=1,...,n, (hi)i=1,...,n such that f(xy)= P gi(x)hi(y). Here, gi, hi i=1 are again representative functions, and this allows the introduction of a ”coassociative” comultiplication n ∆ : R(A) → R(A) ⊗ R(A) by ∆(f) = P gi ⊗ hi which is well defined. Another motivation for the i=1 study of coalgebras is the following observation of Takeuchy, which characterizes linear categories which ”behave” like the category of rational modules over an algebra A. If a linear category C, which has directed colimits, is generated by objects of finite length and hom(S, S) is finite dimensional for S simple (i.e. C is a linear category of finite type), then C is equivalent to the category of right comodules MC over a coalgebra C. Furthermore, if both an algebra and a coalgebra structure are simultaneously present on a vector space B and they are compatible, i.e. one has a bialgebra B, then the category of B-comodules (and also that of finite dimensional B-modules) has a tensor product, i.e. it is monoidal. Thus, in such a situation, one can employ methods similar to those from the representation theory of groups. Hopf algebras are bialgebras with a map S : H → H called antipode, satisfying certain compatibility conditions; S plays the role of the inversion of elements in groups. Given a coalgebra C, its linear dual C∗ is a topological algebra with the so called ”finite topology”, and has a basis of neighborhoods for 0 consisting of cofinite ideals. It is complete and separated under this topology (pseudo-compact algebra), and it is also a profinite algebra. Such algebras can be thought as the algebra analogues of profinite groups. The category of right C-comodules is naturally identified with that of rational left C∗-modules, and thus it is often useful to consider C∗ in the study of C. Part of my research is concerned with corepresentation theory (or infinite dimensional rational represen- tation theory), more precisely, properties of categories of comodules. One important class of properties is Frobenius type properties. A category is Frobenius if it has enough injectives and projectives and the two classes coincide. Historically, this notion emerged from the work of Frobenius. In modular represen- tation theory, when the characteristic of K divides |G|, the group algebra KG is no longer semisimple but it is Frobenius, i.e. KG is isomorphic to its dual as left or as right modules (representations). Such ∗ an algebra A for which A =∼ A as left (or equivalently, right) modules is called a Frobenius algebra; such an algebra is always finite dimensional. This concept was extended and dualized for coalgebras: a coalgebra is left (right) co-Frobenius if C embeds in C∗ as left (right) C∗-module. This concept is not symmetric, and such coalgebras can be infinite dimensional; a finite dimensional coalgebra is left or right co-Frobenius if and only if C∗ is Frobenius. The concept is also motivated by the following: a Hopf Page 1 Research Synopsis M.C.Iovanov algebra is left or right co-Frobenius if and only if it has a non-zero integral. Such Hopf algebras are a generalization and a quantization of algebras of representative functions of a compact group, and can thus be thought of as compact quantum groups. Indeed, if G is a compact group, the left Haar integral restricted to the algebra R(G) of continuous representative functions on G is an integral in the Hopf algebraic sense. A coalgebra is left (right) quasi-Frobenius if C embeds in a coproduct power of C∗ as left (right) C∗-modules. My research provides new understanding and interpretations of these notions. I showed in [4] that the left and right co-Frobenius coalgebras can be understood as coalgebras which are isomorphic to their rational dual as left, or equivalently, as right comodules. This together with categorical consequences, shows that this can indeed be understood as not only a mere dualization, but the right symmetric generalization of the Frobenius algebras. Moreover, in [17], I showed that a similar result can be proved about quasi-co- Frobenius (QCF) coalgebras, that left and right QCF is equivalent to C being weakly isomorphic to its rational (left or right) dual (in the sense that some countable coproduct powers are isomorphic). One of the results of [16] shows that further the left and right QCF coalgebras are exactly those coalgebras for which the category of right (equivalently, that of left) comodules is a Frobenius category. These allowed us to introduce a new generalized Frobenius algebra concept in [17]: a topological algebra A with a topology of algebraic type (basis of 0 consisting of cofinite ideals) is Frobenius if A is isomorphic to its complete dual A∨ as left (equivalently, right) A-modules. A is Frobenius if and only if it is the dual of a co-Frobenius coalgebra, and such an algebra is always profinite. This further motivates and explains the theory of generalized Frobenius algebras and co-Frobenius coalgebras. Also, we showed in [16] that co-Frobenius coalgebras are close to Hopf algebras from another perspective too, that is, there is a theory of integrals. This takes the form of an ”integral” functor hom(C, −) on finite dimensional comodules. It is shown that the existence and uniqueness of integrals - i.e. dim hom(C,M) = dim(M) for all f.d. M’s - is equivalent to C being co-Frobenius; this extends results from Hopf algebras, where existence of integrals implies uniqueness and is equivalent to the Frobenius property. These generalized integrals have concrete interesting interpretations for compact groups as certain quantum invariant vector valued integrals R with a ”quantum invariance” with respect to a representation η of G: R a · fdx = η(a) · R f. Other results concern connections between injective/projective and generator/cogenerator conditions. For a ring A, the properties of being left self-injective and left or right cogenerator does not imply each other, while for finite dimensional algebras they do. For the infinite dimensional case, i.e. for coalgebras and their comodules, it was known that if C is projective in MC (equivalently, C is left QCF) implies that C is a generator for left comodules, but it was unknown if the converse holds. The results of [11] completely solve this problem and show that in this is true if and only if indecomposable injectives have finite Loewy length, and in general, it is false; the examples constructed prove that any coalgebra C can be embedded in a coalgebra D which is a generator for MD. This example is constructed by an infinite process in which at each step, using a certain matrix extension, a new coalgebra is formed containing the previous one and which ”generates” all the f.d. representations of the previous one. Another direction of results involves representations of combinatorial objects such as quivers and PO sets, and related notions, such as quiver algebras and path coalgebras. In [21] the one sided serial coalgebras (i.e. coalgebras which decompose as a direct sum of uniserial modules each of which have a unique composition series) over an algebraically closed field are classified; they are the support for serial linear categories of finite type. These coalgebras are constructed by some combinatorial data: are embedded in quiver Γ which is obtained by attaching trees at the vertices of a cycle, depend on a family of paths (pv)v where for each vertex v, pv is a path ending at v. Also, the representation theory of (left and right) serial coalgebras is developed. It is shown that this is very close and parallel to the theory of infinite abelian primary groups. It is shown that for such a linear category - MC over a serial coalgebra, every object decomposes into a direct sum of indecomposables if and only if injective indecomposables in MC have finite length. Also, the indecomposable objects are precisely the indecomposable injectives and the terms of their Loewy series. As a consequence, it is shown that it is possible that over an algebra A all Page 2 Research Synopsis M.C.Iovanov right rational modules decompose as direct sum of indecomposable modules (even uniserial), while not all left rational modules decompose as such; this is the quiver algebra of the right infinite line A∞.
Load more

Recommended publications
  • Introduction to Linear Bialgebra
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N.
    [Show full text]
  • QUIVER BIALGEBRAS and MONOIDAL CATEGORIES 3 K N ≥ 0
    QUIVER BIALGEBRAS AND MONOIDAL CATEGORIES HUA-LIN HUANG (JINAN) AND BLAS TORRECILLAS (ALMER´IA) Abstract. We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories. 1. Introduction This paper is devoted to the study of natural bialgebra structures on the path coalgebra of an arbitrary quiver and monoidal structures on the category of its locally nilpotent and locally finite representations. A further purpose is to establish a quiver setting for general pointed bialgebras and pointed monoidal categories. Our original motivation is to extend the Hopf quiver theory [4, 7, 8, 12, 13, 25, 31] to the setting of generalized Hopf structures. As bialgebras are a fundamental generalization of Hopf algebras, we naturally initiate our study from this case. The basic problem is to determine what kind of quivers can give rise to bialgebra structures on their associated path algebras or coalgebras. It turns out that the path coalgebra of an arbitrary quiver admits natural bial- gebra structures, see Theorem 3.2. This seems a bit surprising at first sight by comparison with the Hopf case given in [8], where Cibils and Rosso showed that the path coalgebra of a quiver Q admits a Hopf algebra structure if and only if Q is a Hopf quiver which is very special.
    [Show full text]
  • Characterization of Hopf Quasigroups
    Characterization of Hopf Quasigroups Wei WANG and Shuanhong WANG ∗ School of Mathematics, Southeast University, Jiangsu Nanjing 210096, China E-mail: [email protected], [email protected] Abstract. In this paper, we first discuss some properties of the Galois linear maps. We provide some equivalent conditions for Hopf algebras and Hopf (co)quasigroups as its applications. Then let H be a Hopf quasigroup with bijective antipode and G be the set of all Hopf quasigroup automorphisms of H. We introduce a new category CH(α, β) with α, β ∈ G over H and construct a new braided π-category C (H) with all the categories CH (α, β) as components. Key words: Galois linear map; Antipode; Hopf (co)quasigroup; Braided π-category. Mathematics Subject Classification 2010: 16T05. 1. Introduction The most well-known examples of Hopf algebras are the linear spans of (arbitrary) groups. Dually, also the vector space of linear functionals on a finite group carries the structure of a Hopf algebra. In the case of quasigroups (nonassociative groups)(see [A]) however, it is no longer a Hopf algebra but, more generally, a Hopf quasigroup. In 2010, Klim and Majid in [KM] introduced the notion of a Hopf quasigroup which was arXiv:1902.10141v1 [math.QA] 26 Feb 2019 not only devoted to the development of this missing link: Hopf quasigroup but also to under- stand the structure and relevant properties of the algebraic 7-sphere, here Hopf quasigroups are not associative but the lack of this property is compensated by some axioms involving the antipode S. This mathematical object was studied in [FW] for twisted action as a gen- eralization of Hopf algebras actions.
    [Show full text]
  • TENSOR TOPOLOGY 1. Introduction Categorical Approaches Have Been Very Successful in Bringing Topological Ideas Into Other Areas
    TENSOR TOPOLOGY PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL Abstract. A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions complet- ing any monoidal category to one whose subunits universally form a lattice, preframe, or frame. 1. Introduction Categorical approaches have been very successful in bringing topological ideas into other areas of mathematics. A major example is the category of sheaves over a topological space, from which the open sets of the space can be reconstructed as subobjects of the terminal object. More generally, in any topos such subobjects form a frame. Frames are lattices with properties capturing the behaviour of the open sets of a space, and form the basis of the constructive theory of pointfree topology [35]. In this article we study this inherent notion of space in categories more general than those with cartesian products. Specifically, we argue that a semblance of this topological intuition remains in categories with mere tensor products. For the special case of toposes this exposes topological aspects in a different direction than considered previously. The aim of this article is to lay foundations for this (ambitiously titled) `tensor topology'. Boyarchenko and Drinfeld [10, 11] have already shown how to equate the open sets of a space with certain morphisms in its monoidal category of sheaves of vector spaces.
    [Show full text]
  • Left Hopf Algebras
    JOURNAL OF ALGEBRA 65, 399-411 (1980) Left Hopf Algebras J. A. GREEN Mathematics Institute, University of Warwick, Coventry CV4 7AL, England WARREN D. NICHOLS Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802; and Department of Mathematics, Florida State University, Tallahassee, Florida 32306 AND EARL J. TAFT Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903; and School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 Communicated by N. Jacobson Received March 5, 1979 I. INTRODUCTION Let k be a field. If (C, A, ~) is a k-coalgebra with comultiplication A and co- unit ~, and (A, m,/L) is an algebra with multiplication m and unit/z: k-+ A, then Homk(C, A) is an algebra under the convolution productf • g = m(f (~)g)A. The unit element of this algebra is/zE. A bialgebra B is simultaneously a coalgebra and an algebra such that A and E are algebra homomorphisms. Thus Homk(B, B) is an algebra under convolution. B is called a Hopf algebra if the identify map Id of B is invertible in Homk(B, B), i.e., there is an S in Homk(B, B) such that S * Id =/zE = Id • S. Such an S is called the antipode of the Hopf algebra B. Using the notation Ax = ~ x 1 @ x~ for x ~ B (see [8]), the antipode condition is that Y'. S(xa) x 2 = E(x)l = • XlS(X~) for all x E B. A bialgebra B is called a left Hopf algebra if it has a left antipode S, i.e., S ~ Homk(B, B) and S • Id =/,E.
    [Show full text]
  • Quasitriangular Structure of Myhill–Nerode Bialgebras
    Axioms 2012, 1, 155-172; doi:10.3390/axioms1020155 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Article Quasitriangular Structure of Myhill–Nerode Bialgebras Robert G. Underwood Department of Mathematics/Informatics Institute, Auburn University Montgomery, P.O. Box 244023, Montgomery, AL 36124, USA; E-Mail: [email protected]; Tel.: +1-334-244-3325; Fax: +1-334-244-3826 Received: 20 June 2012; in revised form: 15 July 2012 / Accepted: 17 July 2012 / Published: 24 July 2012 Abstract: In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz 2 L exactly when yz 2 L; 8z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras. Keywords: algebra; coalgebra; bialgebra; Myhill–Nerode theorem; Myhill–Nerode bialgebra; quasitriangular structure 1. Introduction ^ Let Σ0 be a finite alphabet and let Σ0 denote the set of words formed from the letters in Σ0. Let ^ L ⊆ Σ0 be a language, and let ∼L be the equivalence relation defined as x ∼L y if and only if xz 2 L ^ exactly when yz 2 L; 8z 2 Σ0. The Myhill–Nerode Theorem of computer science states that L is accepted by a finite automaton if and only if ∼L has finite index (cf.
    [Show full text]
  • From Combinatorial Monoids to Bialgebras and Hopf Algebras, Functorially
    From combinatorial monoids to bialgebras and Hopf algebras, functorially Laurent Poinsot Abstract. This contribution provides a study of some combinatorial monoids, namely finite decomposition and locally finite monoids, using some tools from category theory. One corrects the lack of functoriality of the construction of the large algebras of finite decomposition monoids by considering the later as monoid objects of the category of sets with finite-fiber maps. Moreover it is proved that an algebraic monoid (i.e., a commutative bialgebra) may be as- sociated to any finite decomposition monoid, and that locally finite monoids furthermore provide algebraic groups (i.e., commutative Hopf algebras), by attaching in the first case a monoid scheme to the large algebra, and in the second case a group scheme to a subgroup of invertible elements in the large algebra. 1. Introduction The field of algebraic combinatorics deals with the interface between algebra and combinatorics, i.e., it solves problems from algebra with help of combinatorial methods, and vice versa. The algebras considered are often algebras of \combinato- rial" monoids or their deformations. A combinatorial monoid is roughly speaking a usual monoid together with an informal notion of a natural integer-valued size. The size may be the length of elements of a monoid (when this makes sense), but also the number of decompositions of an element into a product of \smaller" pieces. Hence in some sense in a combinatorial monoid the construction or the decomposition of the elements is combinatorially controlled. Two main classes of such combinatorial monoids have been recognized, namely finite decomposition and locally finite monoids.
    [Show full text]
  • Cocommutative Vertex Bialgebras
    COCOMMUTATIVE VERTEX BIALGEBRAS JIANZHI HAN, HAISHENG LI, YUKUN XIAO Abstract. In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra V , it is proved that the set G(V ) of group-like elements is naturally an abelian semigroup, whereas the set P (V ) of primitive elements is a vertex Lie algebra. For g ∈ G(V ), denote by Vg the connected component containing g. Among the main results, it is proved that if V is a cocommutative vertex bialgebra, then V = ⊕g∈G(V )Vg, where V1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra VP (V ) associated to the vertex Lie algebra P (V ), and Vg is a V1-module for g ∈ G(V ). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP (V ) and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Furthermore, under the condition that G(V ) is a group and lies in the center of V , it is proved that V = VP (V ) ⊗ C[G(V )] as a coalgebra where the vertex algebra structure is explicitly determined. 1. Introduction Vertex algebras are analogous and closely related to classical algebraic systems such as Lie algebras and associative algebras on one hand, and they are also highly nonclassical in nature on the other hand. Especially, vertex algebras can be associated to (infinite-dimensional) Lie algebras of a certain type, including the Virasoro algebra and the affine Lie algebras. Vertex algebra analogues of Lie algebras, called vertex Lie algebras, were studied by Primc (see arXiv:2107.07290v1 [math.QA] 15 Jul 2021 [17]), and the same structures, called Lie conformal algebras, were studied independently by Kac (see [11]).
    [Show full text]
  • Coalgebras from Formulas
    Coalgebras from Formulas Serban Raianu California State University Dominguez Hills Department of Mathematics 1000 E Victoria St Carson, CA 90747 e-mail:[email protected] Abstract Nichols and Sweedler showed in [5] that generic formulas for sums may be used for producing examples of coalgebras. We adopt a slightly different point of view, and show that the reason why all these constructions work is the presence of certain representative functions on some (semi)group. In particular, the indeterminate in a polynomial ring is a primitive element because the identity function is representative. Introduction The title of this note is borrowed from the title of the second section of [5]. There it is explained how each generic addition formula naturally gives a formula for the action of the comultiplication in a coalgebra. Among the examples chosen in [5], this situation is probably best illus- trated by the following two: Let C be a k-space with basis {s, c}. We define ∆ : C −→ C ⊗ C and ε : C −→ k by ∆(s) = s ⊗ c + c ⊗ s ∆(c) = c ⊗ c − s ⊗ s ε(s) = 0 ε(c) = 1. 1 Then (C, ∆, ε) is a coalgebra called the trigonometric coalgebra. Now let H be a k-vector space with basis {cm | m ∈ N}. Then H is a coalgebra with comultiplication ∆ and counit ε defined by X ∆(cm) = ci ⊗ cm−i, ε(cm) = δ0,m. i=0,m This coalgebra is called the divided power coalgebra. Identifying the “formulas” in the above examples is not hard: the for- mulas for sin and cos applied to a sum in the first example, and the binomial formula in the second one.
    [Show full text]
  • Endomorphism Bialgebras of Diagrams and of Non-Commutative Algebras and Spaces
    1 Endomorphism Bialgebras of Diagrams and of Non-Commutative Algebras and Spaces BODO PAREIGIS Department of Mathematics, UniversityofMunich Munich, Germany Bialgebras and Hopf algebras havea very complicated structure. It is not easy to construct explicit examples of suchandcheck all the necessary prop erties. This gets even more complicated if wehavetoverify that something like a como dule algebra over a bialgebra is given. Bialgebras and como dule algebras, however, arise in a very natural way in non-commu- tative geometry and in representation theory.Wewant to study some general principles on how to construct such bialgebras and como dule algebras. The leading idea throughout this pap er is the notion of an action as can b een seen most clearly in the example of vector spaces. Given a vector space V we can asso ciate with it its endomorphism algebra End(V ) that, in turn, de nes an action End (V ) V ! V . There is also the general linear group GL (V ) that de nes an action GL (V ) V ! V . In the case of the endomorphism algebra we are in the pleasant situation that End(V )isavector space itself so that we can write the action also as End (V ) V ! V . The action of GL (V )on V can also b e describ ed using the tensor pro duct by expanding the group GL(V ) to the group algebra K (GL (V )) to obtain K (GL (V )) V ! V . We are going to nd analogues of End (V )or K (GL (V )) acting on non-commutative geometric spaces or on certain diagrams.
    [Show full text]
  • Algebraic Structures in Comodule Categories Over Weak Bialgebras
    Algebraic structures in comodule categories over weak bialgebras AMS Western Sectional Meeting Virtual (formerly at University of Utah) Robert Won University of Washington October 25, 2020 1 / 20 Joint work with Chelsea Walton (Rice University) Elizabeth Wicks (Microsoft) arXiv:1911.12847 October 25, 2020 2 / 20 Big picture • Fix a field k. Goal. Study the symmetries of k-algebras A. A = (A; m : A ⊗k A ! A; u : k ! A) satisfying associativity and unit constraints. • Classically, study group actions of G on A such that g · (ab) = (g · a)(g · b) and g · 1A = 1A; i.e., A is a G-module and m and u are G-module morphisms. October 25, 2020 Symmetry 3 / 20 Bialgebras • A bialgebra H over k is a k-algebra (H; m; u) and a k-coalgebra (H; ∆;") such that (a) m and u are coalgebra morphisms ∆(ab) = ∆(a)∆(b) and ∆(1) = 1 ⊗ 1 (a’) ∆ and " are algebra morphisms ∆(ab) = ∆(a)∆(b) and "(ab) = "(a)"(b): • Sweedler Notation. ∆(a) = a1 ⊗ a2. • A Hopf algebra is a bialgebra with a k-linear antipode S S(a1)a2 = a1S(a2) = "(a): October 25, 2020 Symmetry 4 / 20 Quantum symmetries • Bialgebra (co)actions capture quantum symmetries. • Let HM be the category of left H-modules. • A is called a left H-module algebra if A 2 HM such that h · (ab) = (h1 · a)(h2 · b) h · 1A = "(h)1A: Theorem. [Etingof–Walton, 2013] Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. If A is a commutative domain that is an H-module algebra, then the H-action factors through a group action.
    [Show full text]
  • Burnside's Theorem for Matrix Rings Over Division Rings
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 383 (2004) 29–44 www.elsevier.com/locate/laa Burnside’s theorem for matrix rings over division ringsୋ M. Radjabalipour a,b,∗, P. Rosenthal b, B.R. Yahaghi b aDepartment of Mathematics, Mahani Mathematical Research Center, University of Kerman, Kerman 76169, Iran bDepartment of Mathematics, University of Toronto, Toronto, Ont., Canada M5S 3G3 Received 6 February 2003; accepted 5 August 2003 Submitted by M. Omladicˇ To Heydar Radjavi, an inspirational mathematician and a wonderful friend, on his 70th birthday Abstract A version of Burnside’s theorem states that if F is an arbitrary field and A ⊂ Mn(F ) is an irreducible (or, equivalently, transitive) subalgebra containing a rank-one matrix, then A = Mn(F ). The present paper shows that if F is replaced by a division ring D, then every transitive left subalgebra of Mn(D) containing a rank-one matrix is equal to Mn(D).(Here, by a left algebra we mean a ring which is also a left D-module.) Counterexamples are given in case A is irreducible but not transitive. Moreover, it is shown that irreducible left algebras of quaternionic matrices contain rank-one idempotents and their structures are classified. © 2003 Elsevier Inc. All rights reserved. AMS classification: 15A33; 15A30; 16S50 Keywords: Division ring; Matrix (left) algebra; Submodule; Irreducible (left) algebra; Burnside’s theo- rem; Invariant (right) subspace ୋ Authors are supported by NSERC grants and fellowships; the first author is on sabbatical leave from University of Kerman.
    [Show full text]