Relevant bibliographies by topics / Representation of algebras

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Representation of algebras'

Author: Grafiati
Published: 4 June 2021
Last updated: 14 February 2022

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Journal articles on the topic "Representation of algebras"

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Rouquier, Raphaël. "Quiver Hecke Algebras and 2-Lie Algebras." Algebra Colloquium 19, no. 02 (May 3, 2012): 359–410. http://dx.doi.org/10.1142/s1005386712000247.

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We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.
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Behn, Antonio, Alicia Labra, and Cristián Reyes. "Irreducible Representations of Power-associative Train Algebras." Algebra Colloquium 22, spec01 (November 6, 2015): 903–8. http://dx.doi.org/10.1142/s1005386715000759.

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Train algebras were introduced by Etherington in 1939 as an algebraic framework for treating genetic problems. The aim of this paper is to study the representations and irreducible representations of power-associative train algebras of rank 4. There are three families of such algebras and for two of them we prove that every irreducible representation has dimension one over the ground field. For the third family we give an example of an irreducible representation of dimension three.
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Goodwin, Simon M., Gerhard Röhrle, and Glenn Ubly. "On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type." LMS Journal of Computation and Mathematics 13 (September 2, 2010): 357–69. http://dx.doi.org/10.1112/s1461157009000205.

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AbstractWe consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.
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Benjumea, J. C., J. Núnez, and A. F. Tenorio. "Minimal linear representations of the low-dimensional nilpotent Lie algebras." MATHEMATICA SCANDINAVICA 102, no. 1 (March 1, 2008): 17. http://dx.doi.org/10.7146/math.scand.a-15048.

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The main goal of this paper is to compute a minimal matrix representation for each non-isomorphic nilpotent Lie algebra of dimension less than $6$. Indeed, for each of these algebras, we search the natural number $n\in\mathsf{N}\setminus\{1\}$ such that the linear algebra $\mathfrak{g}_n$, formed by all the $n \times n$ complex strictly upper-triangular matrices, contains a representation of this algebra. Besides, we show an algorithmic procedure which computes such a minimal representation by using the Lie algebras $\mathfrak{g}_n$. In this way, a classification of such algebras according to the dimension of their minimal matrix representations is obtained. In this way, we improve some results by Burde related to the value of the minimal dimension of the matrix representations for nilpotent Lie algebras.
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Zhou, Jian. "Representation Rings of Classical Groups and Hopf Algebras." International Journal of Mathematics 14, no. 05 (July 2003): 461–77. http://dx.doi.org/10.1142/s0129167x03001922.

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We prove a double coset formula for induced representations of compact Lie groups. We apply it to the representation rings of unitary and symplectic groups to obtain Hopf algebras. We also construct a Heisenberg algebra representation based on the restiction and induction of representations of unitary groups.
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Dräxler, Peter. "Normal Forms for Representations of Representation-finite Algebras." Journal of Symbolic Computation 32, no. 5 (November 2001): 491–97. http://dx.doi.org/10.1006/jsco.2000.0480.

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Gonçalves, Daniel, Hui Li, and Danilo Royer. "Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C*-algebras." International Journal of Mathematics 27, no. 10 (September 2016): 1650083. http://dx.doi.org/10.1142/s0129167x1650083x.

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We give a notion of branching systems on ultragraphs. From this, we build concrete representations of ultragraph [Formula: see text]-algebras on the bounded linear operators of Hilbert spaces. To each branching system of an ultragraph, we describe the associated Perron–Frobenius operator in terms of the induced representation. We show that every permutative representation of an ultragraph [Formula: see text]-algebra is unitary equivalent to a representation arising from a branching system. We give a sufficient condition on ultragraphs such that a large class of representations of the [Formula: see text]-algebras of these ultragraphs is permutative. To give a sufficient condition on branching systems, so that their induced representations are faithful, we generalize Szymański’s version of the Cuntz–Krieger uniqueness theorem to ultragraph [Formula: see text]-algebras.
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Dräxler, Peter. "Representation-Directed Diamonds." LMS Journal of Computation and Mathematics 4 (2001): 14–21. http://dx.doi.org/10.1112/s1461157000000784.

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AbstractA module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.
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Gifford, James A. "Operator algebras with a reduction property." Journal of the Australian Mathematical Society 80, no. 3 (June 2006): 297–315. http://dx.doi.org/10.1017/s1446788700014026.

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AbstractGiven a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A—module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property.We show that a C*—algebra has the total reduction property if and only if all its representations are similar to *—representations. The question of whether all C*-algebras have this property is the famous ‘similarity problem’ of Kadison.We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.
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Massuyeau, Gwénaël, and Vladimir Turaev. "Brackets in representation algebras of Hopf algebras." Journal of Noncommutative Geometry 12, no. 2 (July 2, 2018): 577–636. http://dx.doi.org/10.4171/jncg/286.

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Dissertations / Theses on the topic "Representation of algebras"

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Banjo, Elizabeth. "Representation theory of algebras related to the partition algebra." Thesis, City University London, 2013. http://openaccess.city.ac.uk/2360/.

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The main objective of this thesis is to determine the complex generic representation theory of the Juyumaya algebra. We do this by showing that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by Martin (the representation theory of which is computable by a combination of classical and category theoretic techniques). We then use this result and general arguments of Cline, Parshall and Scott to prove that the Juyumaya algebra En(x) over the complex field is generically semisimple for all n 2 N. The theoretical background which will facilitate an understanding of the construction process is developed in suitable detail. We also review a result of Martin on the representation theory of the small ramified partition algebra, and fill in some gaps in the proof of this result by providing proofs to results leading to it.
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Hmaida, Mufida Mohamed A. "Representation theory of algebras related to the bubble algebra." Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/15987/.

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In this thesis we study several algebras which are related to the bubble algebra, including the bubble algebra itself. We introduce a new class of multi-parameter algebras, called the multi-colour partition algebra $ P_{n,m} ( \breve{\delta} )$, which is a generalization of both the partition algebra and the bubble algebra. We also define the bubble algebra and the multi-colour symmetric groupoid algebra as sub-algebras of the algebra $ P_{n,m} ( \breve{\delta} ) $. We investigate the representation theory of the multi-colour symmetric groupoid algebra $ \F S_{n,m} $. We show that $ \F S_{n,m} $ is a cellular algebra and it is isomorphic to the generalized symmetric group algebra $ \F \mathbb{Z}_m \wr S_n $ when $ m $ is invertible and $ \F $ is an algebraically closed field. We then prove that the algebra $ P_{n,m} ( \breve{\delta} ) $ is also a cellular algebra and define its cell modules. We are therefore able to classify all the irreducible modules of the algebra $ P_{n,m} ( \breve{\delta} ) $. We also study the semi-simplicity of the algebra $ P_{n,m} ( \breve{\delta} ) $ and the restriction rules of the cell modules to lower rank $ n $ over the complex field. The main objective of this thesis is to solve some open problems in the representation theory of the bubble algebra $ T_{n,m} ( \breve{\delta} ) $. The algebra $ T_{n,m} ( \breve{\delta} ) $ is known to be cellular. We use many results on the representation theory of the Temperley-Lieb algebra to compute bases of the radicals of cell modules of the algebra $ T_{n,m} ( \breve{\delta} ) $ over an arbitrary field. We then restrict our attention to study representations of $ T_{n,m} ( \breve{\delta} ) $ over the complex field, and we determine the entire Loewy structure of cell modules of the algebra $ T_{n,m} ( \breve{\delta} ) $. In particular, the main theorem is Theorem 5.41.
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Speyer, Liron. "Representation theory of Khovanov-Lauda-Rouquier algebras." Thesis, Queen Mary, University of London, 2015. http://qmro.qmul.ac.uk/xmlui/handle/123456789/9114.

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This thesis concerns representation theory of the symmetric groups and related algebras. In recent years, the study of the “quiver Hecke algebras”, constructed independently by Khovanov and Lauda and by Rouquier, has become extremely popular. In this thesis, our motivation for studying these graded algebras largely stems from a result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which generalise the Hecke algebras of type A, and thus the group algebras of the symmetric groups. This has allowed the study of the graded representation theory of these algebras. In particular, the Specht modules for the Ariki–Koike algebras can be graded; in this thesis we investigate graded Specht modules in the KLR setting. First, we conduct a lengthy investigation of the (graded) homomorphism spaces between Specht modules. We generalise the rowand column removal results of Lyle and Mathas, producing graded analogues which apply to KLR algebras of arbitrary level. These results are obtained by studying a class of homomorphisms we call dominated. Our study provides us with a new result regarding the indecomposability of Specht modules for the Ariki–Koike algebras. Next, we use homomorphisms to produce some decomposability results pertaining to the Hecke algebra of type A in quantum characteristic two. In the remainder of the thesis, we use homogeneous homomorphisms to study some graded decomposition numbers for the Hecke algebra of type A. We investigate graded decomposition numbers for Specht modules corresponding to two-part partitions. Our investigation also leads to the discovery of some exact sequences of homomorphisms between Specht modules.
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Carr, Andrew Nickolas. "Lie Algebras and Representation Theory." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1988.

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Laking, Rosanna Davison. "String algebras in representation theory." Thesis, University of Manchester, 2016. https://www.research.manchester.ac.uk/portal/en/theses/string-algebras-in-representation-theory(c350436a-db9a-429d-a8a5-470dffc0974f).html.

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The work in this thesis is concerned with three subclasses of the string algebras: domestic string algebras, gentle algebras and derived-discrete algebras (of non-Dynkin type). The various questions we answer are linked by the theme of the Krull-Gabriel dimension of categories of functors. We calculate the Cantor-Bendixson rank of the Ziegler spectrum of the category of modules over a domestic string algebra. Since there is no superdecomposable module over a domestic string algebra, this is also the value of the Krull-Gabriel dimension of the category of finitely presented functors from the category of finitely presented modules to the category of abelian groups. We also give a description of a basis for the spaces of homomorphisms between pairs of indecomposable complexes in the bounded derived category of a gentle algebra. We then use this basis to describe the Hom-hammocks involving (possibly infinite) string objects in the homotopy category of complexes of projective modules over a derived-discrete algebra. Using this description, we prove that the Krull-Gabriel dimension of the category of coherent functors from a derived-discrete algebra (of non-Dynkin type) is equal to 2. Since the Krull-Gabriel dimension is finite, it is equal to the Cantor-Bendixson rank of the Ziegler spectrum of the homotopy category and we use this to identify the points of the Ziegler spectrum. In particular, we prove that the indecomposable pure-injective complexes in the homotopy category are exactly the string complexes. Finally, we prove that every indecomposable complex in the homotopy category is pure-injective, and hence is a string complex.
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Boddington, Paul. "No-cycle algebras and representation theory." Thesis, University of Warwick, 2004. http://wrap.warwick.ac.uk/3482/.

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In the first half of this dissertation we study certain quotient algebras of preprojective algebras called no-cycle algebras N. These are studied via one-cycle algebras, which are introduced here. Results include detailed combinatorial information on N, and in certain special cases a presentation for N as a quiver with relations. In the second half we consider deformations of coordinate algebras of Kleinian singularities. Results include an explicit presentation for the deformations of a type D singularity. These two themes are tied together at the end by some mainly speculative comments about the role the various objects studied have to play in representation theory.
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Vaso, Laertis. "Cluster Tilting for Representation-Directed Algebras." Licentiate thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-364224.

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Nash, David A. 1982. "Graded representation theory of Hecke algebras." Thesis, University of Oregon, 2010. http://hdl.handle.net/1794/10871.

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xii, 76 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We study the graded representation theory of the Iwahori-Hecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory. We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of two-column partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th -root of unity to those of the group algebra of the symmetric group over a field of characteristic p.
Committee in charge: Alexander Kleshchev, Chairperson, Mathematics; Jonathan Brundan, Member, Mathematics; Boris Botvinnik, Member, Mathematics; Victor Ostrik, Member, Mathematics; William Harbaugh, Outside Member, Economics
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King, Oliver. "The representation theory of diagram algebras." Thesis, City University London, 2014. http://openaccess.city.ac.uk/5915/.

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In this thesis we study the modular representation theory of diagram algebras, in particular the Brauer and partition algebras, along with a brief consideration of the Temperley-Lieb algebra. The representation theory of these algebras in characteristic zero is well understood, and we show that it can be described through the action of a reflection group on the set of simple modules (a result previously known for the Temperley-Lieb and Brauer algebras). By considering the action of the corresponding affine reflection group, we give a characterisation of the (limiting) blocks of the Brauer and partition algebras in positive characteristic. In the case of the Brauer algebra, we then show that simple reflections give rise to non-zero decomposition numbers. We then restrict our attention to a particular family of Brauer and partition algebras, and use the block result to determine the entire decomposition matrix of the algebras therein.
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Ahmed, Chwas Abas. "Representation theory of diagram algebras : subalgebras and generalisations of the partition algebra." Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/15997/.

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This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions.
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Books on the topic "Representation of algebras"

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Erdmann, Karin, and Thorsten Holm. Algebras and Representation Theory. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91998-0.

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Stuart, Martin. Schur algebras and representation theory. Cambridge: Cambridge University Press, 1993.

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Auslander, Maurice. Representation theory of Artin algebras. Cambridge, U.K: Cambridge University Press, 1997.

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Assem, Ibrahim, and Flávio U. Coelho. Basic Representation Theory of Algebras. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35118-2.

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Auslander, Maurice. Representation theory of Artin algebras. Cambridge: Cambridge University Press, 1995.

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W, Roggenkamp Klaus, Stefanescu Mirela, and North Atlantic Treaty Organization. Scientific Affairs Division., eds. Algebra, representation theory. Dordrecht: Kluwer Academic Publishers, 2001.

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Unbounded operator algebras and representation theory. Basel: Birkhäuser Verlag, 1990.

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Humphreys, James E. Introduction to Lie algebras and representation theory. 7th ed. New York: Springer, 1997.

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Introduction to Lie algebras and representation theory. 6th ed. New York: Springer-Verlag, 1994.

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Representation theory of group graded algebras. Commack, NY: Nova Science, 1999.

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Book chapters on the topic "Representation of algebras"

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Assem, Ibrahim, and Flávio U. Coelho. "Representation-finite algebras." In Graduate Texts in Mathematics, 271–304. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35118-2_6.

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Barot, Michael. "Algebras." In Introduction to the Representation Theory of Algebras, 33–52. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-11475-0_3.

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Finkelberg, Michael, and Victor Ginzburg. "Cherednik Algebras for Algebraic Curves." In Representation Theory of Algebraic Groups and Quantum Groups, 121–53. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4697-4_6.

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OkniŃski, Jan. "In Search for Noetherian Algebras." In Algebra — Representation Theory, 235–47. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0814-3_11.

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Hassani, Sadri. "Representation of Clifford Algebras." In Mathematical Physics, 987–1007. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-01195-0_31.

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Vladimirov, D. A. "Representation of Boolean Algebras." In Boolean Algebras in Analysis, 125–79. Dordrecht: Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-017-0936-1_4.

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Givant, Steven. "Representation theorems." In Advanced Topics in Relation Algebras, 201–314. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-65945-9_4.

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Howe, Roger. "Pieri algebras and Hibi algebras in representation theory." In Symmetry: Representation Theory and Its Applications, 353–84. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-1590-3_13.

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Montgomery, S. "Representation Theory of Semisimple Hopf Algebras." In Algebra — Representation Theory, 189–218. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0814-3_9.

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Chriss, Neil, and Victor Ginzburg. "Representations of Convolution Algebras." In Representation Theory and Complex Geometry, 411–86. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4938-8_9.

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Conference papers on the topic "Representation of algebras"

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Duntsch, Ivo, and Michael Winter. "Timed Contact Algebras." In 2009 16th International Symposium on Temporal Representation and Reasoning (TIME). IEEE, 2009. http://dx.doi.org/10.1109/time.2009.22.

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Leclerc, Bernard. "Cluster Algebras and Representation Theory." In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India, 2011. http://dx.doi.org/10.1142/9789814324359_0154.

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ACCARDI, L., A. BOUKAS, and J. MISIEWICZ. "EXISTENCE OF THE FOCK REPRESENTATION FOR CURRENT ALGEBRAS OF THE GALILEI ALGEBRA." In Proceedings of the 30th Conference. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814338745_0001.

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Rajan, G. Susinder, and B. Sundar Rajan. "STBCs from Representation of Extended Clifford Algebras." In 2007 IEEE International Symposium on Information Theory. IEEE, 2007. http://dx.doi.org/10.1109/isit.2007.4557141.

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Niederle, J., and J. Paseka. "Triple Representation Theorem for Homogeneous Effect Algebras." In 2012 IEEE 42nd International Symposium on Multiple-Valued Logic (ISMVL). IEEE, 2012. http://dx.doi.org/10.1109/ismvl.2012.27.

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Kawazoe, T., T. Oshima, and S. Sano. "Representation Theory of Lie Groups and Lie Algebras." In Fuji-Kawaguchiko Conference on Representation Theory of Lie Groups and Lie Algebras. WORLD SCIENTIFIC, 1992. http://dx.doi.org/10.1142/9789814537162.

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Rivieccio, Umberto, Tommaso Flaminio, and Thiago Nascimento. "On the representation of (weak) nilpotent minimum algebras." In 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). IEEE, 2020. http://dx.doi.org/10.1109/fuzz48607.2020.9177641.

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Hanxing Lin. "On the representation dimension of triangular matrix algebras." In 2011 International Conference on Multimedia Technology (ICMT). IEEE, 2011. http://dx.doi.org/10.1109/icmt.2011.6002464.

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Chajda, Ivan, and Jan Paseka. "Set Representation of Partial Dynamic De Morgan Algebras." In 2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL). IEEE, 2016. http://dx.doi.org/10.1109/ismvl.2016.14.

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Paseka, Jan, and Radek Slesinger. "A Representation Theorem for Quantale Valued sup-algebras." In 2018 IEEE 48th International Symposium on Multiple-Valued Logic (ISMVL). IEEE, 2018. http://dx.doi.org/10.1109/ismvl.2018.00024.

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Reports on the topic "Representation of algebras"

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Krýsl, Svatopluk. Analysis Over $C^*$-Algebras and the Oscillatory Representation. GIQ, 2014. http://dx.doi.org/10.7546/giq-15-2014-173-195.

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Krysl, Svatopluk Krysl. Analysis Over $C^*$-Algebras and the Oscillatory Representation. Jgsp, 2014. http://dx.doi.org/10.7546/jgsp-33-2014-1-25.

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Berceanu, Stefan. A Holomorphic Representation of the Semidirect Sum of Symplectic and Heisenberg Lie Algebras. Journal of Geometry and Symmetry in Physics, 2012. http://dx.doi.org/10.7546/jgsp-5-2006-5-13.

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Birenbaum, Menucha, Anthony E. Kelly, and KiKumi K. Tatsuoka. Toward a Stable Diagnostic Representation of Students' Errors in Algebra. Fort Belvoir, VA: Defense Technical Information Center, October 1992. http://dx.doi.org/10.21236/ada257319.

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Mundy, Joseph L. Representation and Recognition with Algebraic Invariants and Geometric Constraint Models. Fort Belvoir, VA: Defense Technical Information Center, December 1993. http://dx.doi.org/10.21236/ada282926.

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Mundy, Joseph L. Representation and Recognition with Algebraic Invariants and Geometric Constraint Models. Fort Belvoir, VA: Defense Technical Information Center, September 1993. http://dx.doi.org/10.21236/ada271395.

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7

Goodman, I. R. Algebraic Representations of Linguistic and Numerical Modifications of Probability Statements and Inferences Based on a Product Space Construction. Fort Belvoir, VA: Defense Technical Information Center, March 1996. http://dx.doi.org/10.21236/ada306334.

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