555 research outputs found
Invariants of mixed representations of quivers I
We introduce a new concept of mixed representations of quivers that is a
generalization of ordinary representations of quivers and orthogonal
(symplectic) representations of symmetric quivers introduced recently by
Derksen and Weyman. We describe the generating invariants of mixed
representations of quivers (First Fundamental Theorem) and prove additional
results that allow us to describe the defining relations between them in the
second article.Comment: 42 page
Affine quotients of supergroups
In this article we consider sheaf quotients of affine superschemes by affine
supergroups that act on them freely. The necessary and sufficient conditions
for such quotients to be affine are given. If is an affine supergroup and
is its normal supersubgroup, then we prove that a dur -sheaf
is again affine supergroup. Additionally, if is
algebraic, then a -sheaf is also algebraic supergroup and it
coincides with . In particular, any normal supersubgroup
of an affine supergroup is faithfully exact.Comment: 31 page
Invariants of mixed representations of quivers II : defining relations and applications
In the previous article we introduced the new concept of mixed
representations of quivers and described the generators of their algebras of
invariants. In this article we describe the defining relations of these
algebras. Some applications for the invariants of orthogonal or symplectic
groups acting on several matrices are given.Comment: 27 page
On the notion of Krull super-dimension
We introduce the notion of Krull super-dimension of a super-commutative
super-ring. This notion is used to describe regular super-rings and calculate
Krull super-dimensions of completions of super-rings. Moreover, we use this
notion to introduce the notion of super-dimension of any irreducible
superscheme of finite type. Finally, we describe nonsingular superschemes in
terms of sheaves of K\"{a}hler superdifferentials.Comment: 30 page
Semi-invariants of mixed representations of quivers
The notion of mixed representations of quivers can be derived from ordinary
quiver representations by considering the dual action of groups on "vertex"
vector spaces together with the usual action. A generating system for the
algebra of semi-invariants of mixed representations of a quiver is determined.
This is done by reducing the problem to the case of bipartite quivers of the
special form and by introducing a function DP on three matrices, which is a
mixture of the determinant and two pfaffians.Comment: 37 pages; v2. notations are improved; v3. final version; v4. Now the
numeration of statements is the same as in the printed version of the pape
Representations of quivers, their generalizations and invariants
This paper is a survey on invariants of representations of quivers and their
generalizations. We present the description of generating systems for
invariants and relations between generators.Comment: 31 pages; v2. Formulations of Theorems 3.16 and 5.9 are correcte
Generators of supersymmetric polynomials in positive characteristic
Kantor and Trishin described the algebra of polynomial invariants of the
adjoint representation of the Lie supergalgebra and a related algebra
of what they called pseudosymmetric polynomials over an algebraically
closed field of characteristic zero. The algebra was investigated
earlier by Stembridge who called the elements of supersymmetric
polynomials and determined generators of . The case of positive
characteristic has been recently investigated by La Scala and Zubkov. They
formulated two conjectures describing generators of polynomial invariants of
the adjoint action of the general linear supergroup and generators of
, respectively. In the present paper we prove both conjectures.Comment: 10 page
Invariants of and in positive characteristic
Invariants of and , both acting on several copies of
octonions, have been decribed in \cite{schw2} over a ground field of
characteristic zero. In the current manuscript, we extend this result to an
arbitrary infinite field of odd characteristic. More precisely, we prove that
the corresponding algebras of invariants are generated by the same invariants
of degree at most as in the case of a field of characteristic zero.Comment: 30 page
Scale Magnetic Effect in Quantum Electrodynamics and the Wigner-Weyl Formalism
The Scale Magnetic Effect (SME) is the generation of electric current due to
conformal anomaly in external magnetic field in curved spacetime. The effect
appears in a vacuum with electrically charged massless particles. Similarly to
the Hall effect, the direction of the induced anomalous current is
perpendicular to the direction of the external magnetic field and to
the gradient of the conformal factor , while the strength of the current
is proportional to the beta function of the theory. In massive electrodynamics
the SME remains valid, but the value of the induced current differs from the
current generated in the system of massless fermions. In the present paper we
use the Wigner--Weyl formalism to demonstrate that in accordance with the
decoupling property of heavy fermions the corresponding anomalous conductivity
vanishes in the large-mass limit with and .Comment: 12 pages, accepted for publication in Phys.Rev.
On quotients of affine superschemes over finite supergroups
In this article we consider sheaf quotients of affine superschemes by finite
supergroups that act on them freely. More precisely, if a finite supergroup
acts on an affine superscheme freely, then the quotient -sheaf
is again an affine superscheme , where .
Besides, is a finitely presented projective -module.Comment: 12 page
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