Infinite root systems, representations of graphs and invariant theory

Kač, Victor G.

doi:10.1007/bf01403155

Cited by 375 publications

(374 citation statements)

References 20 publications

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“…The bounds given in Theorem 1.3 depend on dim(SI(Q, α)). Kac gave a formula (see [15]) for dim(SI(Q, α)) in terms of the canonical decomposition. There is in fact an efficient algorithm to compute the canonical decomposition due to the first author and Weyman, see [5].…”

Section: Bounds For Generating Semi-invariantsmentioning

confidence: 99%

Degree bounds for semi-invariant rings of quivers

Derksen

Makam

2018

Journal of Pure and Applied Algebra

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Section: Bounds For Generating Semi-invariantsmentioning

confidence: 99%

Degree bounds for semi-invariant rings of quivers

Derksen

Makam

2018

Journal of Pure and Applied Algebra

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“…But we can describe salient pieces. We start now from a multiple Actually, for quiver without relations we could have resorted to the more handy Hom complex provided by Kac [30] (for a review see [15]). So the method comes into its own for the case with relations.…”

Section: Conclusion and Further Directionsmentioning

confidence: 99%

D-branes on Calabi–Yau Manifolds and Superpotentials

Douglas

Govindarajan

Jayaraman

et al. 2004

Commun. Math. Phys.

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“…Proof. We refer the reader to Kac [6] for undefined terms in the following proof. We deal first with the case where E and F are separable field extensions in the division algebra D. Over the algebraic closure k of k , we are considering = e =f copies of Ac and Ac embedded in Mn(k) so that minimal idempotents in = e =f k all have rank n/e and minimal idempotents in k have rank n/f.…”

Section: =1mentioning

confidence: 99%

Division algebra coproducts of index 𝑛

Bergh¹,

Schofield²

1994

Trans. Amer. Math. Soc.

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Abstract.Given a family of separable finite dimensional extensions {L,} of a field A:, we construct a division algebra n2 over its center which is freely generated over k by the fields {L,} . IntroductionWe fix a ground field k. This paper arose from the wish to understand what restrictions, if any, there are on the collections of algebraic elements in a division algebra of index n over a field extension K of k. The example of Cohn's work on skew field products suggested the approach of studying the ring coproduct of a family of field extensions {L, : / g 7} of k modulo the identities of « by « matrices. This study is carried out in §2 subject to the results that are proved in § §3 and 4. We show that there is a division algebra that deserves the name of a division algebra coproduct of index n, and we describe its algebraic elements.In order to prove that a prime polynomial identity algebra is a domain, it is usually necessary to find a large number of homomorphisms from the given algebra to division algebras of the correct index. In our case, the problem is to find a division algebra D of index n of center K z> k such that L, <&k K embeds in D as a Ä-algebra. We solve this problem in §3. Let D be a central division algebra over k of index and order n . Let L be a separable extension of k of dimension dividing n . We construct a field K d k such that L®kK embeds in the division algebra D ®k K. The construction is geometric and the algebraic varieties we need are closely related to Brauer-Severi varieties.One consequence of the above stated results is that there exist division algebras D and E of index n over some field K such that subfields of D and E are the same, but the subgroup of the Brauer group generated by the classes of D and E is isomorphic to (Z/nZ)2. The referee pointed out to us that B. Fein has constructed such examples where K is a number field [4].The other problem left in §2 to be dealt with in §4 is the question of the polynomial identity degree of the ring coproduct of the fields L¡ amalgamating k modulo the identities of « by « matrices. Is it really « ? If there are only two fields Lx and L2 both of which are quadratic, the answer is no, because

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Infinite root systems, representations of graphs and invariant theory

Cited by 375 publications

References 20 publications

Degree bounds for semi-invariant rings of quivers

Degree bounds for semi-invariant rings of quivers

D-branes on Calabi–Yau Manifolds and Superpotentials

Division algebra coproducts of index 𝑛

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