Let V⊆A be a conformal inclusion of vertex operator algebras and let C be a category of grading-restricted generalized V-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which C inherits semisimplicity from the category of grading-restricted generalized A-modules in C, and vice versa. The most important condition is that A be a rigid V-module in C with non-zero categorical dimension, that is, we assume the index of V as a subalgebra of A is finite and non-zero. As a consequence, we show that if A is strongly rational, then V is also strongly rational under the following conditions: A contains V as a V-module direct summand, V is C_2-cofinite with a rigid tensor category of modules, and A has non-zero categorical dimension as a V-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that A is a vertex operator superalgebra.

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Let $\mathbf{U}$ be the quantum group and $\mathbf{f}$ be the Lusztig's algebra associated with a symmetrizable generalized Cartan matrix. The algebra $\mathbf{f}$ can be viewed as the positive part of $\mathbf{U}$. Lusztig introduced some symmetries $T_i$ on $\mathbf{U}$ for all $i\in I$. Since $T_i(\mathbf{f})$ is not contained in $\mathbf{f}$, Lusztig considered two subalgebras ${_i\mathbf{f}}$ and ${^i\mathbf{f}}$ of $\mathbf{f}$ for any $i\in I$, where ${_i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T_i(x)\in\mathbf{f}\}$ and ${^i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T^{-1}_i(x)\in\mathbf{f}\}$. The restriction of $T_i$ on ${_i\mathbf{f}}$ is also denoted by $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$. The geometric realization of $\mathbf{f}$ and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig's symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of ${_i\mathbf{f}}$, ${^i\mathbf{f}}$ and $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$ by using the language 'quiver with automorphism' introduced by Lusztig.

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

Let$p$be a prime integer, 1≤$s$≤$r$be integers and$F$be a field of characteristic different from$p$. We find upper and lower bounds for the essential$p$-dimension ed_{$p$}( $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ ) of the class $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ of central simple algebras of degree$p$^{$r$}and exponent dividing$p$^{$s$}. In particular, we show that ed($Alg$_{8,2})=ed_{2}($Alg$_{8,2})=8 and ed_{$p$}( $$ Al{{g}_{{{{p}^2},p}}} $$ )=$p$^{2}+$p$for$p$odd.