Polynomial invariants of representations of quivers

We introduce the notion of a super-representation of a quiver. For super-representations of quivers over a field of characteristic zero, we describe the corresponding (super)algebras of polynomial semi-invariants and polynomial invariants.

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Polynomial invariants for virtual marked graphs and virtual surface-link theory I

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821651750081x ◽

2017 ◽

Vol 26 (12) ◽

pp. 1750081

Author(s):

Sang Youl Lee

Keyword(s):

Natural Extension ◽

Polynomial Invariants ◽

Kauffman Bracket ◽

Virtual Links ◽

Link Theory ◽

Marked Graphs ◽

Bracket Polynomial ◽

The Ideal

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

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Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

Integral Equations and Operator Theory ◽

10.1007/s00020-015-2228-3 ◽

2015 ◽

Vol 83 (4) ◽

pp. 563-587

Author(s):

Masatoshi Enomoto ◽

Yasuo Watatani

Keyword(s):

Hilbert Spaces ◽

Infinite Dimensional ◽

Representations Of Quivers ◽

Indecomposable Representations ◽

Strongly Irreducible Operators ◽

Strongly Irreducible

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Projective representations of quivers

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202108192 ◽

2002 ◽

Vol 31 (2) ◽

pp. 97-101 ◽

Cited By ~ 1

Author(s):

Sangwon Park

Keyword(s):

Projective Representation ◽

Representations Of Quivers ◽

Direct Sum ◽

Projective Representations

We prove thatP1 →f P2is a projective representation of a quiverQ=•→•if and only ifP1andP2are projective leftR-modules,fis an injection, andf (P 1)⊂P 2is a summand. Then, we generalize the result so that a representationM1 →f1 M2 →f2⋯→fn−2 Mn−1→fn−1 Mnof a quiverQ=•→•→•⋯•→•→•is projective representation if and only if eachMiis a projective leftR-module and the representation is a direct sum of projective representations.

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Polynomial invariants and the integral homology of coverings of knots and links

Inventiones mathematicae ◽

10.1007/bf01418645 ◽

1972 ◽

Vol 15 (1) ◽

pp. 78-90 ◽

Cited By ~ 5

Author(s):

D. W. Sumners

Keyword(s):

Integral Homology ◽

Polynomial Invariants ◽

Knots And Links

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Jones polynomial invariants for knots and satellites

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100069589 ◽

1991 ◽

Vol 109 (1) ◽

pp. 83-103 ◽

Cited By ~ 9

Author(s):

H. R. Morton ◽

P. Strickland

Keyword(s):

Quantum Group ◽

Linear Combination ◽

Simple Formula ◽

Jones Polynomial ◽

Polynomial Invariants ◽

Satellite Link ◽

Representation Ring ◽

Parallel Links ◽

Jones Polynomials ◽

Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

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