Cross product

crossed product, of a group $G$ and a ring $K$

of a vector $a$ by a vector $b$ in $\mathbb{R}^3$, see Vector product.

An associative ring defined as follows. Suppose one is given a mapping $\sigma$ of a group $G$ into the isomorphism group of an associative ring $K$ with identity, and a family

$$ \rho = \{ \rho_{g,h} | g,h \in G\} $$

of invertible elements of $K$, satisfying the conditions

$$ \rho_{g_1,g_2}\rho_{g_1g_2,g_3} = \rho^{\sigma(g_1)}_{g_2,g_3}\rho_{g_1,g_2g_3} $$ $$ \alpha^{\sigma(g_2)\sigma(g_1)} = \rho_{g_1,g_2}\alpha^{\sigma(g_1g_2)}\rho^{-1}_{g_1,g_2} $$

for all $\alpha\in K$ and $g_1,g_2,g_3\in G$. The family $\rho$ is called a factor system. Then the cross product of $G$ and $K$ with respect to the factor system $\rho$ and the mapping $\sigma$ is the set of all formal finite sums of the form

$$ \sum_{g\in G} \alpha_g t_g $$

where $\alpha_g \in K$ and the $t_g$ are symbols uniquely assigned to every element $g\in G$, with binary operations defined by

$$ \sum_{g\in G} \alpha_g t_g + \sum_{g\in G} \beta_g t_g = \sum_{g\in G} (\alpha_g+\beta_g)t_g,$$ $$ \left(\sum_{g\in G}\alpha_gt_g\right) \left(\sum_{g\in G}\beta_gt_g\right) = \sum_{g\in G} \left(\sum_{h_1h_2=g}\alpha_{h_1}\beta^{\sigma(h_1)}_{h_2}\rho_{h_1,h_2}\right) t_g $$

This ring is denoted by $K(G, \rho, \sigma)$; the elements $t_g$ form a $K$-basis for the ring.

If $\sigma$ maps $G$ onto the identity automorphism of $K$, then $K(G, \rho)$ is called a twisted or crossed group ring, and if, in addition, $\rho_{g,h}=1$ for all $g,h\in G$, then $K(G,\rho,\sigma)$ is the group ring of $G$ over $K$ (see Group algebra).

Let $K$ be a field and $\sigma$ a monomorphism. Then $K(G,\rho,\sigma)$ is a simple ring, being the cross product of the field with its Galois group.

Comments

In the defining relations for a factor system above $\rho^{\sigma(g_1)}_{g_2,g_3}$, e.g., of course stands for the result of applying the automorphism $\sigma(g_1)$ to the element $\rho_{g_2,g_3}$. If $\rho_{g,h}=1$ for all $g,h\in G$, then one obtains the skew group ring $K(G,1,\sigma)$. Cross products arise naturally when dealing with extensions. Indeed, let $N$ be a normal subgroup of $G$. Choose a set of representatives $\{\bar{g}\}$ of $G/N$ in $G$. Then every $\alpha\in KG$, the group algebra of $G$, can be written as a unique sum $\alpha=\sum\alpha_{\bar{g}}\bar{g}$, $\alpha_{\bar{g}}\in KN$. Now write

$$ \bar{g}\bar{h}=\rho_{\bar{g},\bar{h}}\bar{g}\bar{h}, \quad \bar{g}\alpha=\sigma_{\bar{g}}(\alpha)\bar{g} . $$

Then the $\rho_{\bar{g},\bar{h}}$ define a factor system (for the group $G/N$ and the ring $KN$ relative to the set of automorphisms $\sigma_{\bar{g}}$) and

$$ KG=KN(G/N,\rho,\sigma) . $$

Up to Brauer equivalence every central simple algebra is a cross product, but not every division algebra is isomorphic to a cross product. Two algebras $A$, $B$ over $K$ are Brauer equivalent if $A\otimes M_{n_1}(K)$ is isomorphic to $B\otimes M_{n_2}(K)$ for suitable $n_1$ and $n_2$. Here $M_n(K)$ is the algebra of $n\times n$ matrices over $K$. Cf. also Brauer group.

References

[1]	S.K. Sehgal, "Topics in group rings" , M. Dekker (1978)
[2]	A.A. Bovdi, "Cross products of semi-groups and rings" Sibirsk. Mat. Zh. , 4 (1963) pp. 481–499 (In Russian)
[3]	A.E. Zalesskii, A.V. Mikhalev, "Group rings" J. Soviet Math. , 4 (1975) pp. 1–74 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 2 (1973) pp. 5–118
[4]	D.S. Passman, "The algebraic structure of group rings" , Wiley (1977)