Posts Tagged: representations

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    football representation theory

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    Unless you never touched a football in your life (that’s a _soccer-ball_ for those of you with an edu account) you will know that the world championship in Germany starts tonight. In the wake of it, the field of ‘football-science’ is booming. The BBC runs its The Science of Football-site and did you know the… Read more »

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    why nag? (1)

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    Let us take a hopeless problem, motivate why something like non-commutative algebraic geometry might help to solve it, and verify whether this promise is kept. Suppose we want to know all solutions in invertible matrices to the braid relation (or Yang-Baxter equation) All such solutions (for varying size of matrices) form an additive Abelian category… Read more »

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    noncommutative topology (4)

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    For a qurve (aka formally smooth algebra) A a *block* is a (possibly infinite dimensional over the basefield) left A-module X such that its endomorphism algebra $D = End_A(X)$ is a division algebra and X (considered as a right D-module) is finite dimensional over D. If a block X is finite dimensional over the basefield,… Read more »

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    noncommutative topology (3)

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    For finite dimensional hereditary algebras, one can describe its noncommutative topology (as developed in part 2) explicitly, using results of Markus Reineke in The monoid of families of quiver representations. Consider a concrete example, say $A = \begin{bmatrix} \mathbb{C} & V \\ 0 & \mathbb{C} \end{bmatrix}$ where $V$ is an n-dimensional complex vectorspace, or equivalently,… Read more »

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    a noncommutative topology 2

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    A *qurve* is an affine algebra such that $~\Omega^1~A$ is a projective $~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of algebra morphisms through nilpotent ideals and as such it is the ‘right’ noncommutative generalization of Grothendieck’s smoothness criterium. Examples of qurves include : semi-simple algebras, coordinate rings of affine smooth curves, hereditary orders… Read more »

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    lulu neverendingbooks

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    Half a year ago, it all started with NeverEndingBooks in which I set out a rather modest goal : Why NeverEndingBooks ? We all complain about exaggerated prices of mathematical books from certain publishers, poor quality of editing and refereeing offered, as well as far too stringent book-contracts. Rather than lamenting about this, NeverEndingBooks gives… Read more »

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    the Klein stack

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    Klein’s quartic $X$is the smooth plane projective curve defined by $x^3y+y^3z+z^3x=0$ and is one of the most remarkable mathematical objects around. For example, it is a Hurwitz curve meaning that the finite group of symmetries (when the genus is at least two this group can have at most $84(g-1)$ elements) is as large as possible,… Read more »

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    sexing up curves

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    Here the story of an idea to construct new examples of non-commutative compact manifolds, the computational difficulties one runs into and, when they are solved, the white noise one gets. But, perhaps, someone else can spot a gem among all gibberish… [Qurves](http://www.neverendingbooks.org/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasi-free algebras, aka formally smooth algebras) are the \’affine\’ pieces of non-commutative… Read more »

  • stories

    back

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    If you recognize where this picture was taken, you will know that I\’m back from France. If you look closer you will see two bikes, my own Bulls mountainbike in front and Stijn\’s lightweight bike behind. If you see the relative position of the saddles, you will know that Stijn is at least 20 cm… Read more »

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    why nag? (3)

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    Here is the construction of this normal space or chart . The sub-semigroup of (all dimension vectors of Q) consisting of those vectors satisfying the numerical condition is generated by six dimension vectors, namely those of the 6 non-isomorphic one-dimensional solutions in In particular, in any component containing an open subset of representations corresponding to… Read more »

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    why nag? (2)

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    Now, can we assign such an non-commutative tangent space, that is a for some quiver Q, to ? As we may restrict any solution in to the finite subgroups and . Now, representations of finite cyclic groups are decomposed into eigen-spaces. For example where with g the generator of . Similarly, where is a primitive… Read more »

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    B for bricks

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    Last time we argued that a noncommutative variety might be an _aggregate_ which locally is of the form $\mathbf{rep}~A$ for some affine (possibly non-commutative) $C$-algebra $A$. However, we didn't specify what we meant by 'locally' as we didn't define a topology on $\mathbf{rep}~A$, let alone on an arbitrary aggregate. Today we will start the construction… Read more »

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    A for aggregates

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    Let us begin with a simple enough question : what are the points of a non-commutative variety? Anyone? Probably you\’d say something like : standard algebra-geometry yoga tells us that we should associate to a non-commutative algebra $A$ on object, say $X_A$ and an arbitrary variety is then build from \’gluing\’ such things together. Ok,… Read more »

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    From Galois to NOG

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    Evariste Galois (1811-1832) must rank pretty high on the all-time list of moving last words. Galois was mortally wounded in a duel he fought with Perscheux d\’Herbinville on May 30th 1832, the reason for the duel not being clear but certainly linked to a girl called Stephanie, whose name appears several times as a marginal… Read more »

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    double Poisson algebras

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    This morning, Michel Van den Bergh posted an interesting paper on the arXiv entitled Double Poisson Algebras. His main motivation was the construction of a natural Poisson structure on quotient varieties of representations of deformed multiplicative preprojective algebras (introduced by Crawley-Boevey and Shaw in Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem) which he… Read more »

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    hyper-resolutions

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    [Last time][1] we saw that for $A$ a smooth order with center $R$ the Brauer-Severi variety $X_A$ is a smooth variety and we have a projective morphism $X_A \rightarrow \mathbf{max}~R$ This situation is very similar to that of a desingularization $~X \rightarrow \mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$. The top variety $~X$ is a… Read more »

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    smooth Brauer-Severis

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    Around the same time Michel Van den Bergh introduced his Brauer-Severi schemes, [Claudio Procesi][1] (extending earlier work of [Bill Schelter][2]) introduced smooth orders as those orders $A$ in a central simple algebra $\Sigma$ (of dimension $n^2$) such that their representation variety $\mathbf{trep}_n~A$ is a smooth variety. Many interesting orders are smooth : hereditary orders, trace… Read more »

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    Brauer-Severi varieties

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    ![][1] Classical Brauer-Severi varieties can be described either as twisted forms of projective space (Severi\’s way) or as varieties containing splitting information about central simple algebras (Brauer\’s way). If $K$ is a field with separable closure $\overline{K}$, the first approach asks for projective varieties $X$ defined over $K$ such that over the separable closure $X(\overline{K})… Read more »

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    moduli spaces

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    In [the previous part][1] we saw that moduli spaces of suitable representations of the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{} \ar@(ur,dr)} $ locally determine the moduli spaces of vectorbundles over smooth projective curves. There is yet another classical problem related to this quiver (which also illustrates the idea of looking at families of moduli spaces… Read more »

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    quiver representations

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    In what way is a formally smooth algebra a _machine_ producing families of manifolds? Consider the special case of the path algebra $\mathbb{C} Q$ of a quiver and recall that an $n$-dimensional representation is an algebra map $\mathbb{C} Q \rightarrow^{\phi} M_n(\mathbb{C})$ or, equivalently, an $n$-dimensional left $\mathbb{C} Q$-module $\mathbb{C}^n_{\phi}$ with the action determined by the… Read more »

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    representation spaces

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    The previous part of this sequence was [quiver representations][1]. When $A$ is a formally smooth algebra, we have an infinite family of smooth affine varieties $\mathbf{rep}_n~A$, the varieties of finite dimensional representations. On $\mathbf{rep}_n~A$ there is a basechange action of $GL_n$ and we are really interested in _isomorphism classes_ of representations, that is, orbits under… Read more »

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    path algebras

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    The previous post can be found [here][1]. Pierre Gabriel invented a lot of new notation (see his book [Representations of finite dimensional algebras][2] for a rather extreme case) and is responsible for calling a directed graph a quiver. For example, $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@(u,ur) \ar@(d,dr) \ar@/^/[ll]} $ is a quiver. Note than it… Read more »

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    driven by ambition and sloth

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    Here’s a part of yesterday’s post by bitch ph.d. : But first of all I have to figure out what the hell I’m going to teach my graduate students this semester, and really more to the point, what I am not going to bother to try to cram into this class just because it’s my… Read more »

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    the one quiver for GL(2,Z)

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    Before the vacation I finished a rewrite of the One quiver to rule them all note. The main point of that note was to associate to any qurve $A$ (formerly known as a quasi-free algebra in the terminology of Cuntz-Quillen or a formally smooth algebra in the terminology of Kontsevich-Rosenberg) a quiver $Q(A)$ and a… Read more »

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    the necklace Lie bialgebra

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    Today Travis Schedler posted a nice paper on the arXiv “A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver”. I heard the first time about necklace Lie algebras from Jacques Alev who had heard a talk by Kirillov who constructed an infinite dimensional Lie algebra on the monomials in two non-commuting… Read more »