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Register a new userGraded sets, graded groups, and Clifford algebras
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Authors
Wolfgang Bertram
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We define a general notion of centrally $Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $Gamma$ is an arbitrary (generalized) ring. The case $Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).
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We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G. We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G,M,L) and they are closely related to Langlands parameters. Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of logarithms of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper.
We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish grad
We prove a version of the Deligne conjecture for $n$-fold monoidal abelian categories $A$ over a field $k$ of characteristic 0, assuming some compatibility and non-degeneracy conditions for $A$. The output of our construction is a weak Leinster $(n,1)$-algebra over $k$, a relaxed version of the concept of Leinster $n$-algebra in $Alg(k)$. The difference between the Leinster original definition and our relaxed one is apparent when $n>1$, for $n=1$ both concepts coincide. We believe that there exists a functor from weak Leinster $(n,1)$-algebras over $k$ to $C(E_{n+1},k)$-algebras, well-defined when $k=mathbb{Q}$, and preserving weak equivalences. For the case $n=1$ such a functor is constructed in [Sh4] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories. Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster $(n,1)$-algebra over $k$, out of an $n$-fold monoidal $k$-linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for $n>1$) is a passage from weak Leinster $(n,1)$-algebras to $C(E_{n+1},k)$-algebras. As an application, we prove that the Gerstenhaber-Schack complex of a Hopf algebra over a field $k$ of characteristic 0 admits a structure of a weak Leinster (2,1)-algebra over $k$ extending the Yoneda structure. It relies on our earlier construction [Sh1] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra.
Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.
As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding to the identity of the group is a free subalgebra which is graded by the usual degree. We look into its Hilbert series and prove that it is a rational function by giving an explicit formula. As an application, we show that, under suitable conditions, this subalgebra is finitely generated if and only if the grading on the base vector space is trivial.
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