On Chevalley Restriction Theorem for Semi-reductive Algebraic Groups and Its Applications

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.

Which algebraic groups are Picard varieties?

We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points.In contrast,no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.

Construction of complete generalized algebraic groups

With one exception, the holomorph of a finite dimensional abelian connectedalgebraic group is shown to be a complete generalized algebraic group. This result on algebraic group is an analogy to that on Lie algebra.

Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,according to the kinematics analysis of serial mechanisms,the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors.Then,based on the standard ideas of Lie operations,the method for kinematics analysis of parallel mechanisms is derived,and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form.Then,according to the mapping relationship between the parallel joints and corresponding equivalent series joints,a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined.A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example.The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.

LOCAL AUTOMORPHISMS OF SEMISIMPLE ALGEBRAS AND GROUP ALGEBRAS

Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.

On the Structure of the Units of Group Algebra of Dihedral Group

In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.

A Determinant Criterion of Invertible Elements in Group Algebras

An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.

ALGEBRAIC GROUPS ASSOCIATED WITH GRADED LIE ALGEBRAS OF CARTAN TYPE

For a graded simple Lie algebra of Cartan type L=X(m:n) (2) X∈{W,S,H,K} , over a field F of odd characteristic p , the group generated by one-parameter subgroups of the form exp( t ad y )is described, where y∈L+F u satisfying y p=0 , t∈F and u is some fixed element of the p -envelope of L in Der u (m :n).

THE BANACH ALGEBRAS GENERATED BY OPERATORS WITH ONE-POINT SPECTRUM

In this article, we present some results related to the structure of Banach algebras generated by operators with one-point spectrum.

Uniform Boundedness of Torsion Subgroups of Linear Groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L 1(G) is its group algebra and L ∞(G) is the dual of L 1(G). Lau has studied the bounded linear operators T : L ∞(G) → L ∞(G) which commute with convolutions and translations. For a subspace H of L ∞(G), we know that M(L ∞(G),H), the Banach algebra of all bounded linear operators on L ∞(G) into H which commute with convolutions, has been studied by Pym and Lau. In this paper, we generalize these problems to L(K)*, the dual of a hypergroup algebra L(K) in a very general setting, i. e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L 1(G) but also most of the semigroup algebras. Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however, we succeed in getting some interesting results.

Integral Cayley Graphs over Finite Groups

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group(s)is integral.In particular,a Cayley graph of a 2-group generated by a normal set of involutions is integral.We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral.We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form(k i j)with fixed k,a s{-n+1,1-n+1,2^2-n+1,...,(n-1)2-n+1}.

On Connected Components of Skew Group Algebras

Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equivalent to the quotient algebra of path algebra kQGmodulo ideal(ρG).Generally,the quiver QGis not connected.In this paper we develop a method to determine the number of connect components of QG.Meanwhile,we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.

LCD Codes and Self-orthogonal Codes in Finite Dihedral Group Algebras

Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes and self-orthogonal codes in the finite dihedral group algebras Fq[D2n]. Some numerical examples are also presented to illustrate the main results.

Stably Calabi-Yau algebras and skew group algebras

Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.

Braid Group Representations from Twisted Tensor Products of Algebras

We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these representations,focusing on the size of their images,which are typically finite groups.The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework,and we give some new examples to illustrate their ubiquity.Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category C(A,Q)and that of C(A,Q)itself.

Grade zero part of forced graded algebras

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.

Hyperbolicity of unitary involutions

We prove the so-called Unitary Hyperbolicity Theorem,a result on hyperbolicity of unitary involutions.The analogous previously known results for the orthogonal and symplectic involutions are formal consequences of the unitary one.While the original proofs in the orthogonal and symplectic cases were based on the incompressibility of generalized Severi-Brauer varieties,the proof in the unitary case is based on the incompressibility of their Weil transfers.

Entire curves,integral sets and principal bundles

We compare the behaviour of entire curves and integral sets,in particular in relation to locally trivial fiber bundles,algebraic groups and finite ramified covers over semi-abelian varieties.

Selfinjective Koszul Algebras of Finite Complexity

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G（l, m）, the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL（2, C） over the exterior algebra of a 2-dimensional vector space.