Nilpotent Elements Research Articles

Semisimplicity of the DS functor for the orthosymplectic Lie superalgebra

We prove that the Duflo-Serganova functor DSx attached to an odd nilpotent element x of osp(m|2n) is semisimple, i.e. sends a semisimple representation M of osp(m|2n) to a semisimple representation of osp(m−2k|2n−2k) where k is the rank of x. We prove a closed formula for DSx(L(λ)) in terms of the arc diagram attached to λ.

Weak core inverses and pseudo core inverses in a ring with involution

In 2020, Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the m-weak group inverse. Furthermore, we prove that an element is pseudo core invertible if this element is both {1, 3}-invertible and weak group invertible. Then, we show that each nilpotent element is Moore–Penrose invertible if and only if each pseudo core invertible element is weak core invertible. Finally, we consider the conditions under which the weak core inverse of an element coincides with another generalized inverse.

Normal forms of nilpotent elements in semisimple Lie algebras

We find the normal form of nilpotent elements in semisimple Lie algebras that generalizes the Jordan normal form in slN, using the theory of cyclic elements.

Remarks on the group of unıts of a corner ring

The aim of this study is to characterize rings having the following properties for a non-trivial idempotent element e of R, U (eRe) = e + eJ(R)e = e + J (eRe) (and U (eRe) = e + N (eRe)),where U (-), N (-) and J (-) denote the group of units, the set of all nilpotent elements and the Jacobson radical of R, respectively. In the present paper, some characterizations are also obtained in terms of every element is of the form e + u, where e2 = e ∈ R and u ∈ U(eRe).

Centers and Azumaya loci for finite W-algebras in positive characteristic

AbstractIn this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Green’s relation ℋ on the monoid of square matrices over a specific local ring

Let [Formula: see text] be a commutative local ring whose maximal ideal is generated by a nilpotent element, and Mat[Formula: see text] be the multiplicative monoid of the square matrices of order [Formula: see text] over [Formula: see text]. In this paper, we provide the construction of Green’s [Formula: see text]-equivalence classes in the multiplicative monoid Mat[Formula: see text]. Then, we enumerate these classes in the special cases [Formula: see text] and [Formula: see text].

Trinil clean index of a ring

From the concepts of clean index of rings, nil clean index of rings, and weakly nil clean index of rings, we expand them by introducing the new concept, that is, trinil clean index of a ring. From any element a of a ring R with unity, we set τ(a): ={e ∈ R|e 3 = e and a − e ∈ Nil(R)}, where Nil(R) is the set of all nilpotent elements of R; trinil clean index of R is defined by sup{|τ(a)||a ∈ R} and it is denoted by Trinin(R). In this article, it will be investigated some properties of trinil clean index of any ring, ring homomorphism, and direct product. Some applications of determining trinil clean index of integral domains are also provided. We also determined the trinil clean index of ℤ n .

Decomposition of exterior and symmetric squares in characteristic two

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e∈sl(V), we describe the Jordan normal form of e on the sl(V)-modules ∧2(V) and S2(V). In the case where e is a regular nilpotent element, we are able to give a closed formula.We also consider the closely related problem of describing, for every unipotent element u∈SL(V), the Jordan normal form of u on ∧2(V) and S2(V). A recursive formula for the Jordan block sizes of u on ∧2(V) was given by Gow and Laffey [J. Group Theory 9 (2006), 659–672]. We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S2(V).

INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS

We prove that all classical affine W-algebras \U0001d4b2(\U0001d524; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.

On the idempotent and nilpotent sum numbers of matrices over certain indecomposable rings and related concepts

We investigate a few special decompositions in arbitrary rings and matrix rings over indecomposable rings into nilpotent and idempotent elements. Moreover, we also define and study the nilpotent sum trace number of nilpotent matrices over an arbitrary ring. Some related notions are explored as well.

Growth functions of lie algebras associated with associative algebras

Let [Formula: see text] be a finitely generated associative algebra over a field [Formula: see text] of characteristic [Formula: see text] and let [Formula: see text] be its associated Lie algebra. In this paper, we investigate relations between the growth functions of [Formula: see text] and the Lie algebra [Formula: see text]. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

Nilpotent decomposition in integral group rings

A finite group G is said to have the nilpotent decomposition property (ND) if for every nilpotent element α of the integral group ring Z[G] one has that αe also belong to Z[G], for every primitive central idempotent e of the rational group algebra Q[G]. Results of Hales, Passi and Wilson, Liu and Passman show that this property is fundamental in the investigations of the multiplicative Jordan decomposition of integral group rings. If G and all its subgroups have ND then Liu and Passman showed that G has property SSN, that is, for subgroups H, Y and N of G, if N◁H and Y⊆H then N⊆Y or YN is normal in H; and such groups have been described. In this article, we study the nilpotent decomposition property in integral group rings and we classify finite SSN groups G such that the rational group algebra Q[G] has only one Wedderburn component which is not a division ring.

Semi-infinite cohomology and the linkage principle for [formula omitted]-algebras

Let g be a simple Lie algebra, and let Wκ be the affine W-algebra associated to a principal nilpotent element of g and level κ. We explain a duality between the categories of smooth W modules at levels κ+κc and −κ+κc, where κc is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the W-algebra.As an application, we determine all homomorphisms between the Verma modules for W, verifying a conjecture from the conformal field theory literature of de Vos–van Driel. Along the way, we determine the linkage principle for Category O of the W-algebra.

Nilpotent orbits and mixed gradings of semisimple Lie algebras

Let σ be an involution of a complex semisimple Lie algebra g and g=g0⊕g1 the related Z2-grading. We study relations between nilpotent G0-orbits in g0 and the respective G-orbits in g. If e∈g0 is nilpotent and {e,h,f}⊂g0 is an sl2-triple, then the semisimple element h yields a Z-grading of g. Our main tool is the combined Z×Z2-grading of g, which is called a mixed grading. We prove, in particular, that if eσ is a regular nilpotent element of g0, then the weighted Dynkin diagram of eσ, D(eσ), has only isolated zeros. It is also shown that if G⋅eσ∩g1≠∅, then the Satake diagram of σ has only isolated black nodes and these black nodes occur among the zeros of D(eσ). Using mixed gradings related to eσ, we define an inner involution σˇ such that σ and σˇ commute. Here we prove that the Satake diagrams for both σˇ and σσˇ have isolated black nodes.

Nilpotence Varieties

We consider algebraic varieties canonically associated with any Lie superalgebra, and study them in detail for super-Poincaré algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of) the odd part of the algebra. Most of these varieties have appeared in various guises in previous literature, but we study them systematically here, from a new perspective: As the natural moduli spaces parameterizing twists of a super-Poincaré-invariant physical theory. We obtain a classification of all possible twists, as well as a systematic analysis of unbroken symmetry in twisted theories. The natural stratification of the varieties, the identification of strata with twists, and the action of Lorentz and R-symmetry are emphasized. We also include a short and unconventional exposition of the pure spinor superfield formalism, from the perspective of twisting, and demonstrate that it can be applied to construct familiar multiplets in four-dimensional minimally supersymmetric theories. In all dimensions and with any amount of supersymmetry, this technique produces BRST or BV complexes of supersymmetric theories from the Koszul complex of the maximal ideal over the coordinate ring of the nilpotence variety, possibly tensored with any equivariant module over that coordinate ring. In addition, we remark on a natural connection to the Chevalley–Eilenberg complex of the supertranslation algebra, and give two applications related to these ideas: a calculation of Chevalley–Eilenberg cohomology for the (2, 0) algebra in six dimensions, and a degenerate BV complex encoding the type IIB supergravity multiplet.

Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic

Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$.

Supersymmetric W-algebras

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As an application, we produce explicit free generators of the $W$-algebra associated with the odd principal nilpotent element of the Lie superalgebra $\mathfrak{gl}(n+1|n).$

Centers of centralizers of nilpotent elements in exceptional Lie superalgebras

Let [Formula: see text] be a finite-dimensional simple Lie superalgebra of type [Formula: see text], [Formula: see text] or [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the simply connected semisimple algebraic group over [Formula: see text] such that [Formula: see text]. Suppose [Formula: see text] is nilpotent. We describe the centralizer [Formula: see text] of [Formula: see text] in [Formula: see text] and its center [Formula: see text] especially. We also determine the labeled Dynkin diagram for [Formula: see text]. We prove theorems relating the dimension of [Formula: see text] and the labeled Dynkin diagram.

On some generalizations of reduced and rigid modules

In this paper we have introduced the notion of weak reduced module and weak rigid module as the generalizations of reduced module by working on the context of nilpotent elements of module and investigated some of its properties. Various results are extended for this generalization and some counter examples are constructed to validate the generalizations.

Nilpotent Graph

In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.